Optimal. Leaf size=119 \[ -\frac {3 d \left (2 a^2-3 b^2\right ) \sin (e+f x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(e+f x)\right )}{8 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{4/3}}-\frac {15 a b}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {3 b (a+b \tan (e+f x))}{2 f \sqrt [3]{d \sec (e+f x)}} \]
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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} \[ -\frac {3 d \left (2 a^2-3 b^2\right ) \sin (e+f x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\cos ^2(e+f x)\right )}{8 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{4/3}}-\frac {15 a b}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {3 b (a+b \tan (e+f x))}{2 f \sqrt [3]{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3486
Rule 3508
Rule 3772
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^2}{\sqrt [3]{d \sec (e+f x)}} \, dx &=\frac {3 b (a+b \tan (e+f x))}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {3}{2} \int \frac {\frac {2 a^2}{3}-b^2+\frac {5}{3} a b \tan (e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx\\ &=-\frac {15 a b}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {3 b (a+b \tan (e+f x))}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {1}{2} \left (2 a^2-3 b^2\right ) \int \frac {1}{\sqrt [3]{d \sec (e+f x)}} \, dx\\ &=-\frac {15 a b}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {3 b (a+b \tan (e+f x))}{2 f \sqrt [3]{d \sec (e+f x)}}+\frac {1}{2} \left (\left (2 a^2-3 b^2\right ) \left (\frac {\cos (e+f x)}{d}\right )^{2/3} (d \sec (e+f x))^{2/3}\right ) \int \sqrt [3]{\frac {\cos (e+f x)}{d}} \, dx\\ &=-\frac {15 a b}{2 f \sqrt [3]{d \sec (e+f x)}}-\frac {3 \left (2 a^2-3 b^2\right ) \cos ^2(e+f x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(e+f x)\right ) (d \sec (e+f x))^{2/3} \sin (e+f x)}{8 d f \sqrt {\sin ^2(e+f x)}}+\frac {3 b (a+b \tan (e+f x))}{2 f \sqrt [3]{d \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 3.94, size = 209, normalized size = 1.76 \[ \frac {3 d \sin (e+f x) (a+b \tan (e+f x))^2 \left (\frac {\left (\left (2 a^2-3 b^2\right ) \cot (e+f x)+4 a b\right ) \left (\left (2 a^2-3 b^2\right ) \sqrt {\sin ^2(e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sec ^2(e+f x)\right )-4 a b \cos (e+f x) \sqrt {-\tan ^2(e+f x)}\right )}{\sqrt {-\tan ^2(e+f x)} \left (\left (2 a^2-3 b^2\right ) \sqrt {\sin ^2(e+f x)} \cot (e+f x)+4 a b \sin (e+f x)\right )}+b^2\right )}{2 f (d \sec (e+f x))^{4/3} (a \cos (e+f x)+b \sin (e+f x))^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \left (d \sec \left (f x + e\right )\right )^{\frac {2}{3}}}{d \sec \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tan \left (f x +e \right )\right )^{2}}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2}}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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