Optimal. Leaf size=24 \[ \frac {\csc (x)}{3 (a \cos (x)+a)}-\frac {2 \cot (x)}{3 a} \]
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Rubi [A] time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ \frac {\csc (x)}{3 (a \cos (x)+a)}-\frac {2 \cot (x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2672
Rule 3767
Rubi steps
\begin {align*} \int \frac {\csc ^2(x)}{a+a \cos (x)} \, dx &=\frac {\csc (x)}{3 (a+a \cos (x))}+\frac {2 \int \csc ^2(x) \, dx}{3 a}\\ &=\frac {\csc (x)}{3 (a+a \cos (x))}-\frac {2 \operatorname {Subst}(\int 1 \, dx,x,\cot (x))}{3 a}\\ &=-\frac {2 \cot (x)}{3 a}+\frac {\csc (x)}{3 (a+a \cos (x))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 1.25 \[ -\frac {(2 \cos (x)+\cos (2 x)) \csc \left (\frac {x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )}{12 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 26, normalized size = 1.08 \[ -\frac {2 \, \cos \relax (x)^{2} + 2 \, \cos \relax (x) - 1}{3 \, {\left (a \cos \relax (x) + a\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 37, normalized size = 1.54 \[ \frac {a^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )}{12 \, a^{3}} - \frac {1}{4 \, a \tan \left (\frac {1}{2} \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.21 \[ \frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}+2 \tan \left (\frac {x}{2}\right )-\frac {1}{\tan \left (\frac {x}{2}\right )}}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 41, normalized size = 1.71 \[ \frac {\frac {6 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {\sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{12 \, a} - \frac {\cos \relax (x) + 1}{4 \, a \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 35, normalized size = 1.46 \[ \frac {-8\,{\cos \left (\frac {x}{2}\right )}^4+4\,{\cos \left (\frac {x}{2}\right )}^2+1}{12\,a\,{\cos \left (\frac {x}{2}\right )}^3\,\sin \left (\frac {x}{2}\right )} \]
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 \[ \frac {\int \frac {\csc ^{2}{\relax (x )}}{\cos {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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