Optimal. Leaf size=30 \[ \frac {2 b \sqrt {b \tan (e+f x)}}{f \sqrt {a \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 30, 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} \[ \frac {2 b \sqrt {b \tan (e+f x)}}{f \sqrt {a \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2589
Rubi steps
\begin {align*} \int \frac {(b \tan (e+f x))^{3/2}}{\sqrt {a \sin (e+f x)}} \, dx &=\frac {2 b \sqrt {b \tan (e+f x)}}{f \sqrt {a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 30, normalized size = 1.00 \[ \frac {2 b \sqrt {b \tan (e+f x)}}{f \sqrt {a \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 45, normalized size = 1.50 \[ \frac {2 \, \sqrt {a \sin \left (f x + e\right )} b \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{a f \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {a \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.53, size = 308, normalized size = 10.27 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right ) \ln \left (-\frac {2 \left (2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )^{2}}\right )-\cos \left (f x +e \right ) \ln \left (-\frac {2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+2 \cos \left (f x +e \right )-1}{\sin \left (f x +e \right )^{2}}\right )+4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\right ) \cos \left (f x +e \right ) \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{2 f \sqrt {a \sin \left (f x +e \right )}\, \sin \left (f x +e \right )^{3} \sqrt {-\frac {\cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\sqrt {a \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.07, size = 39, normalized size = 1.30 \[ \frac {2\,b\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{2\,{\cos \left (e+f\,x\right )}^2}}}{f\,\sqrt {a\,\sin \left (e+f\,x\right )}} \]
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]
________________________________________________________________________________________