Optimal. Leaf size=60 \[ \frac{4 i a^2 \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 c f} \]
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Rubi [A] time = 0.151926, antiderivative size = 60, 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} \[ \frac{4 i a^2 \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 c f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 43
Rubi steps
\begin{align*} \int (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))^{3/2}} \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{c-x}{\sqrt{c+x}} \, 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}{\sqrt{c+x}}-\sqrt{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{4 i a^2 \sqrt{c-i c \tan (e+f x)}}{f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{3/2}}{3 c f}\\ \end{align*}
Mathematica [A] time = 1.94237, size = 37, normalized size = 0.62 \[ -\frac{2 a^2 (\tan (e+f x)-5 i) \sqrt{c-i c \tan (e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 47, normalized size = 0.8 \begin{align*}{\frac{-2\,i{a}^{2}}{cf} \left ({\frac{1}{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}-2\,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 = 1.42727, size = 61, normalized size = 1.02 \begin{align*} -\frac{2 i \,{\left ({\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}} a^{2} - 6 \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} a^{2} c\right )}}{3 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40583, size = 157, normalized size = 2.62 \begin{align*} \frac{\sqrt{2}{\left (12 i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{2}\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^{2} \left (\int - \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int 2 i \sqrt{- i c \tan{\left (e + f x \right )} + c} \tan{\left (e + f x \right )}\, dx + \int \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{\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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