Optimal. Leaf size=113 \[ -\frac{A+i B}{8 a c^2 f (-\tan (e+f x)+i)}+\frac{B+i A}{8 a c^2 f (\tan (e+f x)+i)^2}+\frac{x (3 A+i B)}{8 a c^2}+\frac{A}{4 a c^2 f (\tan (e+f x)+i)} \]
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Rubi [A] time = 0.189822, antiderivative size = 113, 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} \[ -\frac{A+i B}{8 a c^2 f (-\tan (e+f x)+i)}+\frac{B+i A}{8 a c^2 f (\tan (e+f x)+i)^2}+\frac{x (3 A+i B)}{8 a c^2}+\frac{A}{4 a c^2 f (\tan (e+f x)+i)} \]
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)) (c-i c \tan (e+f x))^2} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^2 (c-i c x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{-A-i B}{8 a^2 c^3 (-i+x)^2}-\frac{i (A-i B)}{4 a^2 c^3 (i+x)^3}-\frac{A}{4 a^2 c^3 (i+x)^2}+\frac{3 A+i B}{8 a^2 c^3 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{A+i B}{8 a c^2 f (i-\tan (e+f x))}+\frac{i A+B}{8 a c^2 f (i+\tan (e+f x))^2}+\frac{A}{4 a c^2 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 c^2 f}\\ &=\frac{(3 A+i B) x}{8 a c^2}-\frac{A+i B}{8 a c^2 f (i-\tan (e+f x))}+\frac{i A+B}{8 a c^2 f (i+\tan (e+f x))^2}+\frac{A}{4 a c^2 f (i+\tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 2.23363, size = 166, normalized size = 1.47 \[ \frac{(\cos (2 (e+f x))+i \sin (2 (e+f x))) (A+B \tan (e+f x)) (2 (A (-3-6 i f x)+B (2 f x+i)) \cos (e+f x)+(A+3 i B) \cos (3 (e+f x))-\sin (e+f x) ((-2 B+6 i A) \cos (2 (e+f x))+12 A f x+9 i A+4 i B f x+B))}{32 a c^2 f (\tan (e+f x)-i) (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.07, size = 209, normalized size = 1.9 \begin{align*}{\frac{A}{8\,af{c}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{8}}B}{af{c}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{3\,i}{16}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{af{c}^{2}}}+{\frac{\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{16\,af{c}^{2}}}+{\frac{A}{4\,af{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{{\frac{3\,i}{16}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{af{c}^{2}}}-{\frac{\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{16\,af{c}^{2}}}+{\frac{{\frac{i}{8}}A}{af{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}+{\frac{B}{8\,af{c}^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}} \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 = 1.72389, size = 217, normalized size = 1.92 \begin{align*} \frac{{\left (4 \,{\left (3 \, A + i \, B\right )} f x e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (-i \, A - B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-6 i \, A - 2 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, A - 2 \, B\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{32 \, a c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.63348, size = 286, normalized size = 2.53 \begin{align*} \begin{cases} \frac{\left (\left (512 i A a^{2} c^{4} f^{2} - 512 B a^{2} c^{4} f^{2}\right ) e^{- 2 i f x} + \left (- 1536 i A a^{2} c^{4} f^{2} e^{4 i e} - 512 B a^{2} c^{4} f^{2} e^{4 i e}\right ) e^{2 i f x} + \left (- 256 i A a^{2} c^{4} f^{2} e^{6 i e} - 256 B a^{2} c^{4} f^{2} e^{6 i e}\right ) e^{4 i f x}\right ) e^{- 2 i e}}{8192 a^{3} c^{6} f^{3}} & \text{for}\: 8192 a^{3} c^{6} f^{3} e^{2 i e} \neq 0 \\x \left (- \frac{3 A + i B}{8 a c^{2}} + \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^{- 2 i e}}{8 a c^{2}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (3 A + i B\right )}{8 a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43216, size = 228, normalized size = 2.02 \begin{align*} \frac{\frac{2 \,{\left (3 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a c^{2}} + \frac{2 \,{\left (-3 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a c^{2}} - \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 c^{2}{\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 c^{2}{\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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