3.107 \(\int \frac{\text{CosIntegral}(b x) \sin (b x)}{x^3} \, dx\)

Optimal. Leaf size=102 \[ -\frac{1}{2} b^2 \text{CannotIntegrate}\left (\frac{\text{CosIntegral}(b x) \sin (b x)}{x},x\right )-b^2 \text{Si}(2 b x)-\frac{\text{CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac{b \text{CosIntegral}(b x) \cos (b x)}{2 x}-\frac{\sin (2 b x)}{8 x^2}-\frac{b \cos ^2(b x)}{2 x}-\frac{b \cos (2 b x)}{4 x} \]

[Out]

-(b^2*CannotIntegrate[(CosIntegral[b*x]*Sin[b*x])/x, x])/2 - (b*Cos[b*x]^2)/(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]

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Rubi [A]  time = 0.250831, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(b*Cos[b*x]^2)/(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*} \int \frac{\text{Ci}(b x) \sin (b x)}{x^3} \, dx &=-\frac{\text{Ci}(b x) \sin (b x)}{2 x^2}+\frac{1}{2} b \int \frac{\cos (b x) \text{Ci}(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) \text{Ci}(b x)}{2 x}-\frac{\text{Ci}(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{\text{Ci}(b x) \sin (b x)}{x} \, dx\\ &=-\frac{b \cos (b x) \text{Ci}(b x)}{2 x}-\frac{\text{Ci}(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{\text{Ci}(b x) \sin (b x)}{x} \, dx\\ &=-\frac{b \cos ^2(b x)}{2 x}-\frac{b \cos (b x) \text{Ci}(b x)}{2 x}-\frac{\text{Ci}(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{\text{Ci}(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) \text{Ci}(b x)}{2 x}-\frac{\text{Ci}(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{\text{Ci}(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) \text{Ci}(b x)}{2 x}-\frac{\text{Ci}(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{\text{Ci}(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) \text{Ci}(b x)}{2 x}-\frac{\text{Ci}(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{\text{Ci}(b x) \sin (b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 1.46596, size = 0, normalized size = 0. \[ \int \frac{\text{CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \sin \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*sin(b*x)/x^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Ci}\left (b x\right ) \sin \left (b x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (b x \right )} \operatorname{Ci}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(b*x)*Ci(b*x)/x**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \sin \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Ci(b*x)*sin(b*x)/x^3, x)