Optimal. Leaf size=39 \[ \frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}+\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}} \]
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Rubi [A] time = 0.0181165, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 3475} \[ \frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}+\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cot ^2(x)\right )^{3/2}} \, dx &=\frac{\cot (x) \int \tan ^3(x) \, dx}{a \sqrt{a \cot ^2(x)}}\\ &=\frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}-\frac{\cot (x) \int \tan (x) \, dx}{a \sqrt{a \cot ^2(x)}}\\ &=\frac{\cot (x) \log (\cos (x))}{a \sqrt{a \cot ^2(x)}}+\frac{\tan (x)}{2 a \sqrt{a \cot ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.027751, size = 30, normalized size = 0.77 \[ \frac{\csc (x) \sec (x)+2 \cot (x) \log (\cos (x))}{2 a \sqrt{a \cot ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 36, normalized size = 0.9 \begin{align*} -{\frac{\cot \left ( x \right ) \left ( \ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) \left ( \cot \left ( x \right ) \right ) ^{2}-2\,\ln \left ( \cot \left ( x \right ) \right ) \left ( \cot \left ( x \right ) \right ) ^{2}-1 \right ) }{2} \left ( a \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5459, size = 30, normalized size = 0.77 \begin{align*} \frac{\tan \left (x\right )^{2}}{2 \, a^{\frac{3}{2}}} - \frac{\log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68903, size = 198, normalized size = 5.08 \begin{align*} \frac{{\left ({\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \sin \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sqrt{-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \,{\left (a^{2} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\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*} \int \frac{1}{\left (a \cot ^{2}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.53959, size = 188, normalized size = 4.82 \begin{align*} -\frac{\frac{\log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}} + 2\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{\mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} - \frac{\log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}} - 2\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{\mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} + \frac{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) - 6 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + \frac{1}{\tan \left (\frac{1}{2} \, x\right )^{2}} - 2\right )} \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )} - \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{2 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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