Optimal. Leaf size=71 \[ -\frac{3 i \sqrt [6]{1+i \tan (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{19}{6},\frac{5}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{4 \sqrt [6]{2} a^2 f \sqrt [3]{d \sec (e+f x)}} \]
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Rubi [A] time = 0.179847, antiderivative size = 71, 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} \[ -\frac{3 i \sqrt [6]{1+i \tan (e+f x)} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{19}{6},\frac{5}{6},\frac{1}{2} (1-i \tan (e+f x))\right )}{4 \sqrt [6]{2} a^2 f \sqrt [3]{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{d \sec (e+f x)} (a+i a \tan (e+f x))^2} \, dx &=\frac{\left (\sqrt [6]{a-i a \tan (e+f x)} \sqrt [6]{a+i a \tan (e+f x)}\right ) \int \frac{1}{\sqrt [6]{a-i a \tan (e+f x)} (a+i a \tan (e+f x))^{13/6}} \, 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} (a+i a x)^{19/6}} \, dx,x,\tan (e+f x)\right )}{f \sqrt [3]{d \sec (e+f x)}}\\ &=\frac{\left (\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}{\left (\frac{1}{2}+\frac{i x}{2}\right )^{19/6} (a-i a x)^{7/6}} \, dx,x,\tan (e+f x)\right )}{8 \sqrt [6]{2} a f \sqrt [3]{d \sec (e+f x)}}\\ &=-\frac{3 i \, _2F_1\left (-\frac{1}{6},\frac{19}{6};\frac{5}{6};\frac{1}{2} (1-i \tan (e+f x))\right ) \sqrt [6]{1+i \tan (e+f x)}}{4 \sqrt [6]{2} a^2 f \sqrt [3]{d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.35768, size = 141, normalized size = 1.99 \[ \frac{(d \sec (e+f x))^{2/3} (-3 \sin (2 (e+f x))-3 i \cos (2 (e+f x))) \left (16 e^{3 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{11}{6},-e^{2 i (e+f x)}\right )-10 \left (7 \cos (e+f x)+5 \cos (3 (e+f x))+18 i \sin (e+f x) \cos ^2(e+f x)\right )\right )}{260 a^2 d f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.15, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{2}}{\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(-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{2^{\frac{2}{3}} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}}{\left (-39 i \, e^{\left (7 i \, f x + 7 i \, e\right )} - 57 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 27 i \, e^{\left (5 i \, f x + 5 i \, e\right )} - 69 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 15 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 15 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, e^{\left (i \, f x + i \, e\right )} - 3 i\right )} e^{\left (\frac{2}{3} i \, f x + \frac{2}{3} i \, e\right )} + 104 \,{\left (a^{2} d f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d f e^{\left (5 i \, f x + 5 i \, e\right )}\right )}{\rm integral}\left (\frac{2^{\frac{2}{3}} \left (\frac{d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac{2}{3}}{\left (-8 i \, e^{\left (2 i \, f x + 2 i \, e\right )} - 8 i \, e^{\left (i \, f x + i \, e\right )} - 8 i\right )} e^{\left (\frac{2}{3} i \, f x + \frac{2}{3} i \, e\right )}}{13 \,{\left (a^{2} d f e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, a^{2} d f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{2} d f e^{\left (i \, f x + i \, e\right )}\right )}}, x\right )}{104 \,{\left (a^{2} d f e^{\left (6 i \, f x + 6 i \, e\right )} - a^{2} d f e^{\left (5 i \, f x + 5 i \, e\right )}\right )}} \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{1}{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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