Optimal. Leaf size=50 \[ \frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{16}{15} \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{64}{15} \tan (x) \sqrt{\cos (x) \cot (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0944576, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4400, 2598, 2594, 2589} \[ \frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{16}{15} \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{64}{15} \tan (x) \sqrt{\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4400
Rule 2598
Rule 2594
Rule 2589
Rubi steps
\begin{align*} \int (\cos (x) \cot (x))^{5/2} \, dx &=\frac{\sqrt{\cos (x) \cot (x)} \int \cos ^{\frac{5}{2}}(x) \cot ^{\frac{5}{2}}(x) \, dx}{\sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}+\frac{\left (8 \sqrt{\cos (x) \cot (x)}\right ) \int \sqrt{\cos (x)} \cot ^{\frac{5}{2}}(x) \, dx}{5 \sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=-\frac{16}{15} \cot (x) \sqrt{\cos (x) \cot (x)}+\frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{\left (32 \sqrt{\cos (x) \cot (x)}\right ) \int \sqrt{\cos (x)} \sqrt{\cot (x)} \, dx}{15 \sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=-\frac{16}{15} \cot (x) \sqrt{\cos (x) \cot (x)}+\frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{64}{15} \sqrt{\cos (x) \cot (x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0997359, size = 29, normalized size = 0.58 \[ -\frac{2}{15} \tan (x) \sqrt{\cos (x) \cot (x)} \left (3 \cos ^2(x)+5 \cot ^2(x)+32\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.155, size = 34, normalized size = 0.7 \begin{align*}{\frac{ \left ( 6\, \left ( \cos \left ( x \right ) \right ) ^{4}+48\, \left ( \cos \left ( x \right ) \right ) ^{2}-64 \right ) \sin \left ( x \right ) }{15\, \left ( \cos \left ( x \right ) \right ) ^{5}} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.90874, size = 576, normalized size = 11.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.54345, size = 103, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (3 \, \cos \left (x\right )^{4} + 24 \, \cos \left (x\right )^{2} - 32\right )} \sqrt{\frac{\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{15 \, \cos \left (x\right ) \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12164, size = 36, normalized size = 0.72 \begin{align*} \frac{2}{15} \,{\left (3 \, \sin \left (x\right )^{\frac{5}{2}} - 30 \, \sqrt{\sin \left (x\right )} - \frac{5}{\sin \left (x\right )^{\frac{3}{2}}}\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]