Optimal. Leaf size=46 \[ a (c-i d) x-\frac {a (i c+d) \log (\cos (e+f x))}{f}+\frac {i a d \tan (e+f x)}{f} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3606, 3556}
\begin {gather*} -\frac {a (d+i c) \log (\cos (e+f x))}{f}+a x (c-i d)+\frac {i a d \tan (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c+d \tan (e+f x)) \, dx &=a (c-i d) x+\frac {i a d \tan (e+f x)}{f}+(a (i c+d)) \int \tan (e+f x) \, dx\\ &=a (c-i d) x-\frac {a (i c+d) \log (\cos (e+f x))}{f}+\frac {i a d \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 66, normalized size = 1.43 \begin {gather*} a c x-\frac {i a d \text {ArcTan}(\tan (e+f x))}{f}-\frac {i a c \log (\cos (e+f x))}{f}-\frac {a d \log (\cos (e+f x))}{f}+\frac {i a d \tan (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 50, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {a \left (i \tan \left (f x +e \right ) d +\frac {\left (i c +d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-i d +c \right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(50\) |
default | \(\frac {a \left (i \tan \left (f x +e \right ) d +\frac {\left (i c +d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-i d +c \right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(50\) |
norman | \(\left (-i a d +a c \right ) x +\frac {i a d \tan \left (f x +e \right )}{f}+\frac {\left (i a c +a d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}\) | \(52\) |
risch | \(\frac {2 i a d e}{f}-\frac {2 a c e}{f}-\frac {2 a d}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) d}{f}-\frac {i a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c}{f}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 55, normalized size = 1.20 \begin {gather*} -\frac {-2 i \, a d \tan \left (f x + e\right ) - 2 \, {\left (a c - i \, a d\right )} {\left (f x + e\right )} + {\left (-i \, a c - a d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 67, normalized size = 1.46 \begin {gather*} -\frac {2 \, a d - {\left (-i \, a c - a d + {\left (-i \, a c - a d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 53, normalized size = 1.15 \begin {gather*} - \frac {2 a d}{f e^{2 i e} e^{2 i f x} + f} - \frac {i a \left (c - i d\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 110 vs. \(2 (42) = 84\).
time = 0.46, size = 110, normalized size = 2.39 \begin {gather*} \frac {-i \, a c e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - a d e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - i \, a c \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - a d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 2 \, a d}{f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.88, size = 38, normalized size = 0.83 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (a\,d+a\,c\,1{}\mathrm {i}\right )}{f}+\frac {a\,d\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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