Optimal. Leaf size=90 \[ -\frac{2 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+\frac{2 a e \sin (c+d x) \sqrt{e \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.0605921, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3486, 3768, 3771, 2639} \[ -\frac{2 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+\frac{2 a e \sin (c+d x) \sqrt{e \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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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)) \, dx &=\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+a \int (e \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+\frac{2 a e \sqrt{e \sec (c+d x)} \sin (c+d x)}{d}-\left (a e^2\right ) \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx\\ &=\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+\frac{2 a e \sqrt{e \sec (c+d x)} \sin (c+d x)}{d}-\frac{\left (a e^2\right ) \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=-\frac{2 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{2 i a (e \sec (c+d x))^{3/2}}{3 d}+\frac{2 a e \sqrt{e \sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 0.801333, size = 102, normalized size = 1.13 \[ \frac{2 a e e^{-2 i d x} \sqrt{e \sec (c+d x)} (\cos (c+3 d x)+i \sin (c+3 d x)) \left (i \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )+\tan (c+d x)-2 i\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.21, size = 351, normalized size = 3.9 \begin{align*}{\frac{2\,a \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) ^{2}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 3\,i\sin \left ( dx+c \right ) \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 ) -3\,i\sin \left ( dx+c \right ) \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 ) +3\,i\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 ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -3\,i\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 ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +i\sin \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3\,\cos \left ( dx+c \right ) \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 )}\,{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 (-6 i \, a e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 i \, a 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 )} + 3 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}{\rm integral}\left (\frac{i \, \sqrt{2} a 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 )}}{d}, x\right )}{3 \,{\left (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] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx + \int i \left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}} \tan{\left (c + d x \right )}\, dx\right ) \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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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