Optimal. Leaf size=64 \[ \frac {16 \cot (x)}{3 a^2}-\frac {7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2766, 2978, 2748, 3768, 3770, 3767, 8} \[ \frac {16 \cot (x)}{3 a^2}-\frac {7 \tanh ^{-1}(\cos (x))}{2 a^2}-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (\sin (x)+1)}+\frac {\cot (x) \csc (x)}{3 (a \sin (x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2766
Rule 2978
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^3(x)}{(a+a \sin (x))^2} \, dx &=\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2}+\frac {\int \frac {\csc ^3(x) (5 a-3 a \sin (x))}{a+a \sin (x)} \, dx}{3 a^2}\\ &=\frac {8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2}+\frac {\int \csc ^3(x) \left (21 a^2-16 a^2 \sin (x)\right ) \, dx}{3 a^4}\\ &=\frac {8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2}-\frac {16 \int \csc ^2(x) \, dx}{3 a^2}+\frac {7 \int \csc ^3(x) \, dx}{a^2}\\ &=-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2}+\frac {7 \int \csc (x) \, dx}{2 a^2}+\frac {16 \operatorname {Subst}(\int 1 \, dx,x,\cot (x))}{3 a^2}\\ &=-\frac {7 \tanh ^{-1}(\cos (x))}{2 a^2}+\frac {16 \cot (x)}{3 a^2}-\frac {7 \cot (x) \csc (x)}{2 a^2}+\frac {8 \cot (x) \csc (x)}{3 a^2 (1+\sin (x))}+\frac {\cot (x) \csc (x)}{3 (a+a \sin (x))^2}\\ \end {align*}
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Mathematica [B] time = 0.66, size = 203, normalized size = 3.17 \[ \frac {\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \left (-16 \sin \left (\frac {x}{2}\right )-160 \sin \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2+8 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )+3 \cos \left (\frac {x}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )^3-3 \sin \left (\frac {x}{2}\right ) \left (\cot \left (\frac {x}{2}\right )+1\right )^3-84 \log \left (\cos \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3+84 \log \left (\sin \left (\frac {x}{2}\right )\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3-24 \tan \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3+24 \cot \left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^3\right )}{24 a^2 (\sin (x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 220, normalized size = 3.44 \[ -\frac {64 \, \cos \relax (x)^{4} + 86 \, \cos \relax (x)^{3} - 54 \, \cos \relax (x)^{2} + 21 \, {\left (\cos \relax (x)^{4} - \cos \relax (x)^{3} - 3 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} + 2 \, \cos \relax (x)^{2} - \cos \relax (x) - 2\right )} \sin \relax (x) + \cos \relax (x) + 2\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 21 \, {\left (\cos \relax (x)^{4} - \cos \relax (x)^{3} - 3 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} + 2 \, \cos \relax (x)^{2} - \cos \relax (x) - 2\right )} \sin \relax (x) + \cos \relax (x) + 2\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (32 \, \cos \relax (x)^{3} - 11 \, \cos \relax (x)^{2} - 38 \, \cos \relax (x) + 2\right )} \sin \relax (x) - 80 \, \cos \relax (x) - 4}{12 \, {\left (a^{2} \cos \relax (x)^{4} - a^{2} \cos \relax (x)^{3} - 3 \, a^{2} \cos \relax (x)^{2} + a^{2} \cos \relax (x) + 2 \, a^{2} - {\left (a^{2} \cos \relax (x)^{3} + 2 \, a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 93, normalized size = 1.45 \[ \frac {7 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{2}} + \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{4}} - \frac {42 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, x\right ) + 1}{8 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {2 \, {\left (12 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 21 \, \tan \left (\frac {1}{2} \, x\right ) + 11\right )}}{3 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 92, normalized size = 1.44 \[ \frac {\tan ^{2}\left (\frac {x}{2}\right )}{8 a^{2}}-\frac {\tan \left (\frac {x}{2}\right )}{a^{2}}-\frac {1}{8 a^{2} \tan \left (\frac {x}{2}\right )^{2}}+\frac {1}{a^{2} \tan \left (\frac {x}{2}\right )}+\frac {7 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a^{2}}+\frac {4}{3 a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 155, normalized size = 2.42 \[ \frac {\frac {15 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {239 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {405 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {216 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - 3}{24 \, {\left (\frac {a^{2} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {a^{2} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}}\right )}} - \frac {\frac {8 \, \sin \relax (x)}{\cos \relax (x) + 1} - \frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}}{8 \, a^{2}} + \frac {7 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.40, size = 111, normalized size = 1.73 \[ \frac {36\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+\frac {135\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{2}+\frac {239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{6}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\frac {1}{2}}{4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+12\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+12\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{a^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a^2}+\frac {7\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2\,a^2} \]
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 \[ \frac {\int \frac {\csc ^{3}{\relax (x )}}{\sin ^{2}{\relax (x )} + 2 \sin {\relax (x )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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