Optimal. Leaf size=176 \[ -\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-2 \tan (x) \sqrt{a \cot ^3(x)} \]
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Rubi [A] time = 0.0872821, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-2 \tan (x) \sqrt{a \cot ^3(x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \sqrt{a \cot ^3(x)} \, dx &=\frac{\sqrt{a \cot ^3(x)} \int \cot ^{\frac{3}{2}}(x) \, dx}{\cot ^{\frac{3}{2}}(x)}\\ &=-2 \sqrt{a \cot ^3(x)} \tan (x)-\frac{\sqrt{a \cot ^3(x)} \int \frac{1}{\sqrt{\cot (x)}} \, dx}{\cot ^{\frac{3}{2}}(x)}\\ &=-2 \sqrt{a \cot ^3(x)} \tan (x)+\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )}{\cot ^{\frac{3}{2}}(x)}\\ &=-2 \sqrt{a \cot ^3(x)} \tan (x)+\frac{\left (2 \sqrt{a \cot ^3(x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\cot (x)}\right )}{\cot ^{\frac{3}{2}}(x)}\\ &=-2 \sqrt{a \cot ^3(x)} \tan (x)+\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (x)}\right )}{\cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (x)}\right )}{\cot ^{\frac{3}{2}}(x)}\\ &=-2 \sqrt{a \cot ^3(x)} \tan (x)+\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (x)}\right )}{2 \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (x)}\right )}{2 \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (x)}\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (x)}\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}\\ &=-\frac{\sqrt{a \cot ^3(x)} \log \left (1-\sqrt{2} \sqrt{\cot (x)}+\cot (x)\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left (1+\sqrt{2} \sqrt{\cot (x)}+\cot (x)\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-2 \sqrt{a \cot ^3(x)} \tan (x)+\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (x)}\right )}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right ) \sqrt{a \cot ^3(x)}}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (x)}\right ) \sqrt{a \cot ^3(x)}}{\sqrt{2} \cot ^{\frac{3}{2}}(x)}-\frac{\sqrt{a \cot ^3(x)} \log \left (1-\sqrt{2} \sqrt{\cot (x)}+\cot (x)\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}+\frac{\sqrt{a \cot ^3(x)} \log \left (1+\sqrt{2} \sqrt{\cot (x)}+\cot (x)\right )}{2 \sqrt{2} \cot ^{\frac{3}{2}}(x)}-2 \sqrt{a \cot ^3(x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.104914, size = 122, normalized size = 0.69 \[ -\frac{\sqrt{a \cot ^3(x)} \left (8 \sqrt{\cot (x)}+\sqrt{2} \log \left (\cot (x)-\sqrt{2} \sqrt{\cot (x)}+1\right )-\sqrt{2} \log \left (\cot (x)+\sqrt{2} \sqrt{\cot (x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (x)}+1\right )\right )}{4 \cot ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 165, normalized size = 0.9 \begin{align*}{\frac{1}{4\,\cot \left ( x \right ) }\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{3}} \left ( \sqrt [4]{{a}^{2}}\sqrt{2}\ln \left ( -{ \left ( a\cot \left ( x \right ) +\sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}+\sqrt{{a}^{2}} \right ) \left ( \sqrt [4]{{a}^{2}}\sqrt{a\cot \left ( x \right ) }\sqrt{2}-a\cot \left ( x \right ) -\sqrt{{a}^{2}} \right ) ^{-1}} \right ) +2\,\sqrt [4]{{a}^{2}}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }+\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) +2\,\sqrt [4]{{a}^{2}}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{a\cot \left ( x \right ) }-\sqrt [4]{{a}^{2}}}{\sqrt [4]{{a}^{2}}}} \right ) -8\,\sqrt{a\cot \left ( x \right ) } \right ){\frac{1}{\sqrt{a\cot \left ( x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62105, size = 127, normalized size = 0.72 \begin{align*} -\frac{1}{4} \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) - \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} \sqrt{a} - \frac{2 \, \sqrt{a}}{\sqrt{\tan \left (x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{a \cot ^{3}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cot \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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