Optimal. Leaf size=37 \[ a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0905474, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3657, 4124, 50, 63, 207} \[ a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4124
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \cot (x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx &=\int \cot (x) \left (a \sec ^2(x)\right )^{3/2} \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{\sqrt{a x}}{-1+x} \, dx,x,\sec ^2(x)\right )\\ &=a \sqrt{a \sec ^2(x)}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a x}} \, dx,x,\sec ^2(x)\right )\\ &=a \sqrt{a \sec ^2(x)}+a \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a \sec ^2(x)}\right )\\ &=-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right )+a \sqrt{a \sec ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0380756, size = 34, normalized size = 0.92 \[ a \sqrt{a \sec ^2(x)} \left (\cos (x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 32, normalized size = 0.9 \begin{align*} \left ( \cos \left ( x \right ) \ln \left ( -{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) +\cos \left ( x \right ) +1 \right ) \left ( \cos \left ( x \right ) \right ) ^{2} \left ({\frac{a}{ \left ( \cos \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 [B] time = 1.90549, size = 181, normalized size = 4.89 \begin{align*} \frac{{\left (4 \, a \cos \left (2 \, x\right ) \cos \left (x\right ) + 4 \, a \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, a \cos \left (x\right ) -{\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) +{\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )\right )} \sqrt{a}}{2 \,{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43828, size = 140, normalized size = 3.78 \begin{align*} \frac{1}{2} \, a^{\frac{3}{2}} \log \left (\frac{a \tan \left (x\right )^{2} - 2 \, \sqrt{a \tan \left (x\right )^{2} + a} \sqrt{a} + 2 \, a}{\tan \left (x\right )^{2}}\right ) + \sqrt{a \tan \left (x\right )^{2} + a} a \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 \left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac{3}{2}} \cot{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08167, size = 50, normalized size = 1.35 \begin{align*}{\left (\frac{a \arctan \left (\frac{\sqrt{a \tan \left (x\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{a \tan \left (x\right )^{2} + a}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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