Optimal. Leaf size=166 \[ \frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac{29 \cot (c+d x) \csc (c+d x)}{128 a^3 d} \]
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Rubi [A] time = 0.405674, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2875, 2873, 2607, 14, 2611, 3768, 3770, 270} \[ \frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}+\frac{29 \cot (c+d x) \csc (c+d x)}{128 a^3 d} \]
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 (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \cot ^2(c+d x) \csc ^7(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \cot ^2(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^5(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^6(c+d x)+a^3 \cot ^2(c+d x) \csc ^7(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac{\int \cot ^2(c+d x) \csc ^7(c+d x) \, dx}{a^3}+\frac{3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^3}-\frac{3 \int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a^3}\\ &=-\frac{\cot (c+d x) \csc ^5(c+d x)}{2 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{\int \csc ^7(c+d x) \, dx}{8 a^3}-\frac{\int \csc ^5(c+d x) \, dx}{2 a^3}-\frac{\operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac{\cot (c+d x) \csc ^3(c+d x)}{8 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{5 \int \csc ^5(c+d x) \, dx}{48 a^3}-\frac{3 \int \csc ^3(c+d x) \, dx}{8 a^3}-\frac{\operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{16 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{5 \int \csc ^3(c+d x) \, dx}{64 a^3}-\frac{3 \int \csc (c+d x) \, dx}{16 a^3}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}-\frac{5 \int \csc (c+d x) \, dx}{128 a^3}\\ &=\frac{29 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}+\frac{4 \cot ^3(c+d x)}{3 a^3 d}+\frac{7 \cot ^5(c+d x)}{5 a^3 d}+\frac{3 \cot ^7(c+d x)}{7 a^3 d}+\frac{29 \cot (c+d x) \csc (c+d x)}{128 a^3 d}+\frac{29 \cot (c+d x) \csc ^3(c+d x)}{192 a^3 d}-\frac{23 \cot (c+d x) \csc ^5(c+d x)}{48 a^3 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a^3 d}\\ \end{align*}
Mathematica [A] time = 4.96387, size = 317, normalized size = 1.91 \[ -\frac{\sin ^7(c+d x) \left (\csc \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{1}{2} (c+d x)\right )\right )^6 \left (15 (7 \csc (c+d x)-24) \csc ^8\left (\frac{1}{2} (c+d x)\right )+4 (455 \csc (c+d x)-276) \csc ^6\left (\frac{1}{2} (c+d x)\right )+(1328-210 \csc (c+d x)) \csc ^4\left (\frac{1}{2} (c+d x)\right )-4 (3045 \csc (c+d x)-4864) \csc ^2\left (\frac{1}{2} (c+d x)\right )-8 \left (\frac{1}{4} (4616 \cos (c+d x)+1907 \cos (2 (c+d x))+304 \cos (3 (c+d x))+2833) \sec ^8\left (\frac{1}{2} (c+d x)\right )+3360 \sin ^8\left (\frac{1}{2} (c+d x)\right ) \csc ^9(c+d x)+14560 \sin ^6\left (\frac{1}{2} (c+d x)\right ) \csc ^7(c+d x)-420 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^5(c+d x)-6090 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)+6090 \csc (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )\right )}{13762560 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.206, size = 322, normalized size = 1.9 \begin{align*}{\frac{1}{2048\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}}-{\frac{3}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{1}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}-{\frac{13}{640\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7}{256\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{7}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{32\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{23}{128\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{3}{896\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}-{\frac{23}{128\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{2048\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-8}}+{\frac{13}{640\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{7}{256\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{29}{128\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{7}{384\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{32\,d{a}^{3}} \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 = 1.07134, size = 478, normalized size = 2.88 \begin{align*} \frac{\frac{\frac{38640 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{6720 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{3920 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{5880 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{4368 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{2240 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{720 \, \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^{3}} - \frac{48720 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac{{\left (\frac{720 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{4368 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{5880 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{3920 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{6720 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{38640 \, \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^{3} \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 = 1.17516, size = 691, normalized size = 4.16 \begin{align*} -\frac{6090 \, \cos \left (d x + c\right )^{7} - 22330 \, \cos \left (d x + c\right )^{5} + 13510 \, \cos \left (d x + c\right )^{3} - 3045 \,{\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 ) + 3045 \,{\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 ) - 256 \,{\left (38 \, \cos \left (d x + c\right )^{7} - 133 \, \cos \left (d x + c\right )^{5} + 140 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) + 6090 \, \cos \left (d x + c\right )}{26880 \,{\left (a^{3} d \cos \left (d x + c\right )^{8} - 4 \, a^{3} d \cos \left (d x + c\right )^{6} + 6 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} 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 = 1.28704, size = 370, normalized size = 2.23 \begin{align*} -\frac{\frac{48720 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{132414 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 38640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 6720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3920 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 5880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 4368 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 105}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}} - \frac{105 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 720 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 2240 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 4368 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 5880 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3920 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6720 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 38640 \, a^{21} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{24}}}{215040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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