Optimal. Leaf size=210 \[ -\frac {\cot ^{11}(c+d x)}{11 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]
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Rubi [A] time = 0.43, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2875, 2873, 2607, 270, 2611, 3768, 3770} \[ -\frac {\cot ^{11}(c+d x)}{11 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2607
Rule 2611
Rule 2873
Rule 2875
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x) \csc ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cot ^4(c+d x) \csc ^8(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cot ^4(c+d x) \csc ^6(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^7(c+d x)+a^2 \cot ^4(c+d x) \csc ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^2}+\frac {\int \cot ^4(c+d x) \csc ^8(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx}{a^2}\\ &=\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}+\frac {3 \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{5 a^2}+\frac {\operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \int \csc ^7(c+d x) \, dx}{40 a^2}+\frac {\operatorname {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {\int \csc ^5(c+d x) \, dx}{16 a^2}\\ &=-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \int \csc ^3(c+d x) \, dx}{64 a^2}\\ &=-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}-\frac {3 \int \csc (c+d x) \, dx}{128 a^2}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {4 \cot ^9(c+d x)}{9 a^2 d}-\frac {\cot ^{11}(c+d x)}{11 a^2 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{80 a^2 d}-\frac {3 \cot (c+d x) \csc ^7(c+d x)}{40 a^2 d}+\frac {\cot ^3(c+d x) \csc ^7(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A] time = 4.23, size = 186, normalized size = 0.89 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (2661120 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cot (c+d x) \csc ^{10}(c+d x) (2457378 \sin (c+d x)+5907132 \sin (3 (c+d x))+656964 \sin (5 (c+d x))-121275 \sin (7 (c+d x))+10395 \sin (9 (c+d x))-5752832 \cos (2 (c+d x))+346112 \cos (4 (c+d x))+583168 \cos (6 (c+d x))-104448 \cos (8 (c+d x))+8704 \cos (10 (c+d x))-5402624)\right )}{113541120 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 324, normalized size = 1.54 \[ -\frac {34816 \, \cos \left (d x + c\right )^{11} - 191488 \, \cos \left (d x + c\right )^{9} + 430848 \, \cos \left (d x + c\right )^{7} - 354816 \, \cos \left (d x + c\right )^{5} - 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10395 \, {\left (\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 1386 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, {\left (a^{2} d \cos \left (d x + c\right )^{10} - 5 \, a^{2} d \cos \left (d x + c\right )^{8} + 10 \, a^{2} d \cos \left (d x + c\right )^{6} - 10 \, a^{2} d \cos \left (d x + c\right )^{4} + 5 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 361, normalized size = 1.72 \[ -\frac {\frac {166320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {502266 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 131670 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 25410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 18711 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6930 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1485 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2695 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1386 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 315}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}} - \frac {315 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1386 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 2695 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1485 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 6930 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 18711 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27720 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 25410 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 13860 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 131670 \, a^{20} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{22}}}{7096320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.76, size = 436, normalized size = 2.08 \[ \frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{22528 d \,a^{2}}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{5120 d \,a^{2}}+\frac {7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{18432 d \,a^{2}}-\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 d \,a^{2}}+\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{14336 d \,a^{2}}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 d \,a^{2}}-\frac {27 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10240 a^{2} d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{256 a^{2} d}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3072 d \,a^{2}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{512 a^{2} d}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1024 d \,a^{2}}-\frac {1}{1024 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {19}{1024 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2}}+\frac {27}{10240 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {3}{14336 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{512 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {7}{18432 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}+\frac {1}{2048 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {1}{256 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{22528 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}+\frac {11}{3072 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{5120 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 474, normalized size = 2.26 \[ \frac {\frac {\frac {131670 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {13860 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {25410 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {27720 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {18711 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {6930 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1485 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3465 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {2695 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1386 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{2}} - \frac {166320 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {1386 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2695 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3465 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {1485 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6930 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {18711 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {27720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {25410 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {13860 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {131670 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - 315\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{11}}{a^{2} \sin \left (d x + c\right )^{11}}}{7096320 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.55, size = 579, normalized size = 2.76 \[ -\frac {315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}-315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}+1386\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}-1386\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2695\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}+3465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-1485\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+18711\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-27720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+25410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+13860\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-131670\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+131670\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-13860\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-25410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+27720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-18711\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1485\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2695\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+166320\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{7096320\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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