Optimal. Leaf size=144 \[ \frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{8 a^3 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \]
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Rubi [A] time = 0.2003, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.047, Rules used = {3588, 77} \[ \frac{2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{8 a^3 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 77
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^{3/2}-\frac{4 a^2 (A-2 i B) (c-i c x)^{5/2}}{c}+\frac{a^2 (A-5 i B) (c-i c x)^{7/2}}{c^2}+\frac{i a^2 B (c-i c x)^{9/2}}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{8 a^3 (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac{8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac{2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac{2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f}\\ \end{align*}
Mathematica [A] time = 11.8246, size = 139, normalized size = 0.97 \[ -\frac{2 a^3 c^2 \sec ^4(e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (2 e-f x)-i \sin (2 e-f x)) (5 (121 A-74 i B) \tan (e+f x)+\cos (2 (e+f x)) ((605 A-685 i B) \tan (e+f x)-781 i A-701 B)+9 (31 B-44 i A))}{3465 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 121, normalized size = 0.8 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{3}} \left ({\frac{i}{11}}B \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{11}{2}}}+{\frac{-5\,iBc+Ac}{9} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{9}{2}}}}+{\frac{-4\, \left ( -iBc+Ac \right ) c+4\,iB{c}^{2}}{7} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{ \left ( -4\,iBc+4\,Ac \right ){c}^{2}}{5} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1458, size = 146, normalized size = 1.01 \begin{align*} \frac{2 i \,{\left (315 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{11}{2}} B a^{3} +{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{9}{2}}{\left (385 \, A - 1925 i \, B\right )} a^{3} c -{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}}{\left (1980 \, A - 3960 i \, B\right )} a^{3} c^{2} +{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}}{\left (2772 \, A - 2772 i \, B\right )} a^{3} c^{3}\right )}}{3465 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05812, size = 498, normalized size = 3.46 \begin{align*} \frac{\sqrt{2}{\left ({\left (22176 i \, A + 22176 \, B\right )} a^{3} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (34848 i \, A + 3168 \, B\right )} a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (15488 i \, A + 1408 \, B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (2816 i \, A + 256 \, B\right )} a^{3} c^{2}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3465 \,{\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \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(-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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