Optimal. Leaf size=35 \[ -\frac {3 \tanh ^{-1}(\cos (x))}{8 a}-\frac {\cot (x) \csc ^3(x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3175, 3768, 3770} \[ -\frac {3 \tanh ^{-1}(\cos (x))}{8 a}-\frac {\cot (x) \csc ^3(x)}{4 a}-\frac {3 \cot (x) \csc (x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\csc ^3(x)}{a-a \cos ^2(x)} \, dx &=\frac {\int \csc ^5(x) \, dx}{a}\\ &=-\frac {\cot (x) \csc ^3(x)}{4 a}+\frac {3 \int \csc ^3(x) \, dx}{4 a}\\ &=-\frac {3 \cot (x) \csc (x)}{8 a}-\frac {\cot (x) \csc ^3(x)}{4 a}+\frac {3 \int \csc (x) \, dx}{8 a}\\ &=-\frac {3 \tanh ^{-1}(\cos (x))}{8 a}-\frac {3 \cot (x) \csc (x)}{8 a}-\frac {\cot (x) \csc ^3(x)}{4 a}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 75, normalized size = 2.14 \[ \frac {-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {3}{32} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {3}{32} \sec ^2\left (\frac {x}{2}\right )+\frac {3}{8} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {3}{8} \log \left (\cos \left (\frac {x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 72, normalized size = 2.06 \[ \frac {6 \, \cos \relax (x)^{3} - 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 3 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 10 \, \cos \relax (x)}{16 \, {\left (a \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.83, size = 47, normalized size = 1.34 \[ -\frac {3 \, \log \left (\cos \relax (x) + 1\right )}{16 \, a} + \frac {3 \, \log \left (-\cos \relax (x) + 1\right )}{16 \, a} + \frac {3 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)}{8 \, {\left (\cos \relax (x)^{2} - 1\right )}^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 66, normalized size = 1.89 \[ -\frac {1}{16 a \left (-1+\cos \relax (x )\right )^{2}}+\frac {3}{16 a \left (-1+\cos \relax (x )\right )}+\frac {3 \ln \left (-1+\cos \relax (x )\right )}{16 a}+\frac {1}{16 a \left (\cos \relax (x )+1\right )^{2}}+\frac {3}{16 a \left (\cos \relax (x )+1\right )}-\frac {3 \ln \left (\cos \relax (x )+1\right )}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 51, normalized size = 1.46 \[ \frac {3 \, \cos \relax (x)^{3} - 5 \, \cos \relax (x)}{8 \, {\left (a \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} + a\right )}} - \frac {3 \, \log \left (\cos \relax (x) + 1\right )}{16 \, a} + \frac {3 \, \log \left (\cos \relax (x) - 1\right )}{16 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 39, normalized size = 1.11 \[ -\frac {3\,\mathrm {atanh}\left (\cos \relax (x)\right )}{8\,a}-\frac {\frac {5\,\cos \relax (x)}{8}-\frac {3\,{\cos \relax (x)}^3}{8}}{a\,{\cos \relax (x)}^4-2\,a\,{\cos \relax (x)}^2+a} \]
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 \[ - \frac {\int \frac {\csc ^{3}{\relax (x )}}{\cos ^{2}{\relax (x )} - 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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