Optimal. Leaf size=92 \[ \frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{2 a^3} \]
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Rubi [A] time = 0.222455, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2709, 3770, 3767, 8, 2638, 2635, 2633} \[ \frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos (c+d x)}{a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{x}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2709
Rule 3770
Rule 3767
Rule 8
Rule 2638
Rule 2635
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \cot ^2(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (2 a^5-3 a^5 \csc (c+d x)+a^5 \csc ^2(c+d x)+2 a^5 \sin (c+d x)-3 a^5 \sin ^2(c+d x)+a^5 \sin ^3(c+d x)\right ) \, dx}{a^8}\\ &=\frac{2 x}{a^3}+\frac{\int \csc ^2(c+d x) \, dx}{a^3}+\frac{\int \sin ^3(c+d x) \, dx}{a^3}+\frac{2 \int \sin (c+d x) \, dx}{a^3}-\frac{3 \int \csc (c+d x) \, dx}{a^3}-\frac{3 \int \sin ^2(c+d x) \, dx}{a^3}\\ &=\frac{2 x}{a^3}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{2 \cos (c+d x)}{a^3 d}+\frac{3 \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{3 \int 1 \, dx}{2 a^3}-\frac{\operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}\\ &=\frac{x}{2 a^3}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{3 \cos (c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{\cot (c+d x)}{a^3 d}+\frac{3 \cos (c+d x) \sin (c+d x)}{2 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.958189, size = 126, normalized size = 1.37 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6 \left (6 (c+d x)+9 \sin (2 (c+d x))-33 \cos (c+d x)+\cos (3 (c+d x))+6 \tan \left (\frac{1}{2} (c+d x)\right )-6 \cot \left (\frac{1}{2} (c+d x)\right )-36 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+36 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{12 d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.155, size = 230, normalized size = 2.5 \begin{align*}{\frac{1}{2\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-3\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-12\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+3\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-{\frac{16}{3\,d{a}^{3}} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}+{\frac{1}{d{a}^{3}}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-3\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56722, size = 385, normalized size = 4.18 \begin{align*} -\frac{\frac{\frac{32 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{72 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{21 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 3}{\frac{a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac{6 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac{18 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{3 \, \sin \left (d x + c\right )}{a^{3}{\left (\cos \left (d x + c\right ) + 1\right )}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12946, size = 289, normalized size = 3.14 \begin{align*} -\frac{9 \, \cos \left (d x + c\right )^{3} -{\left (2 \, \cos \left (d x + c\right )^{3} + 3 \, d x - 18 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 9 \, \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 9 \, \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 3 \, \cos \left (d x + c\right )}{6 \, a^{3} d \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 = 1.2936, size = 198, normalized size = 2.15 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}}{a^{3}} - \frac{18 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{3}} + \frac{3 \,{\left (6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{2 \,{\left (9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 36 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 16\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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