Optimal. Leaf size=69 \[ \frac {12 i \sqrt [6]{2} a^2 \sqrt [3]{d \sec (e+f x)} \, _2F_1\left (-\frac {7}{6},\frac {1}{6};\frac {7}{6};\frac {1}{2} (1-i \tan (e+f x))\right )}{f \sqrt [6]{1+i \tan (e+f x)}} \]
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Rubi [A] time = 0.16, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3505, 3523, 70, 69} \[ \frac {12 i \sqrt [6]{2} a^2 \sqrt [3]{d \sec (e+f x)} \text {Hypergeometric2F1}\left (-\frac {7}{6},\frac {1}{6},\frac {7}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{f \sqrt [6]{1+i \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx &=\frac {\sqrt [3]{d \sec (e+f x)} \int \sqrt [6]{a-i a \tan (e+f x)} (a+i a \tan (e+f x))^{13/6} \, dx}{\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}}\\ &=\frac {\left (a^2 \sqrt [3]{d \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{7/6}}{(a-i a x)^{5/6}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}}\\ &=\frac {\left (2 \sqrt [6]{2} a^3 \sqrt [3]{d \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{7/6}}{(a-i a x)^{5/6}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{\frac {a+i a \tan (e+f x)}{a}}}\\ &=\frac {12 i \sqrt [6]{2} a^2 \, _2F_1\left (-\frac {7}{6},\frac {1}{6};\frac {7}{6};\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [3]{d \sec (e+f x)}}{f \sqrt [6]{1+i \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 128, normalized size = 1.86 \[ -\frac {3 a^2 e^{-2 i e} \sqrt [3]{d \sec (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) \left (7 i \sqrt [3]{1+e^{2 i (e+f x)}} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-e^{2 i (e+f x)}\right )+\sec (e) \sin (f x) \sec (e+f x)+\tan (e)-8 i\right )}{4 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ \frac {2^{\frac {1}{3}} {\left (27 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 21 i \, a^{2}\right )} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {1}{3} i \, f x + \frac {1}{3} i \, e\right )} + 4 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} {\rm integral}\left (-\frac {7 i \cdot 2^{\frac {1}{3}} a^{2} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {1}{3}} e^{\left (-\frac {2}{3} i \, f x - \frac {2}{3} i \, e\right )}}{4 \, f}, x\right )}{4 \, {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +i a \tan \left (f x +e \right )\right )^{2}\, dx \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
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 \[ - a^{2} \left (\int \left (- \sqrt [3]{d \sec {\left (e + f x \right )}}\right )\, dx + \int \sqrt [3]{d \sec {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \sqrt [3]{d \sec {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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