Optimal. Leaf size=100 \[ -\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
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Rubi [A] time = 0.337484, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 2611, 3770, 2607, 30, 3768, 14} \[ -\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{7 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2611
Rule 3770
Rule 2607
Rule 30
Rule 3768
Rule 14
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \cot ^2(c+d x) \csc ^4(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac{\int \left (-a^3 \cot ^2(c+d x) \csc (c+d x)+3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)+a^3 \cot ^2(c+d x) \csc ^4(c+d x)\right ) \, dx}{a^6}\\ &=-\frac{\int \cot ^2(c+d x) \csc (c+d x) \, dx}{a^3}+\frac{\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac{3 \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a^3}-\frac{3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^3}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{\int \csc (c+d x) \, dx}{2 a^3}+\frac{3 \int \csc ^3(c+d x) \, dx}{4 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 \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{\cot ^3(c+d x)}{a^3 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac{3 \int \csc (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{7 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{4 \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}+\frac{\cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac{3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.69672, size = 189, normalized size = 1.89 \[ -\frac{\csc ^5(c+d x) \left (-780 \sin (2 (c+d x))+30 \sin (4 (c+d x))+560 \cos (c+d x)-40 \cos (3 (c+d x))-136 \cos (5 (c+d x))-1050 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+525 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-105 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+1050 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-525 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+105 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1920 a^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 208, normalized size = 2.1 \begin{align*}{\frac{1}{160\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{3}{64\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{13}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{7}{16\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{7}{16\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{160\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{3}{64\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{7}{8\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{13}{96\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{8\,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 = 1.02244, size = 317, normalized size = 3.17 \begin{align*} -\frac{\frac{\frac{420 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{120 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{130 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{840 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac{{\left (\frac{45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{130 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{420 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15458, size = 468, normalized size = 4.68 \begin{align*} \frac{272 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) + 105 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 30 \,{\left (\cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} 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 [B] time = 1.31928, size = 251, normalized size = 2.51 \begin{align*} \frac{\frac{840 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{1918 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 130 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} + \frac{6 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 45 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 130 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 120 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 420 \, a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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