Optimal. Leaf size=90 \[ \frac{2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{5 d \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.0746756, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3500, 3771, 2639} \[ \frac{2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{5 d \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{3/2}}{(a+i a \tan (c+d x))^2} \, dx &=\frac{4 i e^2}{5 d \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{e^2 \int \frac{1}{\sqrt{e \sec (c+d x)}} \, dx}{5 a^2}\\ &=\frac{4 i e^2}{5 d \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{e^2 \int \sqrt{\cos (c+d x)} \, dx}{5 a^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}\\ &=\frac{2 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}}+\frac{4 i e^2}{5 d \sqrt{e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.495644, size = 102, normalized size = 1.13 \[ \frac{i e e^{-3 i (c+d x)} \left (2 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+e^{2 i (c+d x)}+1\right ) \sqrt{e \sec (c+d x)}}{5 a^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.228, size = 341, normalized size = 3.8 \begin{align*} -{\frac{2\,\cos \left ( dx+c \right ) }{5\,{a}^{2}d\sin \left ( dx+c \right ) } \left ( i{\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 ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-i{\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 ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}+i{\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 ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-i{\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 ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}-2\,i\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}- \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{{\left (5 \, a^{2} d e^{\left (3 i \, d x + 3 i \, c\right )}{\rm integral}\left (-\frac{i \, \sqrt{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 \, a^{2} d}, x\right ) + \sqrt{2}{\left (2 i \, e e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e\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 )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{5 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{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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