Optimal. Leaf size=176 \[ \frac{\cot ^9(c+d x)}{9 a d}+\frac{\cot ^7(c+d x)}{7 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d} \]
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Rubi [A] time = 0.247625, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2611, 3768, 3770, 2607, 14} \[ \frac{\cot ^9(c+d x)}{9 a d}+\frac{\cot ^7(c+d x)}{7 a d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \frac{\cot ^8(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^6(c+d x) \csc ^4(c+d x) \, dx}{a}+\frac{\int \cot ^6(c+d x) \csc ^5(c+d x) \, dx}{a}\\ &=-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac{\int \cot ^4(c+d x) \csc ^5(c+d x) \, dx}{2 a}-\frac{\operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}+\frac{3 \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{16 a}-\frac{\operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac{\cot ^7(c+d x)}{7 a d}+\frac{\cot ^9(c+d x)}{9 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac{\int \csc ^5(c+d x) \, dx}{32 a}\\ &=\frac{\cot ^7(c+d x)}{7 a d}+\frac{\cot ^9(c+d x)}{9 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac{3 \int \csc ^3(c+d x) \, dx}{128 a}\\ &=\frac{\cot ^7(c+d x)}{7 a d}+\frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}-\frac{3 \int \csc (c+d x) \, dx}{256 a}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{256 a d}+\frac{\cot ^7(c+d x)}{7 a d}+\frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot (c+d x) \csc (c+d x)}{256 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{128 a d}-\frac{\cot (c+d x) \csc ^5(c+d x)}{32 a d}+\frac{\cot ^3(c+d x) \csc ^5(c+d x)}{16 a d}-\frac{\cot ^5(c+d x) \csc ^5(c+d x)}{10 a d}\\ \end{align*}
Mathematica [B] time = 1.50859, size = 386, normalized size = 2.19 \[ -\frac{\csc ^9(c+d x) \left (\csc \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (-537600 \sin (2 (c+d x))-522240 \sin (4 (c+d x))-207360 \sin (6 (c+d x))-25600 \sin (8 (c+d x))+2560 \sin (10 (c+d x))+2367540 \cos (c+d x)+1307880 \cos (3 (c+d x))+436968 \cos (5 (c+d x))+18270 \cos (7 (c+d x))-1890 \cos (9 (c+d x))+119070 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+198450 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-113400 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+42525 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-9450 \cos (8 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+945 \cos (10 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-119070 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-198450 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+113400 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-42525 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+9450 \cos (8 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-945 \cos (10 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{165150720 a d (\csc (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.188, size = 360, normalized size = 2.1 \begin{align*}{\frac{1}{10240\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{10}}-{\frac{1}{4608\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{1}{4096\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}}+{\frac{3}{3584\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{2048\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}+{\frac{1}{512\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{1}{192\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{1024\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{3}{256\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{10240\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-10}}-{\frac{3}{3584\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}-{\frac{3}{256\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{1}{4096\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-8}}-{\frac{1}{512\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{1}{4608\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-9}}-{\frac{3}{256\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{2048\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{1}{192\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{1}{1024\,da} \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 = 1.05124, size = 532, normalized size = 3.02 \begin{align*} \frac{\frac{\frac{15120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{1260 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{6720 \, \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{630 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{1080 \, \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{280 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{126 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}}{a} - \frac{15120 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{{\left (\frac{280 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{315 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1080 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{630 \, \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 )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{6720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{1260 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{15120 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 126\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}{a \sin \left (d x + c\right )^{10}}}{1290240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24031, size = 768, normalized size = 4.36 \begin{align*} -\frac{1890 \, \cos \left (d x + c\right )^{9} - 8820 \, \cos \left (d x + c\right )^{7} - 16128 \, \cos \left (d x + c\right )^{5} + 8820 \, \cos \left (d x + c\right )^{3} - 945 \,{\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 ) + 945 \,{\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 ) - 2560 \,{\left (2 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right ) - 1890 \, \cos \left (d x + c\right )}{161280 \,{\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a 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 = 1.39131, size = 409, normalized size = 2.32 \begin{align*} -\frac{\frac{15120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{126 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 280 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 315 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 1080 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 630 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 2520 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 6720 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1260 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15120 \, a^{9} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{10}} - \frac{44286 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 15120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 1260 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 6720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 2520 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 630 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1080 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 280 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 126}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10}}}{1290240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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