Optimal. Leaf size=119 \[ -\frac {3 d \left (4 a^2-3 b^2\right ) \sin (e+f x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(e+f x)\right )}{8 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}}+\frac {21 a b \sqrt [3]{d \sec (e+f x)}}{4 f}+\frac {3 b \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x))}{4 f} \]
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Rubi [A] time = 0.14, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, 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 2643
Rule 3486
Rule 3508
Rule 3772
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*}
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Mathematica [A] time = 0.48, size = 83, normalized size = 0.70 \[ \frac {\sqrt [3]{d \sec (e+f x)} \left (\left (4 a^2-3 b^2\right ) \cos ^2(e+f x)^{2/3} \tan (e+f x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\sin ^2(e+f x)\right )+3 b (8 a+b \tan (e+f x))\right )}{4 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +b \tan \left (f x +e \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{d \sec {\left (e + f x \right )}} \left (a + b \tan {\left (e + f x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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