Optimal. Leaf size=45 \[ -\frac{10 \cot (x)}{3 a^2}+\frac{2 \tanh ^{-1}(\cos (x))}{a^2}+\frac{2 \cot (x)}{a^2 (\sin (x)+1)}+\frac{\cot (x)}{3 (a \sin (x)+a)^2} \]
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Rubi [A] time = 0.134503, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{10 \cot (x)}{3 a^2}+\frac{2 \tanh ^{-1}(\cos (x))}{a^2}+\frac{2 \cot (x)}{a^2 (\sin (x)+1)}+\frac{\cot (x)}{3 (a \sin (x)+a)^2} \]
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))^2} \, dx &=\frac{\cot (x)}{3 (a+a \sin (x))^2}+\frac{\int \frac{\csc ^2(x) (4 a-2 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac{2 \cot (x)}{a^2 (1+\sin (x))}+\frac{\cot (x)}{3 (a+a \sin (x))^2}+\frac{\int \csc ^2(x) \left (10 a^2-6 a^2 \sin (x)\right ) \, dx}{3 a^4}\\ &=\frac{2 \cot (x)}{a^2 (1+\sin (x))}+\frac{\cot (x)}{3 (a+a \sin (x))^2}-\frac{2 \int \csc (x) \, dx}{a^2}+\frac{10 \int \csc ^2(x) \, dx}{3 a^2}\\ &=\frac{2 \tanh ^{-1}(\cos (x))}{a^2}+\frac{2 \cot (x)}{a^2 (1+\sin (x))}+\frac{\cot (x)}{3 (a+a \sin (x))^2}-\frac{10 \operatorname{Subst}(\int 1 \, dx,x,\cot (x))}{3 a^2}\\ &=\frac{2 \tanh ^{-1}(\cos (x))}{a^2}-\frac{10 \cot (x)}{3 a^2}+\frac{2 \cot (x)}{a^2 (1+\sin (x))}+\frac{\cot (x)}{3 (a+a \sin (x))^2}\\ \end{align*}
Mathematica [B] time = 0.345518, size = 166, normalized size = 3.69 \[ \frac{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (4 \sin \left (\frac{x}{2}\right )+28 \sin \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2-2 \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )+12 \log \left (\cos \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3-12 \log \left (\sin \left (\frac{x}{2}\right )\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3+3 \tan \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3-3 \cot \left (\frac{x}{2}\right ) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^3\right )}{6 (a \sin (x)+a)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 71, normalized size = 1.6 \begin{align*}{\frac{1}{2\,{a}^{2}}\tan \left ({\frac{x}{2}} \right ) }-{\frac{4}{3\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+2\,{\frac{1}{{a}^{2} \left ( \tan \left ( x/2 \right ) +1 \right ) ^{2}}}-6\,{\frac{1}{{a}^{2} \left ( \tan \left ( x/2 \right ) +1 \right ) }}-{\frac{1}{2\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-2\,{\frac{\ln \left ( \tan \left ( x/2 \right ) \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63623, size = 170, normalized size = 3.78 \begin{align*} -\frac{\frac{41 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{69 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{39 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 3}{6 \,{\left (\frac{a^{2} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{3 \, a^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, a^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{a^{2} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}\right )}} - \frac{2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2}} + \frac{\sin \left (x\right )}{2 \, a^{2}{\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 = 1.43824, size = 512, normalized size = 11.38 \begin{align*} -\frac{10 \, \cos \left (x\right )^{3} - 4 \, \cos \left (x\right )^{2} - 3 \,{\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )^{2} +{\left (\cos \left (x\right )^{2} - \cos \left (x\right ) - 2\right )} \sin \left (x\right ) - \cos \left (x\right ) - 2\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (10 \, \cos \left (x\right )^{2} + 14 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 13 \, \cos \left (x\right ) + 1}{3 \,{\left (a^{2} \cos \left (x\right )^{3} + 2 \, a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2} +{\left (a^{2} \cos \left (x\right )^{2} - a^{2} \cos \left (x\right ) - 2 \, a^{2}\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 ^{2}{\left (x \right )} + 2 \sin{\left (x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.04621, size = 93, normalized size = 2.07 \begin{align*} -\frac{2 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac{\tan \left (\frac{1}{2} \, x\right )}{2 \, a^{2}} + \frac{4 \, \tan \left (\frac{1}{2} \, x\right ) - 1}{2 \, a^{2} \tan \left (\frac{1}{2} \, x\right )} - \frac{2 \,{\left (9 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 15 \, \tan \left (\frac{1}{2} \, x\right ) + 8\right )}}{3 \, a^{2}{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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