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

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]

[In]

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

[Out]

-1/2*(b*Cos[b*x]^2)/x - (b*Cos[2*b*x])/(4*x) - (b*Cos[b*x]*CosIntegral[b*x])/(2*x) - (CosIntegral[b*x]*Sin[b*x
])/(2*x^2) - Sin[2*b*x]/(8*x^2) - b^2*SinIntegral[2*b*x] - (b^2*Defer[Int][(CosIntegral[b*x]*Sin[b*x])/x, x])/
2

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\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 {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} \int \frac {\cos (b x) \sin (b x)}{x^3} \, dx+\frac {1}{2} b^2 \int \frac {\cos ^2(b x)}{b x^2} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx \\ & = -\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} \int \frac {\sin (2 b x)}{2 x^3} \, dx+\frac {1}{2} b \int \frac {\cos ^2(b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx \\ & = -\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx+b^2 \int -\frac {\sin (2 b x)}{2 x} \, dx \\ & = -\frac {b \cos ^2(b x)}{2 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}+\frac {1}{4} b \int \frac {\cos (2 b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\sin (2 b x)}{x} \, 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}-\frac {1}{2} b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\sin (2 b x)}{x} \, 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 \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]

[In]

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

[Out]

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 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(sin(b*x)*Ci(b*x)/x**3, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^3} \,d x \]

[In]

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

[Out]

int((cosint(b*x)*sin(b*x))/x^3, x)