Optimal. Leaf size=64 \[ \frac{6 (b \sin (e+f x))^{3/2} \, _2F_1\left (-\frac{1}{6},\frac{7}{12};\frac{19}{12};\sin ^2(e+f x)\right )}{7 d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0934154, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2602, 2577} \[ \frac{6 (b \sin (e+f x))^{3/2} \, _2F_1\left (-\frac{1}{6},\frac{7}{12};\frac{19}{12};\sin ^2(e+f x)\right )}{7 d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \frac{(b \sin (e+f x))^{3/2}}{(d \tan (e+f x))^{4/3}} \, dx &=\frac{\left (b \sqrt [3]{b \sin (e+f x)}\right ) \int \cos ^{\frac{4}{3}}(e+f x) \sqrt [6]{b \sin (e+f x)} \, dx}{d \sqrt [3]{\cos (e+f x)} \sqrt [3]{d \tan (e+f x)}}\\ &=\frac{6 \, _2F_1\left (-\frac{1}{6},\frac{7}{12};\frac{19}{12};\sin ^2(e+f x)\right ) (b \sin (e+f x))^{3/2}}{7 d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.380261, size = 70, normalized size = 1.09 \[ \frac{2 (b \sin (e+f x))^{3/2} \left (2 \sec ^2(e+f x)^{3/4} \, _2F_1\left (\frac{7}{12},\frac{3}{4};\frac{19}{12};-\tan ^2(e+f x)\right )+7\right )}{21 d f \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \sin \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (d \tan \left (f x + e\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac{2}{3}} b \sin \left (f x + e\right )}{d^{2} \tan \left (f x + e\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \sin \left (f x + e\right )\right )^{\frac{3}{2}}}{\left (d \tan \left (f x + e\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]