Optimal. Leaf size=168 \[ -\frac{\cot ^9(c+d x)}{9 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{64 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{64 a^2 d} \]
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Rubi [A] time = 0.386414, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac{\cot ^9(c+d x)}{9 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{64 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{64 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2607
Rule 14
Rule 2611
Rule 3768
Rule 3770
Rule 270
Rubi steps
\begin{align*} \int \frac{\cot ^8(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \cot ^4(c+d x) \csc ^6(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \cot ^4(c+d x) \csc ^4(c+d x)-2 a^2 \cot ^4(c+d x) \csc ^5(c+d x)+a^2 \cot ^4(c+d x) \csc ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \cot ^4(c+d x) \csc ^4(c+d x) \, dx}{a^2}+\frac{\int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^2}-\frac{2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{a^2}\\ &=\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}+\frac{3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{4 a^2}+\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^2 d}\\ &=-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{\int \csc ^5(c+d x) \, dx}{8 a^2}+\frac{\operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\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{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^9(c+d x)}{9 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{3 \int \csc ^3(c+d x) \, dx}{32 a^2}\\ &=-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^9(c+d x)}{9 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{64 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}-\frac{3 \int \csc (c+d x) \, dx}{64 a^2}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{3 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^9(c+d x)}{9 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{64 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{32 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{8 a^2 d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{4 a^2 d}\\ \end{align*}
Mathematica [A] time = 1.79092, size = 313, normalized size = 1.86 \[ \frac{\csc ^9(c+d x) \left (212940 \sin (2 (c+d x))+195300 \sin (4 (c+d x))+16380 \sin (6 (c+d x))-1890 \sin (8 (c+d x))-451584 \cos (c+d x)-155904 \cos (3 (c+d x))+20736 \cos (5 (c+d x))+14976 \cos (7 (c+d x))-1664 \cos (9 (c+d x))-119070 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+79380 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-34020 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+8505 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-945 \sin (9 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+119070 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-79380 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+34020 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-8505 \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+945 \sin (9 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{5160960 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.203, size = 284, normalized size = 1.7 \begin{align*}{\frac{1}{4608\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{1}{1024\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}}+{\frac{5}{3584\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{320\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{1}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{1}{96\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{9}{256\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{5}{3584\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}-{\frac{9}{256\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{1024\,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{1}{128\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{1}{4608\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-9}}-{\frac{3}{64\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{96\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0409, size = 424, normalized size = 2.52 \begin{align*} \frac{\frac{\frac{11340 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \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{1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{70 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{2}} - \frac{15120 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac{{\left (\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{450 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1008 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{2520 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{3360 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{11340 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 70\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}{a^{2} \sin \left (d x + c\right )^{9}}}{322560 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20781, size = 736, normalized size = 4.38 \begin{align*} -\frac{3328 \, \cos \left (d x + c\right )^{9} - 14976 \, \cos \left (d x + c\right )^{7} + 16128 \, \cos \left (d x + c\right )^{5} - 945 \,{\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 ) \sin \left (d x + c\right ) + 945 \,{\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 ) \sin \left (d x + c\right ) + 630 \,{\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \,{\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 )} \sin \left (d x + c\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 = 1.3495, size = 331, normalized size = 1.97 \begin{align*} -\frac{\frac{15120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{42774 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 11340 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 3360 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 2520 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1008 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 450 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 70}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}} - \frac{70 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 315 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 450 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1008 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2520 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3360 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 11340 \, a^{16} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{18}}}{322560 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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