Optimal. Leaf size=62 \[ \frac{6 \sqrt{b \sin (e+f x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{12};\frac{13}{12};\sin ^2(e+f x)\right )}{d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0959553, antiderivative size = 62, 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} \[ \frac{6 \sqrt{b \sin (e+f x)} \, _2F_1\left (-\frac{1}{6},\frac{1}{12};\frac{13}{12};\sin ^2(e+f x)\right )}{d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sin (e+f x)}}{(d \tan (e+f x))^{4/3}} \, dx &=\frac{\left (b \sqrt [3]{b \sin (e+f x)}\right ) \int \frac{\cos ^{\frac{4}{3}}(e+f x)}{(b \sin (e+f x))^{5/6}} \, dx}{d \sqrt [3]{\cos (e+f x)} \sqrt [3]{d \tan (e+f x)}}\\ &=\frac{6 \, _2F_1\left (-\frac{1}{6},\frac{1}{12};\frac{13}{12};\sin ^2(e+f x)\right ) \sqrt{b \sin (e+f x)}}{d f \sqrt [6]{\cos ^2(e+f x)} \sqrt [3]{d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.389723, size = 64, normalized size = 1.03 \[ \frac{6 \sqrt [4]{\sec ^2(e+f x)} \sqrt{b \sin (e+f x)} \, _2F_1\left (\frac{1}{12},\frac{5}{4};\frac{13}{12};-\tan ^2(e+f x)\right )}{d f \sqrt [3]{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.183, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b\sin \left ( fx+e \right ) } \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{\sqrt{b \sin \left (f x + e\right )}}{\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}}}{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{\sqrt{b \sin \left (f x + e\right )}}{\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]