Optimal. Leaf size=25 \[ \frac{2 i a \sqrt{c-i c \tan (e+f x)}}{f} \]
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Rubi [A] time = 0.100693, 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} \[ \frac{2 i a \sqrt{c-i c \tan (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int (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)}{\sqrt{c-i c \tan (e+f x)}} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac{2 i a \sqrt{c-i c \tan (e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.796619, size = 25, normalized size = 1. \[ \frac{2 i a \sqrt{c-i c \tan (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 22, normalized size = 0.9 \begin{align*}{\frac{2\,ia}{f}\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 = 1.50945, size = 26, normalized size = 1.04 \begin{align*} \frac{2 i \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} a}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27363, size = 69, normalized size = 2.76 \begin{align*} \frac{2 i \, \sqrt{2} a \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.88737, size = 42, normalized size = 1.68 \begin{align*} \begin{cases} \frac{2 i a \sqrt{- i c \tan{\left (e + f x \right )} + c}}{f} & \text{for}\: f \neq 0 \\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 [A] time = 1.62683, size = 27, normalized size = 1.08 \begin{align*} \frac{2 i \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} a}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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