Optimal. Leaf size=57 \[ \frac{\cos ^2(e+f x)^{11/6} \tan ^3(e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left (\frac{3}{2},\frac{11}{6};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0421866, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2617} \[ \frac{\cos ^2(e+f x)^{11/6} \tan ^3(e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left (\frac{3}{2},\frac{11}{6};\frac{5}{2};\sin ^2(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2617
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{2/3} \tan ^2(e+f x) \, dx &=\frac{\cos ^2(e+f x)^{11/6} \, _2F_1\left (\frac{3}{2},\frac{11}{6};\frac{5}{2};\sin ^2(e+f x)\right ) (d \sec (e+f x))^{2/3} \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.303094, size = 80, normalized size = 1.4 \[ \frac{3 (d \sec (e+f x))^{2/3} \left (2 \sqrt [6]{\cos ^2(e+f x)} \tan (e+f x)-\sin (2 (e+f x)) \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};\sin ^2(e+f x)\right )\right )}{10 f \sqrt [6]{\cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{{\frac{2}{3}}} \left ( \tan \left ( fx+e \right ) \right ) ^{2}\, 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 \left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}} \tan \left (f x + e\right )^{2}\,{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 (\left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec{\left (e + f x \right )}\right )^{\frac{2}{3}} \tan ^{2}{\left (e + f x \right )}\, dx \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 \left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}} \tan \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]