Optimal. Leaf size=81 \[ -\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.158046, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3497, 3488} \[ -\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{5/2}} \, dx &=-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}}+\frac{(2 a) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx}{5 e^2}\\ &=-\frac{4 i a \sqrt{a+i a \tan (c+d x)}}{5 d e^2 \sqrt{e \sec (c+d x)}}-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.365108, size = 84, normalized size = 1.04 \[ -\frac{2 a (2 \tan (c+d x)+3 i) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (\cos (c+2 d x)+i \sin (c+2 d x))}{5 d e (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.296, size = 86, normalized size = 1.1 \begin{align*} -{\frac{2\,a \left ( i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2\,i \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{5\,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 = 1.88918, size = 80, normalized size = 0.99 \begin{align*} \frac{{\left (-i \, a \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) - 5 i \, a \cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, a \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{5 \, d e^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11201, size = 227, normalized size = 2.8 \begin{align*} \frac{{\left (-i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 6 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 5 i \, a\right )} \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}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{5 \, 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{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\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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