Integrand size = 28, antiderivative size = 69 \[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {12 i \sqrt [6]{2} a^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {1}{6},\frac {7}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) \sqrt [3]{d \sec (e+f x)}}{f \sqrt [6]{1+i \tan (e+f x)}} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3586, 3604, 72, 71} \[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {12 i \sqrt [6]{2} a^2 \sqrt [3]{d \sec (e+f x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {1}{6},\frac {7}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{f \sqrt [6]{1+i \tan (e+f x)}} \]
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \operatorname {Hypergeometric2F1}\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*}
Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.48 \[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\frac {3 a^2 \sqrt [3]{d \sec (e+f x)} \left (2 i+\cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},\sec ^2(e+f x)\right ) \sqrt {-\tan ^2(e+f x)}+\cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sec ^2(e+f x)\right ) \sqrt {-\tan ^2(e+f x)}\right )}{f} \]
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\[\int \left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (a +i a \tan \left (f x +e \right )\right )^{2}d x\]
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\[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \,d x } \]
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\[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=- a^{2} \left (\int \left (- \sqrt [3]{d \sec {\left (e + f x \right )}}\right )\, dx + \int \sqrt [3]{d \sec {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \sqrt [3]{d \sec {\left (e + f x \right )}} \tan {\left (e + f x \right )}\right )\, dx\right ) \]
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\[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \,d x } \]
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\[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {1}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2 \, dx=\int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{1/3}\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
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