Optimal. Leaf size=58 \[ -\frac{2 i a^2 \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{4 i a^2}{f \sqrt{c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.147839, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3522, 3487, 43} \[ -\frac{2 i a^2 \sqrt{c-i c \tan (e+f x)}}{c f}-\frac{4 i a^2}{f \sqrt{c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+i a \tan (e+f x))^2}{\sqrt{c-i c \tan (e+f x)}} \, dx &=\left (a^2 c^2\right ) \int \frac{\sec ^4(e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{c-x}{(c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \left (\frac{2 c}{(c+x)^{3/2}}-\frac{1}{\sqrt{c+x}}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=-\frac{4 i a^2}{f \sqrt{c-i c \tan (e+f x)}}-\frac{2 i a^2 \sqrt{c-i c \tan (e+f x)}}{c f}\\ \end{align*}
Mathematica [A] time = 1.9878, size = 91, normalized size = 1.57 \[ \frac{2 a^2 \sqrt{c-i c \tan (e+f x)} (-2 \sin (2 e)-2 i \cos (2 e)+\sin (2 f x)-i \cos (2 f x)) (\cos (e+f x)+i \sin (e+f x))^2}{c f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 45, normalized size = 0.8 \begin{align*}{\frac{-2\,i{a}^{2}}{cf} \left ( \sqrt{c-ic\tan \left ( fx+e \right ) }+2\,{\frac{c}{\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 = 1.44841, size = 61, normalized size = 1.05 \begin{align*} -\frac{2 i \,{\left (\sqrt{-i \, c \tan \left (f x + e\right ) + c} a^{2} + \frac{2 \, a^{2} 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 = 1.39956, size = 122, normalized size = 2.1 \begin{align*} \frac{\sqrt{2}{\left (-2 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 i \, a^{2}\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^{2} \left (\int - \frac{\tan ^{2}{\left (e + f x \right )}}{\sqrt{- i c \tan{\left (e + f x \right )} + c}}\, dx + \int \frac{2 i \tan{\left (e + f x \right )}}{\sqrt{- i c \tan{\left (e + f x \right )} + c}}\, dx + \int \frac{1}{\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 (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}{\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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