Optimal. Leaf size=138 \[ -\frac{14 a^2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{14 a^2 e \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d} \]
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Rubi [A] time = 0.117203, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3498, 3486, 3768, 3771, 2639} \[ -\frac{14 a^2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{14 a^2 e \sin (c+d x) \sqrt{e \sec (c+d x)}}{5 d}+\frac{2 i \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 3498
Rule 3486
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^2 \, dx &=\frac{2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac{1}{5} (7 a) \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x)) \, dx\\ &=\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}+\frac{1}{5} \left (7 a^2\right ) \int (e \sec (c+d x))^{3/2} \, dx\\ &=\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{14 a^2 e \sqrt{e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}-\frac{1}{5} \left (7 a^2 e^2\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx\\ &=\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{14 a^2 e \sqrt{e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}-\frac{\left (7 a^2 e^2\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{14 a^2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{14 i a^2 (e \sec (c+d x))^{3/2}}{15 d}+\frac{14 a^2 e \sqrt{e \sec (c+d x)} \sin (c+d x)}{5 d}+\frac{2 i (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}{5 d}\\ \end{align*}
Mathematica [C] time = 2.50229, size = 267, normalized size = 1.93 \[ \frac{(a+i a \tan (c+d x))^2 (e \sec (c+d x))^{3/2} \left (\frac{1}{2} \csc (c) (\cos (2 c)-i \sin (2 c)) \sec ^{\frac{5}{2}}(c+d x) (20 i \sin (2 c+d x)+27 \cos (2 c+d x)+21 \cos (2 c+3 d x)-20 i \sin (d x)+36 \cos (d x))-\frac{14 i \sqrt{2} \left (3 \sqrt{1+e^{2 i (c+d x)}}-\left (-1+e^{2 i c}\right ) e^{2 i d x} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right )}{15 d \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.266, size = 374, normalized size = 2.7 \begin{align*} -{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) } \left ( 21\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) -21\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +21\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) -21\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sin \left ( dx+c \right ) +21\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-10\,i\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -24\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3 \right ) \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}} \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 \left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{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 (-42 i \, a^{2} e e^{\left (5 i \, d x + 5 i \, c\right )} - 32 i \, a^{2} e e^{\left (3 i \, d x + 3 i \, c\right )} - 14 i \, a^{2} e e^{\left (i \, d x + i \, c\right )}\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 )} + 15 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}{\rm integral}\left (\frac{7 i \, \sqrt{2} a^{2} e \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 \, d}, x\right )}{15 \,{\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\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 \left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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