Optimal. Leaf size=38 \[ \frac {4 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac {\tanh ^{-1}(\cos (x))}{a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2766, 2978, 12, 3770} \[ \frac {4 \cos (x)}{3 a^2 (\sin (x)+1)}-\frac {\tanh ^{-1}(\cos (x))}{a^2}+\frac {\cos (x)}{3 (a \sin (x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2766
Rule 2978
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc (x)}{(a+a \sin (x))^2} \, dx &=\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int \frac {\csc (x) (3 a-a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int 3 a^2 \csc (x) \, dx}{3 a^4}\\ &=\frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2}+\frac {\int \csc (x) \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a^2}+\frac {4 \cos (x)}{3 a^2 (1+\sin (x))}+\frac {\cos (x)}{3 (a+a \sin (x))^2}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 129, normalized size = 3.39 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (\cos \left (\frac {3 x}{2}\right ) \left (-3 \log \left (\sin \left (\frac {x}{2}\right )\right )+3 \log \left (\cos \left (\frac {x}{2}\right )\right )+8\right )+\cos \left (\frac {x}{2}\right ) \left (9 \log \left (\sin \left (\frac {x}{2}\right )\right )-9 \log \left (\cos \left (\frac {x}{2}\right )\right )-6\right )-6 \sin \left (\frac {x}{2}\right ) \left (-2 \log \left (\sin \left (\frac {x}{2}\right )\right )+2 \log \left (\cos \left (\frac {x}{2}\right )\right )+\cos (x) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+3\right )\right )}{6 a^2 (\sin (x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 117, normalized size = 3.08 \[ -\frac {8 \, \cos \relax (x)^{2} + 3 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 3 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (4 \, \cos \relax (x) - 1\right )} \sin \relax (x) + 10 \, \cos \relax (x) + 2}{6 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2} - {\left (a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.70, size = 40, normalized size = 1.05 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, x\right ) + 5\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 50, normalized size = 1.32 \[ \frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}+\frac {4}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 89, normalized size = 2.34 \[ \frac {2 \, {\left (\frac {9 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {6 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 5\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {3 \, a^{2} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {a^{2} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}\right )}} + \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.48, size = 38, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}+\frac {4\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {10}{3}}{a^2\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc {\relax (x )}}{\sin ^{2}{\relax (x )} + 2 \sin {\relax (x )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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