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var("t", domain='real')
print integral( sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 ), t )

print integral( sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 ), t, 0, 3*pi )

print numerical_integral( sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 ), 0, 3*pi )
t -40/sqrt(sin(t)^2/(cos(t) + 1)^2 + 1) 40 (120.00000059010817, 3.9621779610080586e-05) 
sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 )._maxima_()
sqrt(100*sin(_SAGE_VAR_t)^2+100*(cos(_SAGE_VAR_t)-1)^2) 
%maxima

integrate(sqrt(100*sin(t)^2+100*(cos(t)-1)^2), t, 0, 3*%pi)
40

︠e485473b-534e-4c23-9180-177272e1ff76s︠
integrate(sqrt(100*sin(t)^2+100*(cos(t)-1)^2), t)
-40/sqrt(sin(t)^2/(cos(t) + 1)^2 + 1) 
g = -40/sqrt(sin(t)^2/(cos(t) + 1)^2 + 1)
g(3*pi)
Error in lines 2-2 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/symbolic/expression.pyx", line 4821, in sage.symbolic.expression.Expression.__call__ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/expression.cpp:27873) return self._parent._call_element_(self, *args, **kwds) File "sage/symbolic/ring.pyx", line 798, in sage.symbolic.ring.SymbolicRing._call_element_ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/ring.cpp:9875) return _the_element.subs(d, **kwds) File "sage/symbolic/expression.pyx", line 4730, in sage.symbolic.expression.Expression.substitute (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/expression.cpp:27401) self._gobj.subs_map(smap, 0)) ValueError: power::eval(): division by zero 
plot(sqrt(100*sin(t)^2+100*(cos(t)-1)^2), (t, 0, 3*pi)) + plot(g.derivative(t),(t,0,3*pi), color='red')

g
-40/sqrt(sin(t)^2/(cos(t) + 1)^2 + 1) 
g.derivative(t)
40*(cos(t)*sin(t)/(cos(t) + 1)^2 + sin(t)^3/(cos(t) + 1)^3)/(sin(t)^2/(cos(t) + 1)^2 + 1)^(3/2) 

t = var("t", domain='real')
print integral( sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 ), t )

print integral( sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 ), t, 0, 3*pi, algorithm='sympy' )

print numerical_integral( sqrt( (10-10*cos(t))^2 + (10*sin(t))^2 ), 0, 3*pi )
-40/sqrt(sin(t)^2/(cos(t) + 1)^2 + 1)
Error in lines 3-3 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/misc/functional.py", line 663, in integral return x.integral(*args, **kwds) File "sage/symbolic/expression.pyx", line 11269, in sage.symbolic.expression.Expression.integral (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/expression.cpp:59975) return integral(self, *args, **kwds) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 757, in integrate return integrator(expression, v, a, b) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/symbolic/integration/external.py", line 40, in sympy_integrator result = sympy.integrate(ex, (v, a._sympy_(), b._sympy_())) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/utilities/decorator.py", line 35, in threaded_func return func(expr, *args, **kwargs) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/integrals.py", line 1232, in integrate risch=risch, manual=manual) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/integrals.py", line 487, in doit conds=conds) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/integrals.py", line 862, in _eval_integral h = heurisch_wrapper(g, x, hints=[]) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 128, in heurisch_wrapper unnecessary_permutations) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 583, in heurisch result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 583, in heurisch result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 583, in heurisch result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 583, in heurisch result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 376, in heurisch result = heurisch(f, x, rewrite=True, hints=hints, unnecessary_permutations=unnecessary_permutations) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 569, in heurisch solution = _integrate() File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/integrals/heurisch.py", line 557, in _integrate solution = solve_lin_sys(numer.coeffs(), coeff_ring) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/polys/solvers.py", line 30, in solve_lin_sys echelon, pivots = matrix.rref(iszerofunc=lambda x: not x, simplify=lambda x: x) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/matrices/matrices.py", line 2703, in rref r.zip_row_op(j, pivot, lambda x, y: x - scale*y) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/matrices/dense.py", line 758, in zip_row_op self._mat[i0: i0 + self.cols] = [ f(x, y) for x, y in zip(ri, rk) ] File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/matrices/matrices.py", line 2703, in <lambda> r.zip_row_op(j, pivot, lambda x, y: x - scale*y) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/polys/domains/expressiondomain.py", line 90, in __mul__ return f.simplify(f.ex*g.ex) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/polys/domains/expressiondomain.py", line 50, in simplify return f.__class__(ex.cancel()) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sympy/core/expr.py", line 2986, in cancel from sympy.polys import cancel File "sage/ext/interrupt/interrupt.pyx", line 203, in sage.ext.interrupt.interrupt.sage_python_check_interrupt (/projects/sage/sage-6.9/src/build/cythonized/sage/ext/interrupt/interrupt.c:1890) sig_check() File "sage/ext/interrupt/interrupt.pyx", line 88, in sage.ext.interrupt.interrupt.sig_raise_exception (/projects/sage/sage-6.9/src/build/cythonized/sage/ext/interrupt/interrupt.c:924) raise KeyboardInterrupt KeyboardInterrupt 
%maxima