Integrand size = 31, antiderivative size = 37 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\frac {x}{2 a c}+\frac {\cos (e+f x) \sin (e+f x)}{2 a c f} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 2715, 8} \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\frac {\sin (e+f x) \cos (e+f x)}{2 a c f}+\frac {x}{2 a c} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^2(e+f x) \, dx}{a c} \\ & = \frac {\cos (e+f x) \sin (e+f x)}{2 a c f}+\frac {\int 1 \, dx}{2 a c} \\ & = \frac {x}{2 a c}+\frac {\cos (e+f x) \sin (e+f x)}{2 a c f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\frac {2 (e+f x)+\sin (2 (e+f x))}{4 a c f} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {x}{2 a c}+\frac {\sin \left (2 f x +2 e \right )}{4 a c f}\) | \(31\) |
norman | \(\frac {\frac {x}{2 a c}+\frac {\tan \left (f x +e \right )}{2 a c f}+\frac {x \left (\tan ^{2}\left (f x +e \right )\right )}{2 a c}}{1+\tan ^{2}\left (f x +e \right )}\) | \(58\) |
derivativedivides | \(\frac {\arctan \left (\tan \left (f x +e \right )\right )}{2 f a c}+\frac {1}{4 f a c \left (\tan \left (f x +e \right )+i\right )}+\frac {1}{4 f a c \left (\tan \left (f x +e \right )-i\right )}\) | \(64\) |
default | \(\frac {\arctan \left (\tan \left (f x +e \right )\right )}{2 f a c}+\frac {1}{4 f a c \left (\tan \left (f x +e \right )+i\right )}+\frac {1}{4 f a c \left (\tan \left (f x +e \right )-i\right )}\) | \(64\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\frac {{\left (4 \, f x e^{\left (2 i \, f x + 2 i \, e\right )} - i \, e^{\left (4 i \, f x + 4 i \, e\right )} + i\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.16 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\begin {cases} \frac {\left (- 8 i a c f e^{4 i e} e^{2 i f x} + 8 i a c f e^{- 2 i f x}\right ) e^{- 2 i e}}{64 a^{2} c^{2} f^{2}} & \text {for}\: a^{2} c^{2} f^{2} e^{2 i e} \neq 0 \\x \left (\frac {\left (e^{4 i e} + 2 e^{2 i e} + 1\right ) e^{- 2 i e}}{4 a c} - \frac {1}{2 a c}\right ) & \text {otherwise} \end {cases} + \frac {x}{2 a c} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\frac {\frac {f x + e}{a c} + \frac {\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} a c}}{2 \, f} \]
[In]
[Out]
Time = 5.79 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))} \, dx=\frac {\frac {\sin \left (2\,e+2\,f\,x\right )}{4\,a\,c}+\frac {f\,x}{2\,a\,c}}{f} \]
[In]
[Out]