Optimal. Leaf size=182 \[ \frac{2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}} \]
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Rubi [A] time = 0.16584, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {3523, 45, 37} \[ \frac{2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}} \]
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))^{9/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{21 a f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{\sqrt{c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{105 a^2 f}\\ &=\frac{i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac{i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac{2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 5.31062, size = 115, normalized size = 0.63 \[ \frac{c (\tan (e+f x)+i) \sec ^2(e+f x) \sqrt{c-i c \tan (e+f x)} (140 \cos (2 (e+f x))+27 i \tan (e+f x)+35 i \sin (3 (e+f x)) \sec (e+f x)+92)}{1260 a^4 f (\tan (e+f x)-i)^4 \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 97, normalized size = 0.5 \begin{align*}{\frac{{\frac{i}{315}}c \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \left ( 2\,i \left ( \tan \left ( fx+e \right ) \right ) ^{3}-33\,i\tan \left ( fx+e \right ) +12\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}-58 \right ) }{f{a}^{5} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{6}}\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 = 2.05959, size = 251, normalized size = 1.38 \begin{align*} \frac{{\left (35 i \, c \cos \left (9 \, f x + 9 \, e\right ) + 135 i \, c \cos \left (\frac{7}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 189 i \, c \cos \left (\frac{5}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 105 i \, c \cos \left (\frac{1}{3} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 35 \, c \sin \left (9 \, f x + 9 \, e\right ) + 135 \, c \sin \left (\frac{7}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 189 \, c \sin \left (\frac{5}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 105 \, c \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right )\right )} \sqrt{c}}{2520 \, a^{\frac{9}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45716, size = 401, normalized size = 2.2 \begin{align*} \frac{{\left (-464 i \, c e^{\left (11 i \, f x + 11 i \, e\right )} - 464 i \, c e^{\left (9 i \, f x + 9 i \, e\right )} + 105 i \, c e^{\left (8 i \, f x + 8 i \, e\right )} + 294 i \, c e^{\left (6 i \, f x + 6 i \, e\right )} + 324 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 170 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 35 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 (-9 i \, f x - 9 i \, e\right )}}{2520 \, a^{5} 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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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