Optimal. Leaf size=88 \[ -\frac{3 i \sqrt [6]{1+i \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \text{Hypergeometric2F1}\left (-\frac{2}{3},\frac{13}{6},\frac{1}{3},\frac{1}{2} (1-i \tan (c+d x))\right )}{8 \sqrt [6]{2} a d (e \sec (c+d x))^{4/3}} \]
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Rubi [A] time = 0.209879, antiderivative size = 88, 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} \[ -\frac{3 i \sqrt [6]{1+i \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} \text{Hypergeometric2F1}\left (-\frac{2}{3},\frac{13}{6},\frac{1}{3},\frac{1}{2} (1-i \tan (c+d x))\right )}{8 \sqrt [6]{2} a d (e \sec (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{4/3} \sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{\left ((a-i a \tan (c+d x))^{2/3} (a+i a \tan (c+d x))^{2/3}\right ) \int \frac{1}{(a-i a \tan (c+d x))^{2/3} (a+i a \tan (c+d x))^{7/6}} \, dx}{(e \sec (c+d x))^{4/3}}\\ &=\frac{\left (a^2 (a-i a \tan (c+d x))^{2/3} (a+i a \tan (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{(a-i a x)^{5/3} (a+i a x)^{13/6}} \, dx,x,\tan (c+d x)\right )}{d (e \sec (c+d x))^{4/3}}\\ &=\frac{\left ((a-i a \tan (c+d x))^{2/3} \sqrt{a+i a \tan (c+d x)} \sqrt [6]{\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 )^{13/6} (a-i a x)^{5/3}} \, dx,x,\tan (c+d x)\right )}{4 \sqrt [6]{2} d (e \sec (c+d x))^{4/3}}\\ &=-\frac{3 i \, _2F_1\left (-\frac{2}{3},\frac{13}{6};\frac{1}{3};\frac{1}{2} (1-i \tan (c+d x))\right ) \sqrt [6]{1+i \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}{8 \sqrt [6]{2} a d (e \sec (c+d x))^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.655092, size = 112, normalized size = 1.27 \[ -\frac{3 i \sec ^2(c+d x) \left (-55 \sqrt [6]{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{6},\frac{1}{6},\frac{5}{6},-e^{2 i (c+d x)}\right )+11 i \sin (2 (c+d x))+3 \cos (2 (c+d x))+3\right )}{112 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\sec \left ( dx+c \right ) \right ) ^{-{\frac{4}{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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{4}{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{2^{\frac{1}{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{2}{3}}{\left (-21 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 42 i \, e^{\left (7 i \, d x + 7 i \, c\right )} + 114 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 60 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 303 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 6 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 180 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 24 i \, e^{\left (i \, d x + i \, c\right )} + 12 i\right )} e^{\left (\frac{5}{3} i \, d x + \frac{5}{3} i \, c\right )} + 112 \,{\left (a d e^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 2 \, a d e^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + a d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )}\right )}{\rm integral}\left (\frac{2^{\frac{1}{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{2}{3}}{\left (55 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 385 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 275 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 385 i \, e^{\left (i \, d x + i \, c\right )} + 220 i\right )} e^{\left (\frac{5}{3} i \, d x + \frac{5}{3} i \, c\right )}}{112 \,{\left (a d e^{2} e^{\left (5 i \, d x + 5 i \, c\right )} - 3 \, a d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a d e^{2} e^{\left (3 i \, d x + 3 i \, c\right )} - a d e^{2} e^{\left (2 i \, d x + 2 i \, c\right )}\right )}}, x\right )}{112 \,{\left (a d e^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 2 \, a d e^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + a d e^{2} e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{4}{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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