Optimal. Leaf size=111 \[ \frac{6 i a^3}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{6 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.101171, antiderivative size = 111, 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 = {3496, 3486, 3771, 2639} \[ \frac{6 i a^3}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{6 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3496
Rule 3486
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{5/2}} \, dx &=-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}}-\frac{\left (3 a^2\right ) \int \frac{a+i a \tan (c+d x)}{\sqrt{e \sec (c+d x)}} \, dx}{5 e^2}\\ &=\frac{6 i a^3}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}}-\frac{\left (3 a^3\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{5 e^2}\\ &=\frac{6 i a^3}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}}-\frac{\left (3 a^3\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{6 i a^3}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{6 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{4 i a (a+i a \tan (c+d x))^2}{5 d (e \sec (c+d x))^{5/2}}\\ \end{align*}
Mathematica [C] time = 1.25711, size = 108, normalized size = 0.97 \[ -\frac{4 i a^3 e^{2 i (c+d x)} \left (-\sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+e^{2 i (c+d x)}+1\right )}{5 d e^2 \left (1+e^{2 i (c+d x)}\right ) \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.245, size = 1086, normalized size = 9.8 \begin{align*} \text{result too large to display} \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 (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{5}{2}}}\,{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{\sqrt{2}{\left (-2 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} + 4 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + 6 i \, a^{3}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )} + 5 \,{\left (d e^{3} e^{\left (i \, d x + i \, c\right )} - d e^{3}\right )}{\rm integral}\left (\frac{\sqrt{2}{\left (3 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, a^{3} e^{\left (i \, d x + i \, c\right )} + 3 i \, a^{3}\right )} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{5 \,{\left (d e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3} e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{5 \,{\left (d e^{3} e^{\left (i \, d x + i \, c\right )} - d e^{3}\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{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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