Integrand size = 26, antiderivative size = 47 \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=\frac {(c-i d) x}{2 a}+\frac {i c-d}{2 f (a+i a \tan (e+f x))} \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3607, 8} \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=\frac {-d+i c}{2 f (a+i a \tan (e+f x))}+\frac {x (c-i d)}{2 a} \]
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Rule 8
Rule 3607
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.19 \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=\frac {(c-i d) \arctan (\tan (e+f x))}{2 a f}-\frac {c+i d}{2 a f (i-\tan (e+f x))} \]
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Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15
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 | \(-\frac {i d \arctan \left (\tan \left (f x +e \right )\right )}{2 f a}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c}{2 f a}+\frac {c}{2 f a \left (\tan \left (f x +e \right )-i\right )}+\frac {i d}{2 f a \left (\tan \left (f x +e \right )-i\right )}\) | \(76\) |
default | \(-\frac {i d \arctan \left (\tan \left (f x +e \right )\right )}{2 f a}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c}{2 f a}+\frac {c}{2 f a \left (\tan \left (f x +e \right )-i\right )}+\frac {i d}{2 f a \left (\tan \left (f x +e \right )-i\right )}\) | \(76\) |
norman | \(\frac {\frac {\left (-i d +c \right ) x}{2 a}-\frac {-i c +d}{2 a f}+\frac {\left (i d +c \right ) \tan \left (f x +e \right )}{2 a f}+\frac {\left (-i d +c \right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{2 a}}{1+\tan ^{2}\left (f x +e \right )}\) | \(81\) |
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none
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=\frac {{\left (2 \, {\left (c - i \, d\right )} f x e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a f} \]
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Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.85 \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=\begin {cases} \frac {\left (i c - d\right ) e^{- 2 i e} e^{- 2 i f x}}{4 a f} & \text {for}\: a f e^{2 i e} \neq 0 \\x \left (- \frac {c - i d}{2 a} + \frac {\left (c e^{2 i e} + c - i d e^{2 i e} + i d\right ) e^{- 2 i e}}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {x \left (c - i d\right )}{2 a} \]
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Exception generated. \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (35) = 70\).
Time = 0.36 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.77 \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=-\frac {\frac {{\left (-i \, c - d\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a} + \frac {{\left (i \, c + d\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a} + \frac {-i \, c \tan \left (f x + e\right ) - d \tan \left (f x + e\right ) - 3 \, c - i \, d}{a {\left (\tan \left (f x + e\right ) - i\right )}}}{4 \, f} \]
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Time = 5.77 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \frac {c+d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx=-\frac {x\,\left (d+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a}+\frac {-\frac {d}{2\,a}+\frac {c\,1{}\mathrm {i}}{2\,a}}{f\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \]
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