Optimal. Leaf size=32 \[ -x \tan ^2(x) \sqrt {a \cot ^4(x)}-\tan (x) \sqrt {a \cot ^4(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 32, 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, 8} \[ -x \tan ^2(x) \sqrt {a \cot ^4(x)}-\tan (x) \sqrt {a \cot ^4(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {a \cot ^4(x)} \, dx &=\left (\sqrt {a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^2(x) \, dx\\ &=-\sqrt {a \cot ^4(x)} \tan (x)-\left (\sqrt {a \cot ^4(x)} \tan ^2(x)\right ) \int 1 \, dx\\ &=-\sqrt {a \cot ^4(x)} \tan (x)-x \sqrt {a \cot ^4(x)} \tan ^2(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 20, normalized size = 0.62 \[ \tan ^2(x) (x+\cot (x)) \left (-\sqrt {a \cot ^4(x)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.41, size = 61, normalized size = 1.91 \[ \frac {{\left (x \cos \left (2 \, x\right ) - x - \sin \left (2 \, x\right )\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{\cos \left (2 \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 21, normalized size = 0.66 \[ -\frac {1}{2} \, \sqrt {a} {\left (2 \, x + \frac {1}{\tan \left (\frac {1}{2} \, x\right )} - \tan \left (\frac {1}{2} \, x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.20, size = 27, normalized size = 0.84 \[ \frac {\sqrt {a \left (\cot ^{4}\relax (x )\right )}\, \left (-\cot \relax (x )+\frac {\pi }{2}-\mathrm {arccot}\left (\cot \relax (x )\right )\right )}{\cot \relax (x )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 16, normalized size = 0.50 \[ -\sqrt {a} x - \frac {\sqrt {a}}{\tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {a\,{\mathrm {cot}\relax (x)}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \cot ^{4}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________