Optimal. Leaf size=90 \[ \frac{i (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.126306, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 45, 37} \[ \frac{i (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{5/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{5 f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{15 a f (a+i a \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.72341, size = 79, normalized size = 0.88 \[ \frac{c (1-i \tan (e+f x)) (\tan (e+f x)-4 i) \sqrt{c-i c \tan (e+f x)}}{15 a^2 f (\tan (e+f x)-i)^2 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 75, normalized size = 0.8 \begin{align*}{\frac{c \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 4\,i-\tan \left ( fx+e \right ) \right ) }{15\,f{a}^{3} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{4}}\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 [A] time = 1.93798, size = 116, normalized size = 1.29 \begin{align*} \frac{{\left (3 i \, c \cos \left (5 \, f x + 5 \, e\right ) + 5 i \, c \cos \left (\frac{3}{5} \, \arctan \left (\sin \left (5 \, f x + 5 \, e\right ), \cos \left (5 \, f x + 5 \, e\right )\right )\right ) + 3 \, c \sin \left (5 \, f x + 5 \, e\right ) + 5 \, c \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 )\right )} \sqrt{c}}{30 \, a^{\frac{5}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44953, size = 302, normalized size = 3.36 \begin{align*} \frac{{\left (-8 i \, c e^{\left (7 i \, f x + 7 i \, e\right )} - 8 i \, c e^{\left (5 i \, f x + 5 i \, e\right )} + 5 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 8 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, c\right )} \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}} e^{\left (-5 i \, f x - 5 i \, e\right )}}{30 \, a^{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{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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