Optimal. Leaf size=83 \[ \frac{3 i \sqrt [3]{1+i \tan (c+d x)} \sqrt [3]{e \sec (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{4}{3},\frac{7}{6},\frac{1}{2} (1-i \tan (c+d x))\right )}{\sqrt [3]{2} d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.182116, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3505, 3523, 70, 69} \[ \frac{3 i \sqrt [3]{1+i \tan (c+d x)} \sqrt [3]{e \sec (c+d x)} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{4}{3},\frac{7}{6},\frac{1}{2} (1-i \tan (c+d x))\right )}{\sqrt [3]{2} d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{\sqrt [3]{e \sec (c+d x)} \int \frac{\sqrt [6]{a-i a \tan (c+d x)}}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{\sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}}\\ &=\frac{\left (a^2 \sqrt [3]{e \sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-i a x)^{5/6} (a+i a x)^{4/3}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}}\\ &=\frac{\left (a \sqrt [3]{e \sec (c+d x)} \sqrt [3]{\frac{a+i a \tan (c+d x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{1}{2}+\frac{i x}{2}\right )^{4/3} (a-i a x)^{5/6}} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [3]{2} d \sqrt [6]{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{3 i \, _2F_1\left (\frac{1}{6},\frac{4}{3};\frac{7}{6};\frac{1}{2} (1-i \tan (c+d x))\right ) \sqrt [3]{e \sec (c+d x)} \sqrt [3]{1+i \tan (c+d x)}}{\sqrt [3]{2} d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.519591, size = 95, normalized size = 1.14 \[ \frac{3 \left (8 i-\frac{2 i e^{2 i (c+d x)} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{5}{3},-e^{2 i (c+d x)}\right )}{\sqrt [6]{1+e^{2 i (c+d x)}}}\right ) \sqrt [3]{e \sec (c+d x)}}{16 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{e\sec \left ( dx+c \right ) }{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \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{\left (e \sec \left (d x + c\right )\right )^{\frac{1}{3}}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{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{{\left (4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )}{\rm integral}\left (-\frac{i \cdot 2^{\frac{5}{6}} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}} e^{\left (\frac{1}{3} i \, d x + \frac{1}{3} i \, c\right )}}{4 \, a d}, x\right ) + 2^{\frac{5}{6}} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac{1}{3}}{\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac{4}{3} i \, d x + \frac{4}{3} i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \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*} \int \frac{\sqrt [3]{e \sec{\left (c + d x \right )}}}{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )}}\, dx \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 (e \sec \left (d x + c\right )\right )^{\frac{1}{3}}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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