Optimal. Leaf size=70 \[ \frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}-a x \tan ^2(x) \sqrt {a \cot ^4(x)}-a \tan (x) \sqrt {a \cot ^4(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3658, 3473, 8} \[ -\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}+\frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-a x \tan ^2(x) \sqrt {a \cot ^4(x)}-a \tan (x) \sqrt {a \cot ^4(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \left (a \cot ^4(x)\right )^{3/2} \, dx &=\left (a \sqrt {a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^6(x) \, dx\\ &=-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}-\left (a \sqrt {a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^4(x) \, dx\\ &=\frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}+\left (a \sqrt {a \cot ^4(x)} \tan ^2(x)\right ) \int \cot ^2(x) \, dx\\ &=\frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}-a \sqrt {a \cot ^4(x)} \tan (x)-\left (a \sqrt {a \cot ^4(x)} \tan ^2(x)\right ) \int 1 \, dx\\ &=\frac {1}{3} a \cot (x) \sqrt {a \cot ^4(x)}-\frac {1}{5} a \cot ^3(x) \sqrt {a \cot ^4(x)}-a \sqrt {a \cot ^4(x)} \tan (x)-a x \sqrt {a \cot ^4(x)} \tan ^2(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 39, normalized size = 0.56 \[ -\frac {1}{15} \tan ^6(x) \left (a \cot ^4(x)\right )^{3/2} \left (15 x+\cot (x) \left (3 \csc ^4(x)-11 \csc ^2(x)+23\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.39, size = 110, normalized size = 1.57 \[ \frac {{\left (23 \, a \cos \left (2 \, x\right )^{3} - a \cos \left (2 \, x\right )^{2} - 11 \, a \cos \left (2 \, x\right ) + 15 \, {\left (a x \cos \left (2 \, x\right )^{2} - 2 \, a x \cos \left (2 \, x\right ) + a x\right )} \sin \left (2 \, x\right ) + 13 \, a\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{15 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 2.41, size = 57, normalized size = 0.81 \[ \frac {1}{480} \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 35 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 480 \, x - \frac {330 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 35 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 3}{\tan \left (\frac {1}{2} \, x\right )^{5}} + 330 \, \tan \left (\frac {1}{2} \, x\right )\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 40, normalized size = 0.57 \[ \frac {\left (a \left (\cot ^{4}\relax (x )\right )\right )^{\frac {3}{2}} \left (-3 \left (\cot ^{5}\relax (x )\right )+5 \left (\cot ^{3}\relax (x )\right )+\frac {15 \pi }{2}-15 \,\mathrm {arccot}\left (\cot \relax (x )\right )-15 \cot \relax (x )\right )}{15 \cot \relax (x )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 37, normalized size = 0.53 \[ -a^{\frac {3}{2}} x - \frac {15 \, a^{\frac {3}{2}} \tan \relax (x)^{4} - 5 \, a^{\frac {3}{2}} \tan \relax (x)^{2} + 3 \, a^{\frac {3}{2}}}{15 \, \tan \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {cot}\relax (x)}^4\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cot ^{4}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________