Optimal. Leaf size=161 \[ -\frac{i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{2 a d}+\frac{\sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}-\frac{i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\log (c+d x)}{2 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.277704, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3726, 3303, 3299, 3302} \[ -\frac{i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{2 a d}+\frac{\sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}-\frac{i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\log (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3726
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+i a \cot (e+f x))} \, dx &=\frac{\log (c+d x)}{2 a d}+\frac{i \int \frac{\sin \left (2 \left (e+\frac{\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{2 a}+\frac{\int \frac{\cos \left (2 \left (e+\frac{\pi }{2}\right )+2 f x\right )}{c+d x} \, dx}{2 a}\\ &=\frac{\log (c+d x)}{2 a d}-\frac{\left (i \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}-\frac{\cos \left (2 e-\frac{2 c f}{d}\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}-\frac{\left (i \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}+\frac{\sin \left (2 e-\frac{2 c f}{d}\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}\\ &=-\frac{\cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{2 a d}+\frac{\log (c+d x)}{2 a d}-\frac{i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{2 a d}+\frac{\sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.274129, size = 77, normalized size = 0.48 \[ \frac{\log (c+d x)-\left (\text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right )+i \text{Si}\left (\frac{2 f (c+d x)}{d}\right )\right ) \left (\cos \left (2 e-\frac{2 c f}{d}\right )+i \sin \left (2 e-\frac{2 c f}{d}\right )\right )}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.142, size = 67, normalized size = 0.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{2\,ad}}+{\frac{1}{2\,ad}{{\rm e}^{{\frac{-2\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,-2\,ifx-2\,ie-2\,{\frac{icf-ide}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.38653, size = 150, normalized size = 0.93 \begin{align*} \frac{f \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (-\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - i \, f E_{1}\left (-\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + f \log \left ({\left (f x + e\right )} d - d e + c f\right )}{2 \, a d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68559, size = 117, normalized size = 0.73 \begin{align*} -\frac{{\rm Ei}\left (\frac{2 i \, d f x + 2 i \, c f}{d}\right ) e^{\left (\frac{2 i \, d e - 2 i \, c f}{d}\right )} - \log \left (\frac{d x + c}{d}\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26013, size = 495, normalized size = 3.07 \begin{align*} -\frac{\cos \left (\frac{2 \, c f}{d}\right ) \cos \left (e\right )^{2} \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - i \, \cos \left (e\right )^{2} \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac{2 \, c f}{d}\right ) + 2 i \, \cos \left (\frac{2 \, c f}{d}\right ) \cos \left (e\right ) \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sin \left (e\right ) + 2 \, \cos \left (e\right ) \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac{2 \, c f}{d}\right ) \sin \left (e\right ) - \cos \left (\frac{2 \, c f}{d}\right ) \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sin \left (e\right )^{2} + i \, \operatorname{Ci}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac{2 \, c f}{d}\right ) \sin \left (e\right )^{2} + i \, \cos \left (\frac{2 \, c f}{d}\right ) \cos \left (e\right )^{2} \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) + \cos \left (e\right )^{2} \sin \left (\frac{2 \, c f}{d}\right ) \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - 2 \, \cos \left (\frac{2 \, c f}{d}\right ) \cos \left (e\right ) \sin \left (e\right ) \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) + 2 i \, \cos \left (e\right ) \sin \left (\frac{2 \, c f}{d}\right ) \sin \left (e\right ) \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - i \, \cos \left (\frac{2 \, c f}{d}\right ) \sin \left (e\right )^{2} \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - \sin \left (\frac{2 \, c f}{d}\right ) \sin \left (e\right )^{2} \operatorname{Si}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - \log \left (d x + c\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]