Optimal. Leaf size=90 \[ -\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 a^2 d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A] time = 0.073417, 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, 2641} \[ -\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 a^2 d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{5/2}}{(a+i a \tan (c+d x))^2} \, dx &=\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{e^2 \int \sqrt{e \sec (c+d x)} \, dx}{3 a^2}\\ &=\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\left (e^2 \sqrt{\cos (c+d x)} \sqrt{e \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \sec (c+d x)}}{3 a^2 d}+\frac{4 i e^2 \sqrt{e \sec (c+d x)}}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.36464, size = 101, normalized size = 1.12 \[ \frac{2 (e \sec (c+d x))^{5/2} (\cos (c+d x)+i \sin (c+d x)) \left (\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (\cos (c+d x)+i \sin (c+d x))-2 i \cos (c+d x)\right )}{3 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.24, size = 173, normalized size = 1.9 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,{a}^{2}d} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}} \left ( i\cos \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 EllipticF} \left ({\frac{i \left ( \cos \left ( dx+c \right ) -1 \right ) }{\sin \left ( dx+c \right ) }},i \right ) +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 ) -2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) } \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 (3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )}{\rm integral}\left (\frac{i \, \sqrt{2} e^{2} \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 \, a^{2} d}, x\right ) + \sqrt{2}{\left (2 i \, e^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, e^{2}\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 (-2 i \, d x - 2 i \, c\right )}}{3 \, 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{5}{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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