Optimal. Leaf size=142 \[ -\frac{2 a^3 (5 B+i A)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{8 a^3 (2 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac{8 a^3 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac{2 a^3 B}{c^3 f \sqrt{c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.205122, antiderivative size = 142, 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} \[ -\frac{2 a^3 (5 B+i A)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{8 a^3 (2 B+i A)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac{8 a^3 (B+i A)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac{2 a^3 B}{c^3 f \sqrt{c-i c \tan (e+f x)}} \]
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))^{7/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{(a+i a x)^2 (A+B x)}{(c-i c x)^{9/2}} \, 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)^{9/2}}-\frac{4 a^2 (A-2 i B)}{c (c-i c x)^{7/2}}+\frac{a^2 (A-5 i B)}{c^2 (c-i c x)^{5/2}}+\frac{i a^2 B}{c^3 (c-i c x)^{3/2}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{8 a^3 (i A+B)}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac{8 a^3 (i A+2 B)}{5 c f (c-i c \tan (e+f x))^{5/2}}-\frac{2 a^3 (i A+5 B)}{3 c^2 f (c-i c \tan (e+f x))^{3/2}}+\frac{2 a^3 B}{c^3 f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 13.3834, size = 141, normalized size = 0.99 \[ \frac{a^3 \cos (e+f x) \sqrt{c-i c \tan (e+f x)} (\cos (4 e+7 f x)+i \sin (4 e+7 f x)) (i (A+13 i B) \cos (e+f x)+(89 B-23 i A) \cos (3 (e+f x))+14 \sin (e+f x) ((A-17 i B) \cos (2 (e+f x))+A-2 i B))}{105 c^4 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 105, normalized size = 0.7 \begin{align*}{\frac{2\,i{a}^{3}}{f{c}^{3}} \left ({-iB{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}}-{\frac{4\,{c}^{3} \left ( A-iB \right ) }{7} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{c \left ( A-5\,iB \right ) }{3} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{c}^{2} \left ( A-2\,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 = 1.17561, size = 143, normalized size = 1.01 \begin{align*} -\frac{2 i \,{\left (105 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{3} B a^{3} +{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{2}{\left (35 \, A - 175 i \, B\right )} a^{3} c -{\left (-i \, c \tan \left (f x + e\right ) + c\right )}{\left (84 \, A - 168 i \, B\right )} a^{3} c^{2} +{\left (60 \, A - 60 i \, B\right )} a^{3} c^{3}\right )}}{105 \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}} c^{3} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17568, size = 333, normalized size = 2.35 \begin{align*} \frac{\sqrt{2}{\left ({\left (-15 i \, A - 15 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-18 i \, A + 24 \, B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (i \, A - 13 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-4 i \, A + 52 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-8 i \, A + 104 \, B\right )} a^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{210 \, c^{4} 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 (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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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