Optimal. Leaf size=36 \[ -\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0182863, antiderivative size = 36, 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} \[ -\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3658
Rule 3473
Rule 3475
Rubi steps
\begin{align*} \int \left (a \cot ^2(x)\right )^{3/2} \, dx &=\left (a \sqrt{a \cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-\left (a \sqrt{a \cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=-\frac{1}{2} a \cot (x) \sqrt{a \cot ^2(x)}-a \sqrt{a \cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0193772, size = 27, normalized size = 0.75 \[ -\frac{1}{2} a \tan (x) \sqrt{a \cot ^2(x)} \left (\csc ^2(x)+2 \log (\sin (x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 29, normalized size = 0.8 \begin{align*}{\frac{- \left ( \cot \left ( x \right ) \right ) ^{2}+\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) }{2\, \left ( \cot \left ( x \right ) \right ) ^{3}} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.54505, size = 41, normalized size = 1.14 \begin{align*} \frac{1}{2} \, a^{\frac{3}{2}} \log \left (\tan \left (x\right )^{2} + 1\right ) - a^{\frac{3}{2}} \log \left (\tan \left (x\right )\right ) - \frac{a^{\frac{3}{2}}}{2 \, \tan \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62114, size = 139, normalized size = 3.86 \begin{align*} \frac{{\left ({\left (a \cos \left (2 \, x\right ) - a\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) - 2 \, a\right )} \sqrt{-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, \sin \left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cot ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20721, size = 42, normalized size = 1.17 \begin{align*} \frac{1}{2} \, a^{\frac{3}{2}}{\left (\frac{1}{\cos \left (x\right )^{2} - 1} - \log \left (-\cos \left (x\right )^{2} + 1\right )\right )} \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]