Optimal. Leaf size=62 \[ \frac{4 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{7/2}}{7 c f} \]
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Rubi [A] time = 0.144795, antiderivative size = 62, 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} \[ \frac{4 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{7/2}}{7 c f} \]
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 (c-i c \tan (e+f x))^{5/2} \, dx &=\left (a^2 c^2\right ) \int \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} \, dx\\ &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int (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 (2 c (c+x)^{3/2}-(c+x)^{5/2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f}\\ &=\frac{4 i a^2 (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 i a^2 (c-i c \tan (e+f x))^{7/2}}{7 c f}\\ \end{align*}
Mathematica [A] time = 3.02448, size = 78, normalized size = 1.26 \[ -\frac{2 a^2 c^2 (\cos (2 e)-i \sin (2 e)) (5 \tan (e+f x)-9 i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)}}{35 f (\cos (f x)+i \sin (f x))^2} \]
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}{7} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{2\,c}{5} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38153, size = 62, normalized size = 1. \begin{align*} -\frac{2 i \,{\left (5 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}} a^{2} - 14 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} a^{2} c\right )}}{35 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48686, size = 242, normalized size = 3.9 \begin{align*} \frac{\sqrt{2}{\left (112 i \, a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{2} c^{2}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{35 \,{\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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