Optimal. Leaf size=81 \[ \frac{4 i a \sqrt{e \sec (c+d x)}}{3 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.132141, 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} \[ \frac{4 i a \sqrt{e \sec (c+d x)}}{3 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx &=-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}+\frac{(2 a) \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{3 e^2}\\ &=\frac{4 i a \sqrt{e \sec (c+d x)}}{3 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.167859, size = 48, normalized size = 0.59 \[ \frac{2 (2 \tan (c+d x)+i) \sqrt{a+i a \tan (c+d x)}}{3 d (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.333, size = 75, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,i\cos \left ( dx+c \right ) +4\,\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d{e}^{3}}\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{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.88359, size = 73, normalized size = 0.9 \begin{align*} \frac{\sqrt{a}{\left (-i \, \cos \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 i \, \cos \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{3 \, d e^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93669, size = 220, normalized size = 2.72 \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 (-i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-\frac{1}{2} i \, d x - \frac{1}{2} i \, c\right )}}{3 \, d e^{2}} \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{a \left (i \tan{\left (c + d x \right )} + 1\right )}}{\left (e \sec{\left (c + d x \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{i \, a \tan \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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