Optimal. Leaf size=80 \[ \frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.144943, antiderivative size = 80, 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 = {3502, 3488} \[ \frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sqrt{e \sec (c+d x)}}{(a+i a \tan (c+d x))^{3/2}} \, dx &=\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{5 a}\\ &=\frac{2 i \sqrt{e \sec (c+d x)}}{5 d (a+i a \tan (c+d x))^{3/2}}+\frac{4 i \sqrt{e \sec (c+d x)}}{5 a d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.120254, size = 63, normalized size = 0.79 \[ \frac{2 (3+2 i \tan (c+d x)) \sqrt{e \sec (c+d x)}}{5 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.308, size = 101, normalized size = 1.3 \begin{align*}{\frac{-{\frac{2\,i}{5}}\cos \left ( dx+c \right ) \left ( 2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2\,i\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) \right ) }{{a}^{2}d}\sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.91847, size = 108, normalized size = 1.35 \begin{align*} \frac{\sqrt{e}{\left (i \, \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 i \, \cos \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right )\right )}}{5 \, a^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10288, size = 219, normalized size = 2.74 \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 (5 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac{5}{2} i \, d x - \frac{5}{2} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e \sec{\left (c + d x \right )}}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \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{e \sec \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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