Optimal. Leaf size=140 \[ -\frac{2 a^3 (5 B+i A)}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{8 a^3 (2 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{8 a^3 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 a^3 B \sqrt{c-i c \tan (e+f x)}}{c^3 f} \]
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Rubi [A] time = 0.201367, antiderivative size = 140, 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} \[ -\frac{2 a^3 (5 B+i A)}{c^2 f \sqrt{c-i c \tan (e+f x)}}+\frac{8 a^3 (2 B+i A)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{8 a^3 (B+i A)}{5 f (c-i c \tan (e+f x))^{5/2}}-\frac{2 a^3 B \sqrt{c-i c \tan (e+f x)}}{c^3 f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 77
Rubi steps
\begin{align*} \int \frac{(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 \frac{(a+i a x)^2 (A+B x)}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{4 a^2 (A-i B)}{(c-i c x)^{7/2}}-\frac{4 a^2 (A-2 i B)}{c (c-i c x)^{5/2}}+\frac{a^2 (A-5 i B)}{c^2 (c-i c x)^{3/2}}+\frac{i a^2 B}{c^3 \sqrt{c-i c x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{8 a^3 (i A+B)}{5 f (c-i c \tan (e+f x))^{5/2}}+\frac{8 a^3 (i A+2 B)}{3 c f (c-i c \tan (e+f x))^{3/2}}-\frac{2 a^3 (i A+5 B)}{c^2 f \sqrt{c-i c \tan (e+f x)}}-\frac{2 a^3 B \sqrt{c-i c \tan (e+f x)}}{c^3 f}\\ \end{align*}
Mathematica [A] time = 13.1025, size = 135, normalized size = 0.96 \[ \frac{a^3 \sqrt{c-i c \tan (e+f x)} (\sin (3 (e+2 f x))-i \cos (3 (e+2 f x))) (3 (A-11 i B) \cos (e+f x)+(11 A-91 i B) \cos (3 (e+f x))-10 i \sin (e+f x) ((A-17 i B) \cos (2 (e+f x))+A-14 i B))}{15 c^3 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 105, normalized size = 0.8 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{3}} \left ( iB\sqrt{c-ic\tan \left ( fx+e \right ) }-{c \left ( A-5\,iB \right ){\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}}+{\frac{4\,{c}^{2} \left ( A-2\,iB \right ) }{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{c}^{3} \left ( A-iB \right ) }{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 = 1.23794, size = 150, normalized size = 1.07 \begin{align*} -\frac{2 i \,{\left (-\frac{15 i \, \sqrt{-i \, c \tan \left (f x + e\right ) + c} B a^{3}}{c^{2}} + \frac{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2}{\left (15 \, A - 75 i \, B\right )} a^{3} -{\left (-i \, c \tan \left (f x + e\right ) + c\right )}{\left (20 \, A - 40 i \, B\right )} a^{3} c +{\left (12 \, A - 12 i \, B\right )} a^{3} c^{2}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{5}{2}} c}\right )}}{15 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16856, size = 270, normalized size = 1.93 \begin{align*} \frac{\sqrt{2}{\left ({\left (-3 i \, A - 3 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (i \, A + 11 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-4 i \, A - 44 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-8 i \, A - 88 \, B\right )} a^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{15 \, c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 (B \tan \left (f x + e\right ) + A\right )}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}{{\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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