Optimal. Leaf size=62 \[ \frac{2 a (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac{2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f} \]
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Rubi [A] time = 0.106525, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049, Rules used = {3588, 43} \[ \frac{2 a (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac{2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 43
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left ((A-i B) (c-i c x)^{5/2}+\frac{i B (c-i c x)^{7/2}}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 a (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac{2 a B (c-i c \tan (e+f x))^{9/2}}{9 c f}\\ \end{align*}
Mathematica [A] time = 6.53682, size = 90, normalized size = 1.45 \[ \frac{2 a c^3 \sec ^3(e+f x) (\cos (f x)-i \sin (f x)) \sqrt{c-i c \tan (e+f x)} (\sin (3 e+2 f x)+i \cos (3 e+2 f x)) (9 A+7 B \tan (e+f x)-2 i B)}{63 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 55, normalized size = 0.9 \begin{align*}{\frac{2\,ia}{cf} \left ({\frac{i}{9}}B \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{9}{2}}}+{\frac{-iBc+Ac}{7} \left ( c-ic\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41209, size = 66, normalized size = 1.06 \begin{align*} \frac{2 i \,{\left (7 i \,{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{9}{2}} B a +{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{7}{2}}{\left (9 \, A - 9 i \, B\right )} a c\right )}}{63 \, c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7984, size = 304, normalized size = 4.9 \begin{align*} \frac{\sqrt{2}{\left ({\left (144 i \, A + 144 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (144 i \, A - 80 \, B\right )} a c^{3}\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{63 \,{\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, 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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