Optimal. Leaf size=60 \[ \frac{2 a (B+i A) \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 a B (c-i c \tan (e+f x))^{3/2}}{3 c f} \]
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Rubi [A] time = 0.0989737, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {3588, 43} \[ \frac{2 a (B+i A) \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 a B (c-i c \tan (e+f x))^{3/2}}{3 c f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) \sqrt{c-i c \tan (e+f x)} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{\sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{A-i B}{\sqrt{c-i c x}}+\frac{i B \sqrt{c-i c x}}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 a (i A+B) \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 a B (c-i c \tan (e+f x))^{3/2}}{3 c f}\\ \end{align*}
Mathematica [A] time = 2.36431, size = 45, normalized size = 0.75 \[ \frac{2 a \sqrt{c-i c \tan (e+f x)} (3 i A+i B \tan (e+f x)+2 B)}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 66, normalized size = 1.1 \begin{align*}{\frac{2\,ia}{cf} \left ({\frac{i}{3}}B \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}-iBc\sqrt{c-ic\tan \left ( fx+e \right ) }+Ac\sqrt{c-ic\tan \left ( fx+e \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19481, size = 66, normalized size = 1.1 \begin{align*} \frac{2 i \,{\left (i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}} B a + \sqrt{-i \, c \tan \left (f x + e\right ) + c}{\left (3 \, A - 3 i \, B\right )} a c\right )}}{3 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10666, size = 177, normalized size = 2.95 \begin{align*} \frac{\sqrt{2}{\left ({\left (6 i \, A + 6 \, B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (6 i \, A + 2 \, B\right )} a\right )} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int A \sqrt{- i c \tan{\left (e + f x \right )} + c}\, dx + \int B \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan{\left (e + f x \right )}\, dx + \int i A \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan{\left (e + f x \right )}\, dx + \int i B \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx\right ) \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{\left (B \tan \left (f x + e\right ) + A\right )}{\left (i \, a \tan \left (f x + e\right ) + a\right )} \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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