Optimal. Leaf size=58 \[ \frac {22 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {\tanh ^{-1}(\cos (x))}{a^3}+\frac {7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]
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Rubi [A] time = 0.16, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2766, 2978, 12, 3770} \[ \frac {22 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}-\frac {\tanh ^{-1}(\cos (x))}{a^3}+\frac {7 \cos (x)}{15 a (a \sin (x)+a)^2}+\frac {\cos (x)}{5 (a \sin (x)+a)^3} \]
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))^3} \, dx &=\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {\int \frac {\csc (x) (5 a-2 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {\int \frac {\csc (x) \left (15 a^2-7 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac {\int 15 a^3 \csc (x) \, dx}{15 a^6}\\ &=\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}+\frac {\int \csc (x) \, dx}{a^3}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a^3}+\frac {\cos (x)}{5 (a+a \sin (x))^3}+\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {22 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 160, normalized size = 2.76 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (-6 \sin \left (\frac {x}{2}\right )-44 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4+7 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3-14 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2+3 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-15 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5+15 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^5\right )}{15 (a \sin (x)+a)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 168, normalized size = 2.90 \[ \frac {44 \, \cos \relax (x)^{3} - 58 \, \cos \relax (x)^{2} - 15 \, {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 15 \, {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} + {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2 \, {\left (22 \, \cos \relax (x)^{2} + 51 \, \cos \relax (x) - 3\right )} \sin \relax (x) - 108 \, \cos \relax (x) - 6}{30 \, {\left (a^{3} \cos \relax (x)^{3} + 3 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3} + {\left (a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 56, normalized size = 0.97 \[ \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (45 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 135 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 185 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 115 \, \tan \left (\frac {1}{2} \, x\right ) + 32\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 76, normalized size = 1.31 \[ \frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}+\frac {8}{5 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {4}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {20}{3 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {6}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {6}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.86, size = 143, normalized size = 2.47 \[ \frac {2 \, {\left (\frac {115 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {185 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {135 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {45 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + 32\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {10 \, a^{3} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {a^{3} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}\right )}} + \frac {\log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.65, size = 54, normalized size = 0.93 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3}+\frac {6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+18\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {74\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+\frac {46\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {64}{15}}{a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \]
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 \[ \frac {\int \frac {\csc {\relax (x )}}{\sin ^{3}{\relax (x )} + 3 \sin ^{2}{\relax (x )} + 3 \sin {\relax (x )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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