Optimal. Leaf size=30 \[ -\frac {\cot ^3(x)}{3 a}-\frac {\csc (x)}{a}+\frac {\csc ^3(x)}{3 a} \]
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Rubi [A]
time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2785, 2687, 30,
2686} \begin {gather*} -\frac {\cot ^3(x)}{3 a}+\frac {\csc ^3(x)}{3 a}-\frac {\csc (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 2687
Rule 2785
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{a+a \cos (x)} \, dx &=-\frac {\int \cot ^3(x) \csc (x) \, dx}{a}+\frac {\int \cot ^2(x) \csc ^2(x) \, dx}{a}\\ &=\frac {\text {Subst}\left (\int x^2 \, dx,x,-\cot (x)\right )}{a}+\frac {\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (x)\right )}{a}\\ &=-\frac {\cot ^3(x)}{3 a}-\frac {\csc (x)}{a}+\frac {\csc ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.83 \begin {gather*} \frac {(-3-4 \cos (x)+\cos (2 x)) \csc (x)}{6 a (1+\cos (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 29, normalized size = 0.97
method | result | size |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}-2 \tan \left (\frac {x}{2}\right )-\frac {1}{\tan \left (\frac {x}{2}\right )}}{4 a}\) | \(29\) |
risch | \(-\frac {2 i \left (3 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{2 i x}+{\mathrm e}^{i x}-1\right )}{3 \left ({\mathrm e}^{i x}+1\right )^{3} a \left ({\mathrm e}^{i x}-1\right )}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 42, normalized size = 1.40 \begin {gather*} -\frac {\frac {6 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{12 \, a} - \frac {\cos \left (x\right ) + 1}{4 \, a \sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 24, normalized size = 0.80 \begin {gather*} \frac {\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 2}{3 \, {\left (a \cos \left (x\right ) + a\right )} \sin \left (x\right )} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\cot ^{2}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 37, normalized size = 1.23 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 35, normalized size = 1.17 \begin {gather*} \frac {4\,{\cos \left (\frac {x}{2}\right )}^4-8\,{\cos \left (\frac {x}{2}\right )}^2+1}{12\,a\,{\cos \left (\frac {x}{2}\right )}^3\,\sin \left (\frac {x}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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