Optimal. Leaf size=176 \[ \frac {2 \cot ^7(c+d x)}{7 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {11 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}+\frac {7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]
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Rubi [A] time = 0.41, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2875, 2873, 2611, 3768, 3770, 2607, 14} \[ \frac {2 \cot ^7(c+d x)}{7 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {11 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}+\frac {7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{128 a^2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 2611
Rule 2873
Rule 2875
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^8(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cot ^4(c+d x) \csc ^5(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \cot ^4(c+d x) \csc ^3(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^4(c+d x)+a^2 \cot ^4(c+d x) \csc ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{a^2}+\frac {\int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx}{a^2}\\ &=-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}-\frac {3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{8 a^2}-\frac {\int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{2 a^2}-\frac {2 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=\frac {\cot (c+d x) \csc ^3(c+d x)}{8 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac {\int \csc ^5(c+d x) \, dx}{16 a^2}+\frac {\int \csc ^3(c+d x) \, dx}{8 a^2}-\frac {2 \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{16 a^2 d}+\frac {7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac {3 \int \csc ^3(c+d x) \, dx}{64 a^2}+\frac {\int \csc (c+d x) \, dx}{16 a^2}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac {3 \int \csc (c+d x) \, dx}{128 a^2}\\ &=-\frac {11 \tanh ^{-1}(\cos (c+d x))}{128 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {2 \cot ^7(c+d x)}{7 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{128 a^2 d}+\frac {7 \cot (c+d x) \csc ^3(c+d x)}{64 a^2 d}-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 a^2 d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{16 a^2 d}-\frac {\cot ^3(c+d x) \csc ^5(c+d x)}{8 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 291, normalized size = 1.65 \[ -\frac {\csc ^8(c+d x) \left (-86016 \sin (2 (c+d x))-64512 \sin (4 (c+d x))-12288 \sin (6 (c+d x))+1536 \sin (8 (c+d x))+158270 \cos (c+d x)+77210 \cos (3 (c+d x))-18130 \cos (5 (c+d x))-2310 \cos (7 (c+d x))-40425 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-64680 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+32340 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-9240 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1155 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+40425 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+64680 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-32340 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+9240 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1155 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{1720320 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 239, normalized size = 1.36 \[ \frac {2310 \, \cos \left (d x + c\right )^{7} + 490 \, \cos \left (d x + c\right )^{5} - 8470 \, \cos \left (d x + c\right )^{3} - 1155 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1155 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 1536 \, {\left (2 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right ) + 2310 \, \cos \left (d x + c\right )}{26880 \, {\left (a^{2} d \cos \left (d x + c\right )^{8} - 4 \, a^{2} d \cos \left (d x + c\right )^{6} + 6 \, a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 273, normalized size = 1.55 \[ \frac {\frac {18480 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {50226 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}} + \frac {105 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 480 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 672 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2520 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3360 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10080 \, a^{14} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{16}}}{215040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.70, size = 322, normalized size = 1.83 \[ \frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2048 d \,a^{2}}-\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{448 d \,a^{2}}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d \,a^{2}}+\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{320 a^{2} d}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 a^{2} d}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d \,a^{2}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a^{2} d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d \,a^{2}}-\frac {1}{384 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {3}{64 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{2}}-\frac {1}{320 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{448 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}+\frac {1}{128 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{2048 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}+\frac {3}{256 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{64 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 355, normalized size = 2.02 \[ -\frac {\frac {\frac {10080 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1680 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3360 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2520 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {672 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {560 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {480 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {105 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a^{2}} - \frac {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {480 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {560 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {672 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2520 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3360 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1680 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {10080 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 105\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a^{2} \sin \left (d x + c\right )^{8}}}{215040 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.96, size = 435, normalized size = 2.47 \[ \frac {105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-480\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+480\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+18480\,\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 )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{215040\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]
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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