Optimal. Leaf size=47 \[ \frac {(c-i d) x}{2 a}+\frac {i c-d}{2 f (a+i a \tan (e+f x))} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3607, 8}
\begin {gather*} \frac {-d+i c}{2 f (a+i a \tan (e+f x))}+\frac {x (c-i d)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3607
Rubi steps
\begin {align*} \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx &=\frac {i c-d}{2 f (a+i a \tan (e+f x))}+\frac {(c-i d) \int 1 \, dx}{2 a}\\ &=\frac {(c-i d) x}{2 a}+\frac {i c-d}{2 f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(102\) vs. \(2(47)=94\).
time = 0.46, size = 102, normalized size = 2.17 \begin {gather*} \frac {\cos (e+f x) (c+d \tan (e+f x)) (c-2 i c f x+d (i-2 f x)+(d-2 i d f x+c (-i+2 f x)) \tan (e+f x))}{4 a f (c \cos (e+f x)+d \sin (e+f x)) (-i+\tan (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 70, normalized size = 1.49
method | result | size |
risch | \(-\frac {i x d}{2 a}+\frac {x c}{2 a}-\frac {{\mathrm e}^{-2 i \left (f x +e \right )} d}{4 a f}+\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} c}{4 a f}\) | \(54\) |
derivativedivides | \(\frac {-\frac {i \left (i d -c \right ) \ln \left (\tan \left (f x +e \right )+i\right )}{4}+\left (-\frac {i c}{4}-\frac {d}{4}\right ) \ln \left (\tan \left (f x +e \right )-i\right )-\frac {-\frac {c}{2}-\frac {i d}{2}}{\tan \left (f x +e \right )-i}}{f a}\) | \(70\) |
default | \(\frac {-\frac {i \left (i d -c \right ) \ln \left (\tan \left (f x +e \right )+i\right )}{4}+\left (-\frac {i c}{4}-\frac {d}{4}\right ) \ln \left (\tan \left (f x +e \right )-i\right )-\frac {-\frac {c}{2}-\frac {i d}{2}}{\tan \left (f x +e \right )-i}}{f a}\) | \(70\) |
norman | \(\frac {\frac {\left (-i d +c \right ) x}{2 a}-\frac {-i c +d}{2 a f}+\frac {\left (i d +c \right ) \tan \left (f x +e \right )}{2 a f}+\frac {\left (-i d +c \right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{2 a}}{1+\tan ^{2}\left (f x +e \right )}\) | \(81\) |
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 = 0.94, size = 44, normalized size = 0.94 \begin {gather*} \frac {{\left (2 \, {\left (c - i \, d\right )} f x e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 87, normalized size = 1.85 \begin {gather*} \begin {cases} \frac {\left (i c - d\right ) e^{- 2 i e} e^{- 2 i f x}}{4 a f} & \text {for}\: a f e^{2 i e} \neq 0 \\x \left (- \frac {c - i d}{2 a} + \frac {\left (c e^{2 i e} + c - i d e^{2 i e} + i d\right ) e^{- 2 i e}}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {x \left (c - i d\right )}{2 a} \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. 90 vs. \(2 (36) = 72\).
time = 0.45, size = 90, normalized size = 1.91 \begin {gather*} -\frac {\frac {{\left (i \, c + d\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a} + \frac {{\left (-i \, c - d\right )} \log \left (-i \, \tan \left (f x + e\right ) + 1\right )}{a} + \frac {-i \, c \tan \left (f x + e\right ) - d \tan \left (f x + e\right ) - 3 \, c - i \, d}{a {\left (\tan \left (f x + e\right ) - i\right )}}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.01, size = 45, normalized size = 0.96 \begin {gather*} -\frac {x\,\left (d+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a}+\frac {-\frac {d}{2\,a}+\frac {c\,1{}\mathrm {i}}{2\,a}}{f\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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