Optimal. Leaf size=31 \[ \frac {8}{3} \csc (x) \sqrt {\sin (x) \tan (x)}-\frac {2}{3} \sin (x) \sqrt {\sin (x) \tan (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4400, 2598, 2589} \[ \frac {8}{3} \csc (x) \sqrt {\sin (x) \tan (x)}-\frac {2}{3} \sin (x) \sqrt {\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2589
Rule 2598
Rule 4400
Rubi steps
\begin {align*} \int (\sin (x) \tan (x))^{3/2} \, dx &=\frac {\sqrt {\sin (x) \tan (x)} \int \sin ^{\frac {3}{2}}(x) \tan ^{\frac {3}{2}}(x) \, dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=-\frac {2}{3} \sin (x) \sqrt {\sin (x) \tan (x)}+\frac {\left (4 \sqrt {\sin (x) \tan (x)}\right ) \int \frac {\tan ^{\frac {3}{2}}(x)}{\sqrt {\sin (x)}} \, dx}{3 \sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=\frac {8}{3} \csc (x) \sqrt {\sin (x) \tan (x)}-\frac {2}{3} \sin (x) \sqrt {\sin (x) \tan (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 23, normalized size = 0.74 \[ \frac {2}{3} \sin (x) \left (4 \csc ^2(x)-1\right ) \sqrt {\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 26, normalized size = 0.84 \[ \frac {2 \, {\left (\cos \relax (x)^{2} + 3\right )} \sqrt {-\frac {\cos \relax (x)^{2} - 1}{\cos \relax (x)}}}{3 \, \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sin \relax (x) \tan \relax (x)\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.26, size = 587, normalized size = 18.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.47, size = 57, normalized size = 1.84 \[ -\frac {8 \, {\left (\frac {\sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - 1\right )}}{3 \, {\left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (\sin \relax (x)\,\mathrm {tan}\relax (x)\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sin {\relax (x )} \tan {\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________