Optimal. Leaf size=45 \[ \frac {A x}{2 a c}-\frac {\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f} \]
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Rubi [A]
time = 0.09, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {3669, 74, 653,
211} \begin {gather*} \frac {A x}{2 a c}-\frac {\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 211
Rule 653
Rule 3669
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))} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^2 (c-i c x)^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{\left (a c+a c x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f}+\frac {A \text {Subst}\left (\int \frac {1}{a c+a c x^2} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {A x}{2 a c}-\frac {\cos ^2(e+f x) (B-A \tan (e+f x))}{2 a c f}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 43, normalized size = 0.96 \begin {gather*} \frac {-2 B \cos ^2(e+f x)+A (2 (e+f x)+\sin (2 (e+f x)))}{4 a c f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 82, normalized size = 1.82
method | result | size |
risch | \(\frac {A x}{2 a c}-\frac {B \cos \left (2 f x +2 e \right )}{4 a c f}+\frac {A \sin \left (2 f x +2 e \right )}{4 a c f}\) | \(54\) |
norman | \(\frac {\frac {A x}{2 a c}-\frac {B}{2 a c f}+\frac {A \tan \left (f x +e \right )}{2 a c f}+\frac {A x \left (\tan ^{2}\left (f x +e \right )\right )}{2 a c}}{1+\tan ^{2}\left (f x +e \right )}\) | \(73\) |
derivativedivides | \(\frac {-\frac {i A \ln \left (-i+\tan \left (f x +e \right )\right )}{4}-\frac {-\frac {A}{4}-\frac {i B}{4}}{-i+\tan \left (f x +e \right )}+\frac {i A \ln \left (i+\tan \left (f x +e \right )\right )}{4}-\frac {-\frac {A}{4}+\frac {i B}{4}}{i+\tan \left (f x +e \right )}}{f a c}\) | \(82\) |
default | \(\frac {-\frac {i A \ln \left (-i+\tan \left (f x +e \right )\right )}{4}-\frac {-\frac {A}{4}-\frac {i B}{4}}{-i+\tan \left (f x +e \right )}+\frac {i A \ln \left (i+\tan \left (f x +e \right )\right )}{4}-\frac {-\frac {A}{4}+\frac {i B}{4}}{i+\tan \left (f x +e \right )}}{f a c}\) | \(82\) |
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 [C] Result contains complex when optimal does not.
time = 3.80, size = 61, normalized size = 1.36 \begin {gather*} \frac {{\left (4 \, A f x e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + i \, A - B\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 165, normalized size = 3.67 \begin {gather*} \frac {A x}{2 a c} + \begin {cases} \frac {\left (\left (8 i A a c f - 8 B a c f\right ) e^{- 2 i f x} + \left (- 8 i A a c f e^{4 i e} - 8 B a c f e^{4 i e}\right ) e^{2 i f x}\right ) e^{- 2 i e}}{64 a^{2} c^{2} f^{2}} & \text {for}\: a^{2} c^{2} f^{2} e^{2 i e} \neq 0 \\x \left (- \frac {A}{2 a c} + \frac {\left (A e^{4 i e} + 2 A e^{2 i e} + A - i B e^{4 i e} + i B\right ) e^{- 2 i e}}{4 a c}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 53, normalized size = 1.18 \begin {gather*} \frac {\frac {{\left (f x + e\right )} A}{a c} + \frac {A \tan \left (f x + e\right ) - B}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} a c}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.45, size = 40, normalized size = 0.89 \begin {gather*} \frac {\frac {A\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {B\,\cos \left (2\,e+2\,f\,x\right )}{2}+A\,f\,x}{2\,a\,c\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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