Optimal. Leaf size=25 \[ -\frac{2 i a}{f \sqrt{c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.103065, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 32} \[ -\frac{2 i a}{f \sqrt{c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{a+i a \tan (e+f x)}{\sqrt{c-i c \tan (e+f x)}} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(c-i c \tan (e+f x))^{3/2}} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac{2 i a}{f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [B] time = 0.903638, size = 64, normalized size = 2.56 \[ \frac{2 a \cos (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (e+2 f x)-i \cos (e+2 f x))}{c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 22, normalized size = 0.9 \begin{align*}{\frac{-2\,ia}{f}{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.32103, size = 26, normalized size = 1.04 \begin{align*} -\frac{2 i \, a}{\sqrt{-i \, c \tan \left (f x + e\right ) + c} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38257, size = 111, normalized size = 4.44 \begin{align*} \frac{\sqrt{2}{\left (-i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.72834, size = 44, normalized size = 1.76 \begin{align*} \begin{cases} - \frac{2 i a}{f \sqrt{- i c \tan{\left (e + f x \right )} + c}} & \text{for}\: f \neq 0 \\\frac{x \left (i a \tan{\left (e \right )} + a\right )}{\sqrt{- i c \tan{\left (e \right )} + c}} & \text{otherwise} \end{cases} \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{i \, a \tan \left (f x + e\right ) + a}{\sqrt{-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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