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 = 0.407361, antiderivative size = 176, normalized size of antiderivative = 1., 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 2875
Rule 2873
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
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*}
Mathematica [A] time = 0.894743, 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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Maple [B] time = 0.194, size = 322, normalized size = 1.8 \begin{align*}{\frac{1}{2048\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}}-{\frac{1}{448\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{1}{384\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}+{\frac{1}{320\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{3}{256\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{1}{64\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{3}{64\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{448\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{3}{64\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{2048\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-8}}-{\frac{1}{320\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{3}{256\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{11}{128\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{384\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}-{\frac{1}{64\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09685, size = 479, normalized size = 2.72 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19628, size = 655, normalized size = 3.72 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37301, size = 369, normalized size = 2.1 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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