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 = 0.0551938, antiderivative size = 35, normalized size of antiderivative = 1., 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*}
Mathematica [B] time = 0.0062478, 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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Maple [B] time = 0.037, size = 66, normalized size = 1.9 \begin{align*}{\frac{1}{16\,a \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}+{\frac{3}{16\,a \left ( 1+\cos \left ( x \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\cos \left ( x \right ) \right ) }{16\,a}}-{\frac{1}{16\,a \left ( \cos \left ( x \right ) -1 \right ) ^{2}}}+{\frac{3}{16\,a \left ( \cos \left ( x \right ) -1 \right ) }}+{\frac{3\,\ln \left ( \cos \left ( x \right ) -1 \right ) }{16\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95155, size = 69, normalized size = 1.97 \begin{align*} \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )}} - \frac{3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{3 \, \log \left (\cos \left (x\right ) - 1\right )}{16 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97039, size = 232, normalized size = 6.63 \begin{align*} \frac{6 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \,{\left (a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\csc ^{3}{\left (x \right )}}{\cos ^{2}{\left (x \right )} - 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12851, size = 63, normalized size = 1.8 \begin{align*} -\frac{3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{3 \, \log \left (-\cos \left (x\right ) + 1\right )}{16 \, a} + \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{2} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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