Optimal. Leaf size=83 \[ \frac{3 i \sqrt [6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3} \text{Hypergeometric2F1}\left (\frac{5}{6},\frac{7}{6},\frac{11}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{5 \sqrt [6]{2} f (a+i a \tan (e+f x))} \]
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Rubi [A] time = 0.175058, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ \frac{3 i \sqrt [6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3} \text{Hypergeometric2F1}\left (\frac{5}{6},\frac{7}{6},\frac{11}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{5 \sqrt [6]{2} f (a+i a \tan (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(d \sec (e+f x))^{5/3}}{a+i a \tan (e+f x)} \, dx &=\frac{(d \sec (e+f x))^{5/3} \int \frac{(a-i a \tan (e+f x))^{5/6}}{\sqrt [6]{a+i a \tan (e+f x)}} \, dx}{(a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}}\\ &=\frac{\left (a^2 (d \sec (e+f x))^{5/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{a-i a x} (a+i a x)^{7/6}} \, dx,x,\tan (e+f x)\right )}{f (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}}\\ &=\frac{\left (a (d \sec (e+f x))^{5/3} \sqrt [6]{\frac{a+i a \tan (e+f x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{1}{2}+\frac{i x}{2}\right )^{7/6} \sqrt [6]{a-i a x}} \, dx,x,\tan (e+f x)\right )}{2 \sqrt [6]{2} f (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))}\\ &=\frac{3 i \, _2F_1\left (\frac{5}{6},\frac{7}{6};\frac{11}{6};\frac{1}{2} (1-i \tan (e+f x))\right ) (d \sec (e+f x))^{5/3} \sqrt [6]{1+i \tan (e+f x)}}{5 \sqrt [6]{2} f (a+i a \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 0.460657, size = 84, normalized size = 1.01 \[ \frac{6 d e^{i (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{2}{3},\frac{5}{6},-e^{2 i (e+f x)}\right ) (d \sec (e+f x))^{2/3}}{a f \sqrt [3]{1+e^{2 i (e+f x)}} (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.136, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+ia\tan \left ( fx+e \right ) } \left ( d\sec \left ( fx+e \right ) \right ) ^{{\frac{5}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{{\left (a f e^{\left (i \, f x + i \, e\right )}{\rm integral}\left (-\frac{i \cdot 2^{\frac{2}{3}} d \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 )}}{a f}, x\right ) + 2^{\frac{2}{3}}{\left (3 i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, d\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 )}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{3}}}{i \, a \tan \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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