Integrand size = 26, antiderivative size = 47 \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {(A-i B) x}{2 a}+\frac {i A-B}{2 d (a+i a \tan (c+d x))} \]
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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} \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {-B+i A}{2 d (a+i a \tan (c+d x))}+\frac {x (A-i B)}{2 a} \]
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Rule 8
Rule 3607
Rubi steps \begin{align*} \text {integral}& = \frac {i A-B}{2 d (a+i a \tan (c+d x))}+\frac {(A-i B) \int 1 \, dx}{2 a} \\ & = \frac {(A-i B) x}{2 a}+\frac {i A-B}{2 d (a+i a \tan (c+d x))} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.19 \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {(A-i B) \arctan (\tan (c+d x))}{2 a d}-\frac {A+i B}{2 a d (i-\tan (c+d x))} \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {i x B}{2 a}+\frac {x A}{2 a}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )} B}{4 a d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A}{4 a d}\) | \(54\) |
derivativedivides | \(\frac {A}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {i B}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {A \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}-\frac {i B \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}\) | \(76\) |
default | \(\frac {A}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {i B}{2 d a \left (\tan \left (d x +c \right )-i\right )}+\frac {A \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}-\frac {i B \arctan \left (\tan \left (d x +c \right )\right )}{2 d a}\) | \(76\) |
norman | \(\frac {\frac {\left (-i B +A \right ) x}{2 a}-\frac {-i A +B}{2 a d}+\frac {\left (-i B +A \right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{2 a}+\frac {\left (i B +A \right ) \tan \left (d x +c \right )}{2 a d}}{1+\tan ^{2}\left (d x +c \right )}\) | \(81\) |
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none
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {{\left (2 \, {\left (A - i \, B\right )} d x e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A - B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \]
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Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.85 \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\begin {cases} \frac {\left (i A - B\right ) e^{- 2 i c} e^{- 2 i d x}}{4 a d} & \text {for}\: a d e^{2 i c} \neq 0 \\x \left (- \frac {A - i B}{2 a} + \frac {\left (A e^{2 i c} + A - i B e^{2 i c} + i B\right ) e^{- 2 i c}}{2 a}\right ) & \text {otherwise} \end {cases} + \frac {x \left (A - i B\right )}{2 a} \]
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Exception generated. \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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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 = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.77 \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {\frac {{\left (-i \, A - B\right )} \log \left (\tan \left (d x + c\right ) + i\right )}{a} + \frac {{\left (i \, A + B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a} + \frac {-i \, A \tan \left (d x + c\right ) - B \tan \left (d x + c\right ) - 3 \, A - i \, B}{a {\left (\tan \left (d x + c\right ) - i\right )}}}{4 \, d} \]
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Time = 7.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \frac {A+B \tan (c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {-\frac {B}{2\,a}+\frac {A\,1{}\mathrm {i}}{2\,a}}{d\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {x\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a} \]
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