Optimal. Leaf size=27 \[ -\frac{2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.104654, 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} \[ -\frac{2 i a}{5 f (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3522
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^{5/2}} \, dx &=(a c) \int \frac{\sec ^2(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx\\ &=\frac{(i a) \operatorname{Subst}\left (\int \frac{1}{(c+x)^{7/2}} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac{2 i a}{5 f (c-i c \tan (e+f x))^{5/2}}\\ \end{align*}
Mathematica [B] time = 2.3041, size = 72, normalized size = 2.67 \[ \frac{2 a \cos ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (3 e+4 f x)-i \cos (3 e+4 f x))}{5 c^3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, 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 = 1.10291, size = 26, normalized size = 0.96 \begin{align*} -\frac{2 i \, a}{5 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.43858, size = 196, normalized size = 7.26 \begin{align*} \frac{\sqrt{2}{\left (-i \, a e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, a e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{20 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 28.2084, size = 46, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{2 i a}{5 f \left (- i c \tan{\left (e + f x \right )} + c\right )^{\frac{5}{2}}} & \text{for}\: f \neq 0 \\\frac{x \left (i a \tan{\left (e \right )} + a\right )}{\left (- i c \tan{\left (e \right )} + c\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \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{i \, a \tan \left (f x + e\right ) + a}{{\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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