Optimal. Leaf size=64 \[ \frac{2 \sqrt [12]{\cos ^2(e+f x)} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac{1}{12},\frac{3}{4};\frac{7}{4};\sin ^2(e+f x)\right )}{3 d f (b \sec (e+f x))^{4/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0541346, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2617} \[ \frac{2 \sqrt [12]{\cos ^2(e+f x)} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac{1}{12},\frac{3}{4};\frac{7}{4};\sin ^2(e+f x)\right )}{3 d f (b \sec (e+f x))^{4/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2617
Rubi steps
\begin{align*} \int \frac{\sqrt{d \tan (e+f x)}}{(b \sec (e+f x))^{4/3}} \, dx &=\frac{2 \sqrt [12]{\cos ^2(e+f x)} \, _2F_1\left (\frac{1}{12},\frac{3}{4};\frac{7}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f (b \sec (e+f x))^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.123525, size = 64, normalized size = 1. \[ -\frac{3 d \sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac{2}{3},\frac{1}{4};\frac{1}{3};\sec ^2(e+f x)\right )}{4 f (b \sec (e+f x))^{4/3} \sqrt{d \tan (e+f x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{d\tan \left ( fx+e \right ) } \left ( b\sec \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{d \tan \left (f x + e\right )}}{\left (b \sec \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{\left (b \sec \left (f x + e\right )\right )^{\frac{2}{3}} \sqrt{d \tan \left (f x + e\right )}}{b^{2} \sec \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{d \tan \left (f x + e\right )}}{\left (b \sec \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]