Optimal. Leaf size=119 \[ -\frac{3 d \left (4 a^2-3 b^2\right ) \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\cos ^2(e+f x)\right )}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}}+\frac{21 a b \sqrt [3]{d \sec (e+f x)}}{4 f}+\frac{3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f} \]
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Rubi [A] time = 0.136399, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3508, 3486, 3772, 2643} \[ -\frac{3 d \left (4 a^2-3 b^2\right ) \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},\cos ^2(e+f x)\right )}{8 f \sqrt{\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}}+\frac{21 a b \sqrt [3]{d \sec (e+f x)}}{4 f}+\frac{3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f} \]
Antiderivative was successfully verified.
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Rule 3508
Rule 3486
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))^2 \, dx &=\frac{3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f}+\frac{3}{4} \int \sqrt [3]{d \sec (e+f x)} \left (\frac{4 a^2}{3}-b^2+\frac{7}{3} a b \tan (e+f x)\right ) \, dx\\ &=\frac{21 a b \sqrt [3]{d \sec (e+f x)}}{4 f}+\frac{3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f}+\frac{1}{4} \left (4 a^2-3 b^2\right ) \int \sqrt [3]{d \sec (e+f x)} \, dx\\ &=\frac{21 a b \sqrt [3]{d \sec (e+f x)}}{4 f}+\frac{3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f}+\frac{1}{4} \left (\left (4 a^2-3 b^2\right ) \sqrt [3]{\frac{\cos (e+f x)}{d}} \sqrt [3]{d \sec (e+f x)}\right ) \int \frac{1}{\sqrt [3]{\frac{\cos (e+f x)}{d}}} \, dx\\ &=\frac{21 a b \sqrt [3]{d \sec (e+f x)}}{4 f}-\frac{3 \left (4 a^2-3 b^2\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \sin (e+f x)}{8 f \sqrt{\sin ^2(e+f x)}}+\frac{3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f}\\ \end{align*}
Mathematica [A] time = 0.460638, size = 83, normalized size = 0.7 \[ \frac{\sqrt [3]{d \sec (e+f x)} \left (\left (4 a^2-3 b^2\right ) \cos ^2(e+f x)^{2/3} \tan (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{2}{3},\frac{3}{2},\sin ^2(e+f x)\right )+3 b (8 a+b \tan (e+f x))\right )}{4 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.141, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{d\sec \left ( fx+e \right ) } \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}\, 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{1}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{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^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{d \sec{\left (e + f x \right )}} \left (a + b \tan{\left (e + f x \right )}\right )^{2}\, dx \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{1}{3}}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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