Optimal. Leaf size=76 \[ \frac {3 b \sqrt [3]{d \sec (e+f x)}}{f}-\frac {3 a d \sin (e+f x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(e+f x)\right )}{2 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}} \]
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Rubi [A] time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3486, 3772, 2643} \[ \frac {3 b \sqrt [3]{d \sec (e+f x)}}{f}-\frac {3 a d \sin (e+f x) \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(e+f x)\right )}{2 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3486
Rule 3772
Rubi steps
\begin {align*} \int \sqrt [3]{d \sec (e+f x)} (a+b \tan (e+f x)) \, dx &=\frac {3 b \sqrt [3]{d \sec (e+f x)}}{f}+a \int \sqrt [3]{d \sec (e+f x)} \, dx\\ &=\frac {3 b \sqrt [3]{d \sec (e+f x)}}{f}+\left (a \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 {3 b \sqrt [3]{d \sec (e+f x)}}{f}-\frac {3 a \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)}{2 f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 58, normalized size = 0.76 \[ \frac {\sqrt [3]{d \sec (e+f x)} \left (a \cos ^2(e+f x)^{2/3} \tan (e+f x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};\sin ^2(e+f x)\right )+3 b\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}, x\right ) \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +b \tan \left (f x +e \right )\right )\, dx \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \sqrt [3]{d \sec {\left (e + f x \right )}} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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