3.115 \(\int \frac{\cos (b x) \text{CosIntegral}(b x)}{x^2} \, dx\)

Optimal. Leaf size=50 \[ -b \text{CannotIntegrate}\left (\frac{\text{CosIntegral}(b x) \sin (b x)}{x},x\right )-\frac{\text{CosIntegral}(b x) \cos (b x)}{x}-b \text{Si}(2 b x)-\frac{\cos ^2(b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0930662, 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 = {} \[ \int \frac{\cos (b x) \text{CosIntegral}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[b*x]*CosIntegral[b*x])/x^2,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\cos (b x) \text{Ci}(b x)}{x^2} \, dx &=-\frac{\cos (b x) \text{Ci}(b x)}{x}+b \int \frac{\cos ^2(b x)}{b x^2} \, dx-b \int \frac{\text{Ci}(b x) \sin (b x)}{x} \, dx\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{x}-b \int \frac{\text{Ci}(b x) \sin (b x)}{x} \, dx+\int \frac{\cos ^2(b x)}{x^2} \, dx\\ &=-\frac{\cos ^2(b x)}{x}-\frac{\cos (b x) \text{Ci}(b x)}{x}-b \int \frac{\text{Ci}(b x) \sin (b x)}{x} \, dx+(2 b) \int -\frac{\sin (2 b x)}{2 x} \, dx\\ &=-\frac{\cos ^2(b x)}{x}-\frac{\cos (b x) \text{Ci}(b x)}{x}-b \int \frac{\text{Ci}(b x) \sin (b x)}{x} \, dx-b \int \frac{\sin (2 b x)}{x} \, dx\\ &=-\frac{\cos ^2(b x)}{x}-\frac{\cos (b x) \text{Ci}(b x)}{x}-b \text{Si}(2 b x)-b \int \frac{\text{Ci}(b x) \sin (b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.444453, size = 0, normalized size = 0. \[ \int \frac{\cos (b x) \text{CosIntegral}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^2,x]

[Out]

Integrate[(Cos[b*x]*CosIntegral[b*x])/x^2, x]

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Maple [A]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x)*cos(b*x)/x^2,x)

[Out]

int(Ci(b*x)*cos(b*x)/x^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*cos(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(Ci(b*x)*cos(b*x)/x^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (b x\right ) \operatorname{Ci}\left (b x\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*cos(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(cos(b*x)*cos_integral(b*x)/x^2, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (b x \right )} \operatorname{Ci}{\left (b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*cos(b*x)/x**2,x)

[Out]

Integral(cos(b*x)*Ci(b*x)/x**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \cos \left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*cos(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(Ci(b*x)*cos(b*x)/x^2, x)