Optimal. Leaf size=117 \[ \frac{A-i B}{8 a^2 c f (\tan (e+f x)+i)}-\frac{-B+i A}{8 a^2 c f (-\tan (e+f x)+i)^2}+\frac{x (3 A-i B)}{8 a^2 c}-\frac{A}{4 a^2 c f (-\tan (e+f x)+i)} \]
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Rubi [A] time = 0.186812, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {3588, 77, 203} \[ \frac{A-i B}{8 a^2 c f (\tan (e+f x)+i)}-\frac{-B+i A}{8 a^2 c f (-\tan (e+f x)+i)^2}+\frac{x (3 A-i B)}{8 a^2 c}-\frac{A}{4 a^2 c f (-\tan (e+f x)+i)} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 77
Rule 203
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^3 (c-i c x)^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{i (A+i B)}{4 a^3 c^2 (-i+x)^3}-\frac{A}{4 a^3 c^2 (-i+x)^2}+\frac{-A+i B}{8 a^3 c^2 (i+x)^2}+\frac{3 A-i B}{8 a^3 c^2 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A-B}{8 a^2 c f (i-\tan (e+f x))^2}-\frac{A}{4 a^2 c f (i-\tan (e+f x))}+\frac{A-i B}{8 a^2 c f (i+\tan (e+f x))}+\frac{(3 A-i B) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 a^2 c f}\\ &=\frac{(3 A-i B) x}{8 a^2 c}-\frac{i A-B}{8 a^2 c f (i-\tan (e+f x))^2}-\frac{A}{4 a^2 c f (i-\tan (e+f x))}+\frac{A-i B}{8 a^2 c f (i+\tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 2.03392, size = 129, normalized size = 1.1 \[ -\frac{2 (A-3 i B) \cos (2 (e+f x))+(B+3 i A) \sin (3 (e+f x)) \sec (e+f x)-12 A f x \tan (e+f x)+6 i A \tan (e+f x)+12 i A f x-7 A-2 B \tan (e+f x)+4 i B f x \tan (e+f x)+4 B f x+i B}{32 a^2 c f (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.089, size = 209, normalized size = 1.8 \begin{align*}{\frac{B}{8\,f{a}^{2}c \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{\frac{i}{8}}A}{f{a}^{2}c \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{A}{4\,f{a}^{2}c \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{3\,i}{16}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{f{a}^{2}c}}-{\frac{\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{16\,f{a}^{2}c}}+{\frac{A}{8\,f{a}^{2}c \left ( \tan \left ( fx+e \right ) +i \right ) }}-{\frac{{\frac{i}{8}}B}{f{a}^{2}c \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{{\frac{3\,i}{16}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{f{a}^{2}c}}+{\frac{\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{16\,f{a}^{2}c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07083, size = 216, normalized size = 1.85 \begin{align*} \frac{{\left (4 \,{\left (3 \, A - i \, B\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-2 i \, A - 2 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (6 i \, A - 2 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, A - B\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{32 \, a^{2} c f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.54824, size = 298, normalized size = 2.55 \begin{align*} \begin{cases} \frac{\left (\left (256 i A a^{4} c^{2} f^{2} e^{2 i e} - 256 B a^{4} c^{2} f^{2} e^{2 i e}\right ) e^{- 4 i f x} + \left (1536 i A a^{4} c^{2} f^{2} e^{4 i e} - 512 B a^{4} c^{2} f^{2} e^{4 i e}\right ) e^{- 2 i f x} + \left (- 512 i A a^{4} c^{2} f^{2} e^{8 i e} - 512 B a^{4} c^{2} f^{2} e^{8 i e}\right ) e^{2 i f x}\right ) e^{- 6 i e}}{8192 a^{6} c^{3} f^{3}} & \text{for}\: 8192 a^{6} c^{3} f^{3} e^{6 i e} \neq 0 \\x \left (- \frac{3 A - i B}{8 a^{2} c} + \frac{\left (A e^{6 i e} + 3 A e^{4 i e} + 3 A e^{2 i e} + A - i B e^{6 i e} - i B e^{4 i e} + i B e^{2 i e} + i B\right ) e^{- 4 i e}}{8 a^{2} c}\right ) & \text{otherwise} \end{cases} + \frac{x \left (3 A - i B\right )}{8 a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38387, size = 228, normalized size = 1.95 \begin{align*} \frac{\frac{2 \,{\left (3 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{2} c} + \frac{2 \,{\left (-3 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{2} c} - \frac{2 \,{\left (3 \, A \tan \left (f x + e\right ) - i \, B \tan \left (f x + e\right ) + 5 i \, A + 3 \, B\right )}}{a^{2} c{\left (-i \, \tan \left (f x + e\right ) + 1\right )}} + \frac{9 i \, A \tan \left (f x + e\right )^{2} + 3 \, B \tan \left (f x + e\right )^{2} + 26 \, A \tan \left (f x + e\right ) - 6 i \, B \tan \left (f x + e\right ) - 21 i \, A + B}{a^{2} c{\left (\tan \left (f x + e\right ) - i\right )}^{2}}}{32 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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