Optimal. Leaf size=100 \[ -\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d} \]
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Rubi [A] time = 0.171977, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2839, 2607, 14, 2611, 3768, 3770} \[ -\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2607
Rule 14
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cot ^4(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a}+\frac{\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a}\\ &=\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{\int \csc ^3(c+d x) \, dx}{4 a}+\frac{\operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac{\int \csc (c+d x) \, dx}{8 a}+\frac{\operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.585639, size = 189, normalized size = 1.89 \[ -\frac{\csc ^5(c+d x) \left (-180 \sin (2 (c+d x))-30 \sin (4 (c+d x))+320 \cos (c+d x)+80 \cos (3 (c+d x))-16 \cos (5 (c+d x))-150 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+75 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-15 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+150 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-75 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+15 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1920 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 170, normalized size = 1.7 \begin{align*}{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{1}{96\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{16\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{16\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{160\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{1}{64\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{1}{8\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{96\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13519, size = 263, normalized size = 2.63 \begin{align*} -\frac{\frac{\frac{60 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \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} - \frac{120 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{{\left (\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{60 \, \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 \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 = 1.10251, size = 452, normalized size = 4.52 \begin{align*} \frac{32 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 15 \,{\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 ) + 15 \,{\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} + \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \,{\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a 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 = 1.35326, size = 212, normalized size = 2.12 \begin{align*} \frac{\frac{120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{6 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 10 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 60 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{5}} - \frac{274 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 60 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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