Optimal. Leaf size=79 \[ \frac {2 \sqrt [4]{\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{4},\frac {1}{4} (1+2 m);\frac {1}{4} (5+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m \sqrt {b \tan (e+f x)}}{b f (1+2 m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2682, 2657}
\begin {gather*} \frac {2 \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)} (a \sin (e+f x))^m \, _2F_1\left (\frac {1}{4},\frac {1}{4} (2 m+1);\frac {1}{4} (2 m+5);\sin ^2(e+f x)\right )}{b f (2 m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^m}{\sqrt {b \tan (e+f x)}} \, dx &=\frac {\left (a \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \sqrt {\cos (e+f x)} (a \sin (e+f x))^{-\frac {1}{2}+m} \, dx}{b \sqrt {a \sin (e+f x)}}\\ &=\frac {2 \sqrt [4]{\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{4},\frac {1}{4} (1+2 m);\frac {1}{4} (5+2 m);\sin ^2(e+f x)\right ) (a \sin (e+f x))^m \sqrt {b \tan (e+f x)}}{b f (1+2 m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.00, size = 87, normalized size = 1.10 \begin {gather*} \frac {2 \, _2F_1\left (\frac {2+m}{2},\frac {1}{4} (1+2 m);\frac {1}{4} (5+2 m);-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{m/2} (a \sin (e+f x))^m \sqrt {b \tan (e+f x)}}{b f (1+2 m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\left (a \sin \left (f x +e \right )\right )^{m}}{\sqrt {b \tan \left (f x +e \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \sin {\left (e + f x \right )}\right )^{m}}{\sqrt {b \tan {\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________