Optimal. Leaf size=112 \[ -\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d} \]
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Rubi [A] time = 0.240746, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2869, 2757, 3767, 3768, 3770} \[ -\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2869
Rule 2757
Rule 3767
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))^2} \, dx &=\frac{\int \csc ^6(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac{\int \left (a^2 \csc ^4(c+d x)-2 a^2 \csc ^5(c+d x)+a^2 \csc ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \csc ^4(c+d x) \, dx}{a^2}+\frac{\int \csc ^6(c+d x) \, dx}{a^2}-\frac{2 \int \csc ^5(c+d x) \, dx}{a^2}\\ &=\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac{3 \int \csc ^3(c+d x) \, dx}{2 a^2}-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{a^2 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}-\frac{3 \int \csc (c+d x) \, dx}{4 a^2}\\ &=\frac{3 \tanh ^{-1}(\cos (c+d x))}{4 a^2 d}-\frac{2 \cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{a^2 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}+\frac{3 \cot (c+d x) \csc (c+d x)}{4 a^2 d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.736342, size = 189, normalized size = 1.69 \[ \frac{\csc ^5(c+d x) \left (140 \sin (2 (c+d x))-30 \sin (4 (c+d x))-160 \cos (c+d x)+120 \cos (3 (c+d x))-24 \cos (5 (c+d x))-150 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+75 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-15 \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+150 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-75 \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+15 \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{320 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.183, size = 208, normalized size = 1.9 \begin{align*}{\frac{1}{160\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{1}{32\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}+{\frac{3}{32\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{1}{4\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{11}{16\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{11}{16\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{160\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}+{\frac{1}{32\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{3}{4\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{3}{32\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{4\,d{a}^{2}} \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 = 1.0929, size = 315, normalized size = 2.81 \begin{align*} \frac{\frac{\frac{110 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{15 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{2}} - \frac{120 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac{{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{110 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15899, size = 491, normalized size = 4.38 \begin{align*} -\frac{48 \, \cos \left (d x + c\right )^{5} - 120 \, \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 ) + 10 \,{\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 80 \, \cos \left (d x + c\right )}{40 \,{\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{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 = 1.38664, size = 251, normalized size = 2.24 \begin{align*} -\frac{\frac{120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{274 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 110 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 5 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}} - \frac{a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 5 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 15 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 110 \, a^{8} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{10}}}{160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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