Optimal. Leaf size=49 \[ \frac {a}{8 (a \cos (x)+a)^2}-\frac {1}{8 (a-a \cos (x))}+\frac {1}{4 (a \cos (x)+a)}-\frac {3 \tanh ^{-1}(\cos (x))}{8 a} \]
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Rubi [A] time = 0.07, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2667, 44, 206} \[ \frac {a}{8 (a \cos (x)+a)^2}-\frac {1}{8 (a-a \cos (x))}+\frac {1}{4 (a \cos (x)+a)}-\frac {3 \tanh ^{-1}(\cos (x))}{8 a} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\csc ^3(x)}{a+a \cos (x)} \, dx &=-\left (a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \cos (x)\right )\right )\\ &=-\left (a^3 \operatorname {Subst}\left (\int \left (\frac {1}{8 a^3 (a-x)^2}+\frac {1}{4 a^2 (a+x)^3}+\frac {1}{4 a^3 (a+x)^2}+\frac {3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \cos (x)\right )\right )\\ &=-\frac {1}{8 (a-a \cos (x))}+\frac {a}{8 (a+a \cos (x))^2}+\frac {1}{4 (a+a \cos (x))}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cos (x)\right )\\ &=-\frac {3 \tanh ^{-1}(\cos (x))}{8 a}-\frac {1}{8 (a-a \cos (x))}+\frac {a}{8 (a+a \cos (x))^2}+\frac {1}{4 (a+a \cos (x))}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 60, normalized size = 1.22 \[ \frac {-2 \cot ^2\left (\frac {x}{2}\right )+\sec ^2\left (\frac {x}{2}\right )-12 \cos ^2\left (\frac {x}{2}\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+4}{16 a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 83, normalized size = 1.69 \[ \frac {6 \, \cos \relax (x)^{2} - 3 \, {\left (\cos \relax (x)^{3} + \cos \relax (x)^{2} - \cos \relax (x) - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 3 \, {\left (\cos \relax (x)^{3} + \cos \relax (x)^{2} - \cos \relax (x) - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 6 \, \cos \relax (x) - 4}{16 \, {\left (a \cos \relax (x)^{3} + a \cos \relax (x)^{2} - a \cos \relax (x) - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 52, normalized size = 1.06 \[ -\frac {3 \, \log \left (\cos \relax (x) + 1\right )}{16 \, a} + \frac {3 \, \log \left (-\cos \relax (x) + 1\right )}{16 \, a} + \frac {3 \, \cos \relax (x)^{2} + 3 \, \cos \relax (x) - 2}{8 \, a {\left (\cos \relax (x) + 1\right )}^{2} {\left (\cos \relax (x) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 55, normalized size = 1.12 \[ \frac {1}{8 a \left (-1+\cos \relax (x )\right )}+\frac {3 \ln \left (-1+\cos \relax (x )\right )}{16 a}+\frac {1}{8 a \left (\cos \relax (x )+1\right )^{2}}+\frac {1}{4 a \left (\cos \relax (x )+1\right )}-\frac {3 \ln \left (\cos \relax (x )+1\right )}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 58, normalized size = 1.18 \[ \frac {3 \, \cos \relax (x)^{2} + 3 \, \cos \relax (x) - 2}{8 \, {\left (a \cos \relax (x)^{3} + a \cos \relax (x)^{2} - a \cos \relax (x) - a\right )}} - \frac {3 \, \log \left (\cos \relax (x) + 1\right )}{16 \, a} + \frac {3 \, \log \left (\cos \relax (x) - 1\right )}{16 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 45, normalized size = 0.92 \[ -\frac {\frac {3\,{\cos \relax (x)}^2}{8}+\frac {3\,\cos \relax (x)}{8}-\frac {1}{4}}{-a\,{\cos \relax (x)}^3-a\,{\cos \relax (x)}^2+a\,\cos \relax (x)+a}-\frac {3\,\mathrm {atanh}\left (\cos \relax (x)\right )}{8\,a} \]
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 \[ \frac {\int \frac {\csc ^{3}{\relax (x )}}{\cos {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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