Optimal. Leaf size=194 \[ -\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.350867, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2611, 3768, 3770, 2607, 14, 270} \[ -\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 14
Rule 270
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+a^3 \cot ^4(c+d x) \csc ^6(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{1}{2} a^3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx-\frac{1}{8} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{8} a^3 \int \csc ^3(c+d x) \, dx+\frac{1}{16} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{16} a^3 \int \csc (c+d x) \, dx+\frac{1}{64} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{128} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{17 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{4 a^3 \cot ^5(c+d x)}{5 d}-\frac{5 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{9 d}-\frac{17 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end{align*}
Mathematica [A] time = 1.32716, size = 313, normalized size = 1.61 \[ -\frac{a^3 \csc ^9(c+d x) \left (669060 \sin (2 (c+d x))+676620 \sin (4 (c+d x))-14700 \sin (6 (c+d x))-10710 \sin (8 (c+d x))+1161216 \cos (c+d x)+247296 \cos (3 (c+d x))-198144 \cos (5 (c+d x))-71424 \cos (7 (c+d x))+7936 \cos (9 (c+d x))-674730 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+449820 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-192780 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+48195 \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-5355 \sin (9 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+674730 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-449820 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+192780 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-48195 \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+5355 \sin (9 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{10321920 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 224, normalized size = 1.2 \begin{align*} -{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{384\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}+{\frac{17\,{a}^{3}\cos \left ( dx+c \right ) }{128\,d}}+{\frac{17\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{31\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{62\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{3\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15675, size = 362, normalized size = 1.87 \begin{align*} \frac{945 \, a^{3}{\left (\frac{2 \,{\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 )}}{\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} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 840 \, a^{3}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \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} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{6912 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac{256 \,{\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{80640 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25196, size = 803, normalized size = 4.14 \begin{align*} -\frac{15872 \, a^{3} \cos \left (d x + c\right )^{9} - 71424 \, a^{3} \cos \left (d x + c\right )^{7} + 64512 \, a^{3} \cos \left (d x + c\right )^{5} + 5355 \,{\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 ) \sin \left (d x + c\right ) - 5355 \,{\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 ) \sin \left (d x + c\right ) - 210 \,{\left (51 \, a^{3} \cos \left (d x + c\right )^{7} - 59 \, a^{3} \cos \left (d x + c\right )^{5} - 187 \, a^{3} \cos \left (d x + c\right )^{3} + 51 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \,{\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 )} \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 = 1.36776, size = 439, normalized size = 2.26 \begin{align*} \frac{140 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 945 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 2340 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1680 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 4032 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 16800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5040 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 85680 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 52920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{242386 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 52920 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 5040 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 16800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 12600 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4032 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 1680 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2340 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 945 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 140 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9}}}{645120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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