3.72 \(\int x \text{CosIntegral}(b x) \, dx\)

Optimal. Leaf size=35 \[ -\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)-\frac{x \sin (b x)}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0235583, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6504, 12, 3296, 2638} \[ -\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)-\frac{x \sin (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x*CosIntegral[b*x],x]

[Out]

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

Rule 6504

Int[CosIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*CosIntegr
al[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Cos[a + b*x])/(a + b*x), x], x] /; F
reeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0079792, size = 35, normalized size = 1. \[ -\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)-\frac{x \sin (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*CosIntegral[b*x],x]

[Out]

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

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Maple [A]  time = 0.05, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it Ci} \left ( bx \right ) }{2}}-{\frac{\sin \left ( bx \right ) bx}{2}}-{\frac{\cos \left ( bx \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Ci(b*x),x)

[Out]

1/b^2*(1/2*b^2*x^2*Ci(b*x)-1/2*sin(b*x)*b*x-1/2*cos(b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(b*x),x, algorithm="maxima")

[Out]

integrate(x*Ci(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(b*x),x, algorithm="fricas")

[Out]

integral(x*cos_integral(b*x), x)

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Sympy [A]  time = 1.74918, size = 53, normalized size = 1.51 \begin{align*} - \frac{x^{2} \log{\left (b x \right )}}{2} + \frac{x^{2} \log{\left (b^{2} x^{2} \right )}}{4} + \frac{x^{2} \operatorname{Ci}{\left (b x \right )}}{2} - \frac{x \sin{\left (b x \right )}}{2 b} - \frac{\cos{\left (b x \right )}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(b*x),x)

[Out]

-x**2*log(b*x)/2 + x**2*log(b**2*x**2)/4 + x**2*Ci(b*x)/2 - x*sin(b*x)/(2*b) - cos(b*x)/(2*b**2)

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Giac [A]  time = 1.16488, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{Ci}\left (b x\right ) - \frac{x \sin \left (b x\right )}{2 \, b} - \frac{\cos \left (b x\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Ci(b*x),x, algorithm="giac")

[Out]

1/2*x^2*cos_integral(b*x) - 1/2*x*sin(b*x)/b - 1/2*cos(b*x)/b^2