Optimal. Leaf size=58 \[ -\frac{2 a (B+i A)}{f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a B \sqrt{c-i c \tan (e+f x)}}{c f} \]
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Rubi [A] time = 0.100029, antiderivative size = 58, 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} \[ -\frac{2 a (B+i A)}{f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a B \sqrt{c-i c \tan (e+f x)}}{c f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int \frac{(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}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{A-i B}{(c-i c x)^{3/2}}+\frac{i B}{c \sqrt{c-i c x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{2 a (i A+B)}{f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a B \sqrt{c-i c \tan (e+f x)}}{c f}\\ \end{align*}
Mathematica [A] time = 2.52525, size = 82, normalized size = 1.41 \[ \frac{2 a (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (e+2 f x)-i \cos (e+2 f x)) (-B \sin (e+f x)+(A-2 i B) \cos (e+f x))}{c f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.133, size = 53, normalized size = 0.9 \begin{align*}{\frac{2\,ia}{cf} \left ( iB\sqrt{c-ic\tan \left ( fx+e \right ) }-{c \left ( A-iB \right ){\frac{1}{\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 = 1.1618, size = 65, normalized size = 1.12 \begin{align*} \frac{2 i \,{\left (i \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} B a - \frac{{\left (A - i \, B\right )} a c}{\sqrt{-i \, c \tan \left (f x + e\right ) + c}}\right )}}{c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06398, size = 136, normalized size = 2.34 \begin{align*} \frac{\sqrt{2}{\left ({\left (-i \, A - B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-i \, A - 3 \, B\right )} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{A}{\sqrt{- i c \tan{\left (e + f x \right )} + c}}\, dx + \int \frac{B \tan{\left (e + f x \right )}}{\sqrt{- i c \tan{\left (e + f x \right )} + c}}\, dx + \int \frac{i A \tan{\left (e + f x \right )}}{\sqrt{- i c \tan{\left (e + f x \right )} + c}}\, dx + \int \frac{i B \tan ^{2}{\left (e + f x \right )}}{\sqrt{- i c \tan{\left (e + f x \right )} + c}}\, 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 \frac{{\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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