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 = 0.23, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 8
Rule 2748
Rule 2766
Rule 2978
Rule 3767
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*}
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Mathematica [B] time = 0.15, 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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fricas [B] time = 0.51, size = 225, normalized size = 3.46 \[ \frac {48 \, \cos \relax (x)^{4} + 114 \, \cos \relax (x)^{3} - 60 \, \cos \relax (x)^{2} + 15 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) + 2 \, \cos \relax (x) + 4\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 15 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) + 2 \, \cos \relax (x) + 4\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (24 \, \cos \relax (x)^{3} - 33 \, \cos \relax (x)^{2} - 63 \, \cos \relax (x) - 1\right )} \sin \relax (x) - 124 \, \cos \relax (x) + 2}{10 \, {\left (a^{3} \cos \relax (x)^{4} - 2 \, a^{3} \cos \relax (x)^{3} - 5 \, a^{3} \cos \relax (x)^{2} + 2 \, a^{3} \cos \relax (x) + 4 \, a^{3} - {\left (a^{3} \cos \relax (x)^{3} + 3 \, a^{3} \cos \relax (x)^{2} - 2 \, a^{3} \cos \relax (x) - 4 \, a^{3}\right )} \sin \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 85, normalized size = 1.31 \[ -\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a^{3}} + \frac {6 \, \tan \left (\frac {1}{2} \, x\right ) - 1}{2 \, a^{3} \tan \left (\frac {1}{2} \, x\right )} - \frac {4 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 50 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 70 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 45 \, \tan \left (\frac {1}{2} \, x\right ) + 12\right )}}{5 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 97, normalized size = 1.49 \[ \frac {\tan \left (\frac {x}{2}\right )}{2 a^{3}}-\frac {1}{2 a^{3} \tan \left (\frac {x}{2}\right )}-\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {8}{5 a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {8}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {8}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {12}{a^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.75, size = 180, normalized size = 2.77 \[ -\frac {\frac {121 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {410 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {610 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {425 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {125 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + 5}{10 \, {\left (\frac {a^{3} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {5 \, a^{3} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {10 \, a^{3} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {5 \, a^{3} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {a^{3} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}}\right )}} - \frac {3 \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{3}} + \frac {\sin \relax (x)}{2 \, a^{3} {\left (\cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.72, size = 129, normalized size = 1.98 \[ \frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^3}-\frac {25\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+85\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+122\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+82\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {121\,\mathrm {tan}\left (\frac {x}{2}\right )}{5}+1}{2\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+10\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+20\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+20\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+10\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3} \]
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 \[ \frac {\int \frac {\csc ^{2}{\relax (x )}}{\sin ^{3}{\relax (x )} + 3 \sin ^{2}{\relax (x )} + 3 \sin {\relax (x )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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