Optimal. Leaf size=90 \[ \frac {2 b \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}{a^2 f}-\frac {2 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{a^2 f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2593, 2601, 2639} \[ \frac {2 b \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}{a^2 f}-\frac {2 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{a^2 f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2593
Rule 2601
Rule 2639
Rubi steps
\begin {align*} \int \frac {(b \tan (e+f x))^{3/2}}{(a \sin (e+f x))^{3/2}} \, dx &=\frac {2 b \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}{a^2 f}-\frac {b^2 \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx}{a^2}\\ &=\frac {2 b \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}{a^2 f}-\frac {\left (b^2 \sqrt {a \sin (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}\\ &=-\frac {2 b^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{a^2 f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}+\frac {2 b \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}{a^2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.27, size = 92, normalized size = 1.02 \[ \frac {(b \tan (e+f x))^{3/2} \left (2 \cos (e+f x) \cos ^2(e+f x)^{3/4}-\cos ^3(e+f x) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right )\right )}{a f \cos ^2(e+f x)^{3/4} \sqrt {a \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} b \tan \left (f x + e\right )}{a^{2} \cos \left (f x + e\right )^{2} - a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.47, size = 316, normalized size = 3.51 \[ -\frac {2 \left (i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-i \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}{f \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sin \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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]
________________________________________________________________________________________