Optimal. Leaf size=86 \[ \frac{3 i 2^{2/3} a \sqrt [3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{7}{6},\frac{13}{6},\frac{1}{2} (1-i \tan (c+d x))\right )}{7 d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.210934, antiderivative size = 86, 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} \[ \frac{3 i 2^{2/3} a \sqrt [3]{1+i \tan (c+d x)} (e \sec (c+d x))^{7/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{7}{6},\frac{13}{6},\frac{1}{2} (1-i \tan (c+d x))\right )}{7 d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{7/3}}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{(e \sec (c+d x))^{7/3} \int (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{2/3} \, dx}{(a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}}\\ &=\frac{\left (a^2 (e \sec (c+d x))^{7/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a-i a x}}{\sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{7/6}}\\ &=\frac{\left (a^2 (e \sec (c+d x))^{7/3} \sqrt [3]{\frac{a+i a \tan (c+d x)}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{a-i a x}}{\sqrt [3]{\frac{1}{2}+\frac{i x}{2}}} \, dx,x,\tan (c+d x)\right )}{\sqrt [3]{2} d (a-i a \tan (c+d x))^{7/6} (a+i a \tan (c+d x))^{3/2}}\\ &=\frac{3 i 2^{2/3} a \, _2F_1\left (\frac{1}{3},\frac{7}{6};\frac{13}{6};\frac{1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{7/3} \sqrt [3]{1+i \tan (c+d x)}}{7 d (a+i a \tan (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.662903, size = 118, normalized size = 1.37 \[ -\frac{3 i \sqrt [3]{2} e e^{i (c+d x)} \left (\frac{e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{4/3} \left (4+\left (1+e^{2 i (c+d x)}\right )^{5/6} \text{Hypergeometric2F1}\left (\frac{2}{3},\frac{5}{6},\frac{5}{3},-e^{2 i (c+d x)}\right )\right )}{5 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\sec \left ( dx+c \right ) \right ) ^{{\frac{7}{3}}}{\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{7}{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{-6 i \cdot 2^{\frac{5}{6}} e^{2} \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{4}{3} i \, d x + \frac{4}{3} i \, c\right )} + 5 \, a d{\rm integral}\left (-\frac{2 i \cdot 2^{\frac{5}{6}} e^{2} \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 )}}{5 \, a d}, x\right )}{5 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{7}{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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