Optimal. Leaf size=36 \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}} \]
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Rubi [A] time = 0.0608407, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {3488} \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}} \, dx &=-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0437845, size = 36, normalized size = 1. \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)}}{d \sqrt{e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.322, size = 56, normalized size = 1.6 \begin{align*}{\frac{-2\,i\cos \left ( dx+c \right ) }{de}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}\sqrt{{\frac{e}{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61616, size = 103, normalized size = 2.86 \begin{align*} -\frac{2 i \, \sqrt{a} \sqrt{-\frac{2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}}{d \sqrt{e} \sqrt{-\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86377, size = 176, normalized size = 4.89 \begin{align*} \frac{2 \, \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 (2 i \, d x + 2 i \, c\right )} - i\right )} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}}{d e} \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 )}}{\sqrt{e \sec{\left (c + d x \right )}}}\, 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}}{\sqrt{e \sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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