Optimal. Leaf size=154 \[ \frac{8 \tan (e+f x)}{15 a^2 c^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{15 a c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{\tan (e+f x)}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.148245, antiderivative size = 154, 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, 40, 39} \[ \frac{8 \tan (e+f x)}{15 a^2 c^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{4 \tan (e+f x)}{15 a c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{\tan (e+f x)}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 40
Rule 39
Rubi steps
\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{7/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x)}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{5/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac{\tan (e+f x)}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac{4 \tan (e+f x)}{15 a c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{8 \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 )}{15 a c f}\\ &=\frac{\tan (e+f x)}{5 f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac{4 \tan (e+f x)}{15 a c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac{8 \tan (e+f x)}{15 a^2 c^2 f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.70909, size = 117, normalized size = 0.76 \[ -\frac{(150 \sin (e+f x)+25 \sin (3 (e+f x))+3 \sin (5 (e+f x))) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (3 (e+f x))+i \sin (3 (e+f x)))}{240 a^2 c^3 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 105, normalized size = 0.7 \begin{align*}{\frac{ \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \tan \left ( fx+e \right ) \left ( 8\, \left ( \tan \left ( fx+e \right ) \right ) ^{4}+20\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}+15 \right ) }{15\,f{a}^{3}{c}^{3} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \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 [A] time = 1.90506, size = 96, normalized size = 0.62 \begin{align*} \frac{3 \, \sin \left (5 \, f x + 5 \, e\right ) + 25 \, \sin \left (\frac{3}{5} \, \arctan \left (\sin \left (5 \, f x + 5 \, e\right ), \cos \left (5 \, f x + 5 \, e\right )\right )\right ) + 150 \, \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (5 \, f x + 5 \, e\right ), \cos \left (5 \, f x + 5 \, e\right )\right )\right )}{240 \, a^{\frac{5}{2}} c^{\frac{5}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43156, size = 344, normalized size = 2.23 \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 (-3 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 28 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 175 i \, e^{\left (8 i \, f x + 8 i \, e\right )} + 175 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 28 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i\right )} e^{\left (-5 i \, f x - 5 i \, e\right )}}{480 \, a^{3} c^{3} 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}}{\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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