Optimal. Leaf size=67 \[ -\frac{3 i 2^{5/6} a \sqrt [6]{1+i \tan (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{6},\frac{5}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{f \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A] time = 0.154474, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3505, 3523, 70, 69} \[ -\frac{3 i 2^{5/6} a \sqrt [6]{1+i \tan (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{6},\frac{5}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{f \sqrt [3]{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{a+i a \tan (e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx &=\frac{\left (\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \int \frac{(a+i a \tan (e+f x))^{5/6}}{\sqrt [6]{a-i a \tan (e+f x)}} \, dx}{\sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\left (a^2 \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-i a x)^{7/6} \sqrt [6]{a+i a x}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\left (a^2 \sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{\frac{a+i a \tan (e+f x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{\frac{1}{2}+\frac{i x}{2}} (a-i a x)^{7/6}} \, dx,x,\tan (e+f x)\right )}{\sqrt [6]{2} f \sqrt [3]{d \sec (e+f x)}}\\ &=-\frac{3 i 2^{5/6} a \, _2F_1\left (-\frac{1}{6},\frac{1}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{f \sqrt [3]{d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.440552, size = 98, normalized size = 1.46 \[ -\frac{3 i 2^{2/3} a e^{2 i (e+f x)} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{11}{6},-e^{2 i (e+f x)}\right )}{5 f \sqrt [3]{1+e^{2 i (e+f x)}} \sqrt [3]{\frac{d e^{i (e+f x)}}{1+e^{2 i (e+f x)}}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.143, size = 0, normalized size = 0. \begin{align*} \int{(a+ia\tan \left ( fx+e \right ) ){\frac{1}{\sqrt [3]{d\sec \left ( fx+e \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{i \, a \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2^{\frac{2}{3}}{\left (-3 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, a\right )} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}} e^{\left (\frac{2}{3} i \, f x + \frac{2}{3} i \, e\right )} +{\left (d f e^{\left (i \, f x + i \, e\right )} - d f\right )}{\rm integral}\left (\frac{2^{\frac{2}{3}}{\left (-2 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, a e^{\left (i \, f x + i \, e\right )} - 2 i \, a\right )} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}} e^{\left (\frac{2}{3} i \, f x + \frac{2}{3} i \, e\right )}}{d f e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, d f e^{\left (2 i \, f x + 2 i \, e\right )} + d f e^{\left (i \, f x + i \, e\right )}}, x\right )}{d f e^{\left (i \, f x + i \, e\right )} - d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{1}{\sqrt [3]{d \sec{\left (e + f x \right )}}}\, dx + \int \frac{i \tan{\left (e + f x \right )}}{\sqrt [3]{d \sec{\left (e + f x \right )}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{i \, a \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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