Optimal. Leaf size=117 \[ \frac {(3 A-i B) x}{8 a^2 c}-\frac {i A-B}{8 a^2 c f (i-\tan (e+f x))^2}-\frac {A}{4 a^2 c f (i-\tan (e+f x))}+\frac {A-i B}{8 a^2 c f (i+\tan (e+f x))} \]
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Rubi [A]
time = 0.14, antiderivative size = 117, normalized size of antiderivative = 1.00, 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 = {3669, 78, 209}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 209
Rule 3669
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) \text {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) \text {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) \text {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*}
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Mathematica [A]
time = 1.08, size = 129, normalized size = 1.10 \begin {gather*} -\frac {-7 A+i B+12 i A f x+4 B f x+2 (A-3 i B) \cos (2 (e+f x))+(3 i A+B) \sec (e+f x) \sin (3 (e+f x))+6 i A \tan (e+f x)-2 B \tan (e+f x)-12 A f x \tan (e+f x)+4 i B f x \tan (e+f x)}{32 a^2 c f (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 106, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {\left (\frac {3 i A}{16}+\frac {B}{16}\right ) \ln \left (i+\tan \left (f x +e \right )\right )-\frac {-\frac {A}{8}+\frac {i B}{8}}{i+\tan \left (f x +e \right )}-\frac {-\frac {B}{4}+\frac {i A}{4}}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}+\left (-\frac {3 i A}{16}-\frac {B}{16}\right ) \ln \left (-i+\tan \left (f x +e \right )\right )+\frac {A}{-4 i+4 \tan \left (f x +e \right )}}{f \,a^{2} c}\) | \(106\) |
default | \(\frac {\left (\frac {3 i A}{16}+\frac {B}{16}\right ) \ln \left (i+\tan \left (f x +e \right )\right )-\frac {-\frac {A}{8}+\frac {i B}{8}}{i+\tan \left (f x +e \right )}-\frac {-\frac {B}{4}+\frac {i A}{4}}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}+\left (-\frac {3 i A}{16}-\frac {B}{16}\right ) \ln \left (-i+\tan \left (f x +e \right )\right )+\frac {A}{-4 i+4 \tan \left (f x +e \right )}}{f \,a^{2} c}\) | \(106\) |
risch | \(-\frac {i x B}{8 a^{2} c}+\frac {3 x A}{8 a^{2} c}-\frac {{\mathrm e}^{-4 i \left (f x +e \right )} B}{32 a^{2} c f}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )} A}{32 a^{2} c f}-\frac {\cos \left (2 f x +2 e \right ) B}{8 a^{2} c f}+\frac {i \cos \left (2 f x +2 e \right ) A}{8 a^{2} c f}+\frac {A \sin \left (2 f x +2 e \right )}{4 a^{2} c f}\) | \(130\) |
norman | \(\frac {\frac {\left (-i B +3 A \right ) x}{8 a c}-\frac {-i A +B}{4 a c f}+\frac {\left (-i B +3 A \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 a c f}+\frac {\left (-i B +3 A \right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{4 a c}+\frac {\left (-i B +3 A \right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{8 a c}+\frac {\left (i B +5 A \right ) \tan \left (f x +e \right )}{8 a c f}}{a \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 83, normalized size = 0.71 \begin {gather*} \frac {{\left (4 \, {\left (3 \, A - i \, B\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, {\left (i \, A + B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} - 2 \, {\left (-3 i \, A + 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 296, normalized size = 2.53 \begin {gather*} \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}\: 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 169, normalized size = 1.44 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.77, size = 129, normalized size = 1.10 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {3\,A}{8\,a^2\,c}-\frac {B\,1{}\mathrm {i}}{8\,a^2\,c}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {B}{8\,a^2\,c}+\frac {A\,3{}\mathrm {i}}{8\,a^2\,c}\right )-\frac {B}{4\,a^2\,c}+\frac {A\,1{}\mathrm {i}}{4\,a^2\,c}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}+1\right )}-\frac {x\,\left (B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,a^2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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