Optimal. Leaf size=227 \[ \frac{i f^2 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{a d^3}-\frac{f^2 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{a d^3}+\frac{f^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{i f^2 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{i f}{d^2 (c+d x) (a+i a \tan (e+f x))}-\frac{i f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+i a \tan (e+f x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.321541, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3725, 3724, 3303, 3299, 3302} \[ \frac{i f^2 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{a d^3}-\frac{f^2 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{a d^3}+\frac{f^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{i f^2 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{a d^3}+\frac{i f}{d^2 (c+d x) (a+i a \tan (e+f x))}-\frac{i f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+i a \tan (e+f x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3725
Rule 3724
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^3 (a+i a \tan (e+f x))} \, dx &=-\frac{i f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}-\frac{(i f) \int \frac{1}{(c+d x)^2 (a+i a \tan (e+f x))} \, dx}{d}\\ &=-\frac{i f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}+\frac{i f}{d^2 (c+d x) (a+i a \tan (e+f x))}+\frac{\left (i f^2\right ) \int \frac{\sin (2 e+2 f x)}{c+d x} \, dx}{a d^2}-\frac{f^2 \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{a d^2}\\ &=-\frac{i f}{2 a d^2 (c+d x)}-\frac{1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}+\frac{i f}{d^2 (c+d x) (a+i a \tan (e+f x))}+\frac{\left (i f^2 \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}{a d^2}-\frac{\left (f^2 \cos \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}{a d^2}+\frac{\left (i f^2 \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}{a d^2}+\frac{\left (f^2 \sin \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}{a d^2}\\ &=-\frac{i f}{2 a d^2 (c+d x)}-\frac{f^2 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{a d^3}+\frac{i f^2 \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{a d^3}+\frac{i f^2 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{a d^3}+\frac{f^2 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{a d^3}-\frac{1}{2 d (c+d x)^2 (a+i a \tan (e+f x))}+\frac{i f}{d^2 (c+d x) (a+i a \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 1.046, size = 285, normalized size = 1.26 \[ \frac{\sec (e+f x) \left (\cos \left (\frac{c f}{d}\right )+i \sin \left (\frac{c f}{d}\right )\right ) \left (4 f^2 (c+d x)^2 \text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sin \left (e-\frac{f (c+d x)}{d}\right )+i \cos \left (e-\frac{f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \text{Si}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac{f (c+d x)}{d}\right )-i \sin \left (e-\frac{f (c+d x)}{d}\right )\right )+d \left (-d \sin \left (f \left (x-\frac{c}{d}\right )+e\right )+d \sin \left (f \left (\frac{c}{d}+x\right )+e\right )-2 i c f \sin \left (f \left (\frac{c}{d}+x\right )+e\right )-2 i d f x \sin \left (f \left (\frac{c}{d}+x\right )+e\right )+i d \cos \left (f \left (x-\frac{c}{d}\right )+e\right )+(2 c f+2 d f x+i d) \cos \left (f \left (\frac{c}{d}+x\right )+e\right )\right )\right )}{4 a d^3 (c+d x)^2 (\tan (e+f x)-i)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.217, size = 216, normalized size = 1. \begin{align*} -{\frac{1}{4\,ad \left ( dx+c \right ) ^{2}}}+{\frac{{\frac{i}{2}}{f}^{3}{{\rm e}^{-2\,i \left ( fx+e \right ) }}x}{ad \left ({d}^{2}{x}^{2}{f}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}{{\rm e}^{-2\,i \left ( fx+e \right ) }}}{4\,ad \left ({d}^{2}{x}^{2}{f}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{\frac{i}{2}}{f}^{3}{{\rm e}^{-2\,i \left ( fx+e \right ) }}c}{a{d}^{2} \left ({d}^{2}{x}^{2}{f}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{2}}{a{d}^{3}}{{\rm e}^{{\frac{2\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,2\,ifx+2\,ie+{\frac{2\,i \left ( cf-de \right ) }{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.56882, size = 211, normalized size = 0.93 \begin{align*} -\frac{2 \, f^{3} \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{3}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) + 2 i \, f^{3} E_{3}\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^{3}}{4 \,{\left ({\left (f x + e\right )}^{2} a d^{3} + a d^{3} e^{2} - 2 \, a c d^{2} e f + a c^{2} d f^{2} - 2 \,{\left (a d^{3} e - a c d^{2} f\right )}{\left (f x + e\right )}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6385, size = 300, normalized size = 1.32 \begin{align*} \frac{{\left (2 i \, d^{2} f x + 2 i \, c d f - d^{2} -{\left (4 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )}{\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 )} + d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \,{\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )}} \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 = 1.20679, size = 732, normalized size = 3.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]