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 = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 3473
Rule 3475
Rule 3658
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 = 0.02, 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]
________________________________________________________________________________________
fricas [A] time = 1.32, size = 52, normalized size = 1.44 \[ \frac {{\left ({\left (a \cos \left (2 \, x\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) - 2 \, a\right )} \sqrt {-\frac {a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}}}{2 \, \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.60, size = 31, normalized size = 0.86 \[ \frac {1}{2} \, a^{\frac {3}{2}} {\left (\frac {1}{\cos \relax (x)^{2} - 1} - \log \left (-\cos \relax (x)^{2} + 1\right )\right )} \mathrm {sgn}\left (\cos \relax (x)\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.15, size = 29, normalized size = 0.81 \[ \frac {\left (a \left (\cot ^{2}\relax (x )\right )\right )^{\frac {3}{2}} \left (-\left (\cot ^{2}\relax (x )\right )+\ln \left (\cot ^{2}\relax (x )+1\right )\right )}{2 \cot \relax (x )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 30, normalized size = 0.83 \[ \frac {1}{2} \, a^{\frac {3}{2}} \log \left (\tan \relax (x)^{2} + 1\right ) - a^{\frac {3}{2}} \log \left (\tan \relax (x)\right ) - \frac {a^{\frac {3}{2}}}{2 \, \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a\,{\mathrm {cot}\relax (x)}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cot ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________