Optimal. Leaf size=140 \[ \frac{2 \tan (e+f x)}{5 a^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.139244, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 45, 39} \[ \frac{2 \tan (e+f x)}{5 a^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 45
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{7/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{5/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=\frac{i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt{c-i c \tan (e+f x)}}+\frac{i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt{c-i c \tan (e+f x)}}+\frac{2 \tan (e+f x)}{5 a^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.42467, size = 99, normalized size = 0.71 \[ \frac{\sqrt{c-i c \tan (e+f x)} (-4 \cos (2 (e+f x))+5 i \tan (e+f x)-3 i \sin (3 (e+f x)) \sec (e+f x)+12)}{20 a^2 c f (\tan (e+f x)-i) \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 118, normalized size = 0.8 \begin{align*} -{\frac{4\,i \left ( \tan \left ( fx+e \right ) \right ) ^{4}-2\, \left ( \tan \left ( fx+e \right ) \right ) ^{5}+6\,i \left ( \tan \left ( fx+e \right ) \right ) ^{2}- \left ( \tan \left ( fx+e \right ) \right ) ^{3}+2\,i+\tan \left ( fx+e \right ) }{5\,f{a}^{3}c \left ( -\tan \left ( fx+e \right ) +i \right ) ^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39206, size = 365, normalized size = 2.61 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-5 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 16 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 10 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 16 i \, e^{\left (5 i \, f x + 5 i \, e\right )} + 20 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 6 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-5 i \, f x - 5 i \, e\right )}}{40 \, a^{3} c f} \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 \frac{1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{5}{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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