Optimal. Leaf size=64 \[ \frac {6 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2}}{17 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2682, 2657}
\begin {gather*} \frac {6 \cos ^2(e+f x)^{3/4} (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac {3}{4},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right )}{17 d f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2657
Rule 2682
Rubi steps
\begin {align*} \int (b \sin (e+f x))^{4/3} \sqrt {d \tan (e+f x)} \, dx &=\frac {\left (b \cos ^{\frac {3}{2}}(e+f x) (d \tan (e+f x))^{3/2}\right ) \int \frac {(b \sin (e+f x))^{11/6}}{\sqrt {\cos (e+f x)}} \, dx}{d (b \sin (e+f x))^{3/2}}\\ &=\frac {6 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3} (d \tan (e+f x))^{3/2}}{17 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 10.44, size = 69, normalized size = 1.08 \begin {gather*} \frac {3 \, _2F_1\left (\frac {3}{4},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{4/3} \sin (2 (e+f x)) \sqrt {d \tan (e+f x)}}{17 f \sqrt [4]{\cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (b \sin \left (f x +e \right )\right )^{\frac {4}{3}} \sqrt {d \tan \left (f x +e \right )}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (b\,\sin \left (e+f\,x\right )\right )}^{4/3}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________