Optimal. Leaf size=32 \[ -\frac {2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2589} \[ -\frac {2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2589
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{3/2}}{\sqrt {b \tan (e+f x)}} \, dx &=-\frac {2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 32, normalized size = 1.00 \[ -\frac {2 b (a \sin (e+f x))^{3/2}}{3 f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.62, size = 53, normalized size = 1.66 \[ -\frac {2 \, \sqrt {a \sin \left (f x + e\right )} a \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2}}{3 \, b f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 48, normalized size = 1.50 \[ -\frac {2 \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \cos \left (f x +e \right )}{3 f \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.63, size = 69, normalized size = 2.16 \[ -\frac {a\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (\sin \left (e+f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\right )\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{6\,b\,f\,{\sin \left (e+f\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________