Optimal. Leaf size=78 \[ \frac{3 a d \sin (e+f x) (d \sec (e+f x))^{2/3} \text{Hypergeometric2F1}\left (-\frac{1}{3},\frac{1}{2},\frac{2}{3},\cos ^2(e+f x)\right )}{2 f \sqrt{\sin ^2(e+f x)}}+\frac{3 b (d \sec (e+f x))^{5/3}}{5 f} \]
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Rubi [A] time = 0.0642202, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3486, 3772, 2643} \[ \frac{3 a d \sin (e+f x) (d \sec (e+f x))^{2/3} \text{Hypergeometric2F1}\left (-\frac{1}{3},\frac{1}{2},\frac{2}{3},\cos ^2(e+f x)\right )}{2 f \sqrt{\sin ^2(e+f x)}}+\frac{3 b (d \sec (e+f x))^{5/3}}{5 f} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int (d \sec (e+f x))^{5/3} (a+b \tan (e+f x)) \, dx &=\frac{3 b (d \sec (e+f x))^{5/3}}{5 f}+a \int (d \sec (e+f x))^{5/3} \, dx\\ &=\frac{3 b (d \sec (e+f x))^{5/3}}{5 f}+\left (a \left (\frac{\cos (e+f x)}{d}\right )^{2/3} (d \sec (e+f x))^{2/3}\right ) \int \frac{1}{\left (\frac{\cos (e+f x)}{d}\right )^{5/3}} \, dx\\ &=\frac{3 b (d \sec (e+f x))^{5/3}}{5 f}+\frac{3 a d \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(e+f x)\right ) (d \sec (e+f x))^{2/3} \sin (e+f x)}{2 f \sqrt{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.506981, size = 126, normalized size = 1.62 \[ \frac{d (d \sec (e+f x))^{2/3} (a+b \tan (e+f x)) \left (3 \cos ^2(e+f x)^{2/3} (5 a \sin (2 (e+f x))+4 b)-10 a \sin (e+f x) \cos ^3(e+f x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{3}{2},\sin ^2(e+f x)\right )\right )}{20 f \cos ^2(e+f x)^{2/3} (a \cos (e+f x)+b \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.1, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sec \left ( fx+e \right ) \right ) ^{{\frac{5}{3}}} \left ( a+b\tan \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{\frac{5}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b d \sec \left (f x + e\right ) \tan \left (f x + e\right ) + a d \sec \left (f x + e\right )\right )} \left (d \sec \left (f x + e\right )\right )^{\frac{2}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \sec \left (f x + e\right )\right )^{\frac{5}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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