Optimal. Leaf size=27 \[ \frac{2 i a (c-i c \tan (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.0987998, antiderivative size = 27, 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} \[ \frac{2 i a (c-i c \tan (e+f x))^{3/2}}{3 f} \]
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)) (c-i c \tan (e+f x))^{3/2} \, dx &=(a c) \int \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \sqrt{c+x} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac{2 i a (c-i c \tan (e+f x))^{3/2}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.997671, size = 54, normalized size = 2. \[ \frac{2 a c (\sin (e)+i \cos (e)) \sec (e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 22, normalized size = 0.8 \begin{align*}{\frac{{\frac{2\,i}{3}}a}{f} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21802, size = 26, normalized size = 0.96 \begin{align*} \frac{2 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}} a}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.37182, size = 109, normalized size = 4.04 \begin{align*} \frac{4 i \, \sqrt{2} a c \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \,{\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.8423, size = 44, normalized size = 1.63 \begin{align*} \begin{cases} \frac{2 i a \left (- i c \tan{\left (e + f x \right )} + c\right )^{\frac{3}{2}}}{3 f} & \text{for}\: f \neq 0 \\x \left (i a \tan{\left (e \right )} + a\right ) \left (- i c \tan{\left (e \right )} + c\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43793, size = 27, normalized size = 1. \begin{align*} \frac{2 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}} a}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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