Optimal. Leaf size=65 \[ -\frac{24 \cot (x)}{5 a^3}+\frac{3 \tanh ^{-1}(\cos (x))}{a^3}+\frac{3 \cot (x)}{a^3 \sin (x)+a^3}+\frac{3 \cot (x)}{5 a (a \sin (x)+a)^2}+\frac{\cot (x)}{5 (a \sin (x)+a)^3} \]
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Rubi [A] time = 0.228711, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2766, 2978, 2748, 3767, 8, 3770} \[ -\frac{24 \cot (x)}{5 a^3}+\frac{3 \tanh ^{-1}(\cos (x))}{a^3}+\frac{3 \cot (x)}{a^3 \sin (x)+a^3}+\frac{3 \cot (x)}{5 a (a \sin (x)+a)^2}+\frac{\cot (x)}{5 (a \sin (x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2766
Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\csc ^2(x)}{(a+a \sin (x))^3} \, dx &=\frac{\cot (x)}{5 (a+a \sin (x))^3}+\frac{\int \frac{\csc ^2(x) (6 a-3 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac{\cot (x)}{5 (a+a \sin (x))^3}+\frac{3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac{\int \frac{\csc ^2(x) \left (27 a^2-18 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac{\cot (x)}{5 (a+a \sin (x))^3}+\frac{3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac{3 \cot (x)}{a^3+a^3 \sin (x)}+\frac{\int \csc ^2(x) \left (72 a^3-45 a^3 \sin (x)\right ) \, dx}{15 a^6}\\ &=\frac{\cot (x)}{5 (a+a \sin (x))^3}+\frac{3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac{3 \cot (x)}{a^3+a^3 \sin (x)}-\frac{3 \int \csc (x) \, dx}{a^3}+\frac{24 \int \csc ^2(x) \, dx}{5 a^3}\\ &=\frac{3 \tanh ^{-1}(\cos (x))}{a^3}+\frac{\cot (x)}{5 (a+a \sin (x))^3}+\frac{3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac{3 \cot (x)}{a^3+a^3 \sin (x)}-\frac{24 \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{5 a^3}\\ &=\frac{3 \tanh ^{-1}(\cos (x))}{a^3}-\frac{24 \cot (x)}{5 a^3}+\frac{\cot (x)}{5 (a+a \sin (x))^3}+\frac{3 \cot (x)}{5 a (a+a \sin (x))^2}+\frac{3 \cot (x)}{a^3+a^3 \sin (x)}\\ \end{align*}
Mathematica [B] time = 0.139807, size = 206, normalized size = 3.17 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (4 \sin \left (\frac{x}{2}\right )+76 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4-8 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3+16 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2-2 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )+30 \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-30 \log \left (\sin \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5+5 \tan \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5-5 \cot \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^5\right )}{10 (a \sin (x)+a)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 97, normalized size = 1.5 \begin{align*}{\frac{1}{2\,{a}^{3}}\tan \left ({\frac{x}{2}} \right ) }-{\frac{8}{5\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+4\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{4}}}-8\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{3}}}+8\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}-12\,{\frac{1}{{a}^{3} \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{2\,{a}^{3}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-3\,{\frac{\ln \left ( \tan \left ( x/2 \right ) \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.89945, size = 243, normalized size = 3.74 \begin{align*} -\frac{\frac{121 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{410 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{610 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{425 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{125 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + 5}{10 \,{\left (\frac{a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{5 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{10 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{5 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac{a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}}\right )}} - \frac{3 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} + \frac{\sin \left (x\right )}{2 \, a^{3}{\left (\cos \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.5206, size = 691, normalized size = 10.63 \begin{align*} \frac{48 \, \cos \left (x\right )^{4} + 114 \, \cos \left (x\right )^{3} - 60 \, \cos \left (x\right )^{2} + 15 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 15 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) + 2 \, \cos \left (x\right ) + 4\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \,{\left (24 \, \cos \left (x\right )^{3} - 33 \, \cos \left (x\right )^{2} - 63 \, \cos \left (x\right ) - 1\right )} \sin \left (x\right ) - 124 \, \cos \left (x\right ) + 2}{10 \,{\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{3} - 5 \, a^{3} \cos \left (x\right )^{2} + 2 \, a^{3} \cos \left (x\right ) + 4 \, a^{3} -{\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\csc ^{2}{\left (x \right )}}{\sin ^{3}{\left (x \right )} + 3 \sin ^{2}{\left (x \right )} + 3 \sin{\left (x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27868, size = 115, normalized size = 1.77 \begin{align*} -\frac{3 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac{\tan \left (\frac{1}{2} \, x\right )}{2 \, a^{3}} + \frac{6 \, \tan \left (\frac{1}{2} \, x\right ) - 1}{2 \, a^{3} \tan \left (\frac{1}{2} \, x\right )} - \frac{4 \,{\left (15 \, \tan \left (\frac{1}{2} \, x\right )^{4} + 50 \, \tan \left (\frac{1}{2} \, x\right )^{3} + 70 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 45 \, \tan \left (\frac{1}{2} \, x\right ) + 12\right )}}{5 \, a^{3}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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