Optimal. Leaf size=31 \[ \frac{2 i a \sec (c+d x)}{d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.0281013, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {3493} \[ \frac{2 i a \sec (c+d x)}{d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3493
Rubi steps
\begin{align*} \int \sec (c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=\frac{2 i a \sec (c+d x)}{d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.143904, size = 39, normalized size = 1.26 \[ \frac{2 \sqrt{a+i a \tan (c+d x)} (\sin (c+d x)+i \cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.242, size = 50, normalized size = 1.6 \begin{align*} 2\,{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{d}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34026, size = 66, normalized size = 2.13 \begin{align*} \frac{2 i \, \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{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 \sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )} \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 \sqrt{i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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