Optimal. Leaf size=27 \[ \frac{2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
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Rubi [A] time = 0.100823, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3522, 3487, 32} \[ \frac{2 i a (c-i c \tan (e+f x))^{5/2}}{5 f} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx &=(a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^{3/2} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int (c+x)^{3/2} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=\frac{2 i a (c-i c \tan (e+f x))^{5/2}}{5 f}\\ \end{align*}
Mathematica [B] time = 1.37617, size = 70, normalized size = 2.59 \[ \frac{2 a c^2 \sec ^2(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (2 e+f x)+i \cos (2 e+f x))}{5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 22, normalized size = 0.8 \begin{align*}{\frac{{\frac{2\,i}{5}}a}{f} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28123, size = 26, normalized size = 0.96 \begin{align*} \frac{2 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} a}{5 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42088, size = 147, normalized size = 5.44 \begin{align*} \frac{8 i \, \sqrt{2} a c^{2} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{5 \,{\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, 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 )}{\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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