Optimal. Leaf size=122 \[ \frac{8 i a}{15 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{15 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.196973, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3497, 3502, 3488} \[ \frac{8 i a}{15 d e^2 \sqrt{a+i a \tan (c+d x)} \sqrt{e \sec (c+d x)}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{15 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx &=-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}+\frac{(4 a) \int \frac{1}{\sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}} \, dx}{5 e^2}\\ &=\frac{8 i a}{15 d e^2 \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}+\frac{8 \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{15 e^2}\\ &=\frac{8 i a}{15 d e^2 \sqrt{e \sec (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}-\frac{16 i \sqrt{a+i a \tan (c+d x)}}{15 d e^2 \sqrt{e \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.185985, size = 63, normalized size = 0.52 \[ \frac{i \sqrt{a+i a \tan (c+d x)} (-4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d e^2 \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.35, size = 85, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -16\,i \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{15\,d{e}^{5}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.94218, size = 176, normalized size = 1.44 \begin{align*} \frac{\sqrt{a}{\left (5 i \, \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) - 3 i \, \cos \left (\frac{5}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) - 30 i \, \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sin \left (\frac{5}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right ) + 30 \, \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ), \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )\right )\right )\right )}}{30 \, d e^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0929, size = 262, normalized size = 2.15 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-3 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 33 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 25 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac{3}{2} i \, d x - \frac{3}{2} i \, c\right )}}{30 \, d e^{3}} \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{\sqrt{i \, a \tan \left (d x + c\right ) + a}}{\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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