\(\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx\) [107]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \text {Int}\left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x},x\right ) \] Output:

-1/2*b*cos(b*x)^2/x-1/4*b*cos(2*b*x)/x-1/2*b*cos(b*x)*Ci(b*x)/x-1/2*Ci(b*x 
)*sin(b*x)/x^2-1/8*sin(2*b*x)/x^2-b^2*Si(2*b*x)-1/2*b^2*Defer(Int)(Ci(b*x) 
*sin(b*x)/x,x)
 

Mathematica [N/A]

Not integrable

Time = 1.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \] Input:

Integrate[(CosIntegral[b*x]*Sin[b*x])/x^3,x]
 

Output:

Integrate[(CosIntegral[b*x]*Sin[b*x])/x^3, x]
 

Rubi [N/A]

Not integrable

Time = 1.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx\)

\(\Big \downarrow \) 7076

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{2} b \int \frac {\cos (b x) \sin (b x)}{b x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\cos (b x) \sin (b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\sin (2 b x)}{2 x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \int \frac {\cos (2 b x)}{x^2}dx-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \int \frac {\sin \left (2 b x+\frac {\pi }{2}\right )}{x^2}dx-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \left (2 b \int -\frac {\sin (2 b x)}{x}dx-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \left (-2 b \int \frac {\sin (2 b x)}{x}dx-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \left (-2 b \int \frac {\sin (2 b x)}{x}dx-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 7070

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+b \int \frac {\cos ^2(b x)}{b x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+\int \frac {\cos ^2(b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+\int \frac {\sin \left (b x+\frac {\pi }{2}\right )^2}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3794

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+2 b \int -\frac {\sin (2 b x)}{2 x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-b \int \frac {\sin (2 b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-b \int \frac {\sin (2 b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-b \text {Si}(2 b x)-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-b \text {Si}(2 b x)-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\)

Input:

Int[(CosIntegral[b*x]*Sin[b*x])/x^3,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )}{x^{3}}d x\]

Input:

int(Ci(b*x)*sin(b*x)/x^3,x)
 

Output:

int(Ci(b*x)*sin(b*x)/x^3,x)
 

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(b*x)*sin(b*x)/x^3,x, algorithm="fricas")
 

Output:

integral(fresnel_cos(b*x)*sin(b*x)/x^3, x)
 

Sympy [N/A]

Not integrable

Time = 2.46 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \] Input:

integrate(Ci(b*x)*sin(b*x)/x**3,x)
 

Output:

Integral(sin(b*x)*Ci(b*x)/x**3, x)
 

Maxima [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(b*x)*sin(b*x)/x^3,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x)*sin(b*x)/x^3, x)
 

Giac [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \] Input:

integrate(fresnel_cos(b*x)*sin(b*x)/x^3,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)*sin(b*x)/x^3, x)
 

Mupad [N/A]

Not integrable

Time = 4.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^3} \,d x \] Input:

int((cosint(b*x)*sin(b*x))/x^3,x)
 

Output:

int((cosint(b*x)*sin(b*x))/x^3, x)
 

Reduce [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\mathit {ci} \left (b x \right ) \sin \left (b x \right )}{x^{3}}d x \] Input:

int(Ci(b*x)*sin(b*x)/x^3,x)
 

Output:

int((ci(b*x)*sin(b*x))/x**3,x)