Optimal. Leaf size=238 \[ -\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]
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Rubi [A] time = 0.355159, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-a^3 x \]
Antiderivative was successfully verified.
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Rule 2873
Rule 3473
Rule 8
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{8} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-a^3 \int \cot ^4(c+d x) \, dx-\frac{1}{2} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac{1}{16} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^2(c+d x) \, dx+\frac{1}{8} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{15 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{64} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{16} \left (15 a^3\right ) \int \csc (c+d x) \, dx-a^3 \int 1 \, dx\\ &=-a^3 x+\frac{15 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac{1}{128} \left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=-a^3 x+\frac{125 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a^3 \cot ^7(c+d x)}{7 d}-\frac{115 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^3 \cot ^3(c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc (c+d x)}{2 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac{a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end{align*}
Mathematica [A] time = 1.17163, size = 279, normalized size = 1.17 \[ \frac{a^3 \left (118784 \tan \left (\frac{1}{2} (c+d x)\right )-118784 \cot \left (\frac{1}{2} (c+d x)\right )-108780 \csc ^2\left (\frac{1}{2} (c+d x)\right )+105 \sec ^8\left (\frac{1}{2} (c+d x)\right )+700 \sec ^6\left (\frac{1}{2} (c+d x)\right )-17010 \sec ^4\left (\frac{1}{2} (c+d x)\right )+108780 \sec ^2\left (\frac{1}{2} (c+d x)\right )-210000 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+210000 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+71936 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-15 (24 \sin (c+d x)+7) \csc ^8\left (\frac{1}{2} (c+d x)\right )+4 (732 \sin (c+d x)-175) \csc ^6\left (\frac{1}{2} (c+d x)\right )+(17010-4496 \sin (c+d x)) \csc ^4\left (\frac{1}{2} (c+d x)\right )+720 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right )-5856 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )-215040 c-215040 d x\right )}{215040 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.096, size = 253, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{a}^{3}x-{\frac{{a}^{3}c}{d}}-{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{25\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d}}-{\frac{125\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}-{\frac{125\,{a}^{3}\cos \left ( dx+c \right ) }{128\,d}}-{\frac{125\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67002, size = 358, normalized size = 1.5 \begin{align*} -\frac{1792 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 35 \, a^{3}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\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} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{3}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{11520 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23547, size = 983, normalized size = 4.13 \begin{align*} -\frac{26880 \, a^{3} d x \cos \left (d x + c\right )^{8} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{6} - 54390 \, a^{3} \cos \left (d x + c\right )^{7} + 161280 \, a^{3} d x \cos \left (d x + c\right )^{4} + 127750 \, a^{3} \cos \left (d x + c\right )^{5} - 107520 \, a^{3} d x \cos \left (d x + c\right )^{2} - 96250 \, a^{3} \cos \left (d x + c\right )^{3} + 26880 \, a^{3} d x + 26250 \, a^{3} \cos \left (d x + c\right ) - 13125 \,{\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 13125 \,{\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 256 \,{\left (116 \, a^{3} \cos \left (d x + c\right )^{7} - 406 \, a^{3} \cos \left (d x + c\right )^{5} + 350 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{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 = 1.26472, size = 408, normalized size = 1.71 \begin{align*} \frac{105 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 720 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1120 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3696 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 14280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 77280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 215040 \,{\left (d x + c\right )} a^{3} - 210000 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 122640 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{570750 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 122640 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 77280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 560 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 14280 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3696 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1120 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 720 \, a^{3} \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}}}{215040 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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