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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 12

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

Rule 2638

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

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*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 1.00 \[ -\frac {\cos (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Ci}(b x)-\frac {x \sin (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Cos[b*x]/b^2 + (x^2*CosIntegral[b*x])/2 - (x*Sin[b*x])/(2*b)

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Ci}\left (b x\right ), x\right ) \]

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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giac [A]  time = 0.17, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} \operatorname {Ci}\left (b x\right ) - \frac {x \sin \left (b x\right )}{2 \, b} - \frac {\cos \left (b x\right )}{2 \, b^{2}} \]

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

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maple [A]  time = 0.02, size = 32, normalized size = 0.91 \[ \frac {\frac {b^{2} x^{2} \Ci \left (b x \right )}{2}-\frac {\cos \left (b x \right )}{2}-\frac {b x \sin \left (b x \right )}{2}}{b^{2}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Ci}\left (b x\right )\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \frac {x^2\,\mathrm {cosint}\left (b\,x\right )}{2}-\frac {\cos \left (b\,x\right )+b\,x\,\sin \left (b\,x\right )}{2\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosint(b*x),x)

[Out]

(x^2*cosint(b*x))/2 - (cos(b*x) + b*x*sin(b*x))/(2*b^2)

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sympy [A]  time = 1.12, size = 53, normalized size = 1.51 \[ - \frac {x^{2} \log {\left (b x \right )}}{2} + \frac {x^{2} \log {\left (b^{2} x^{2} \right )}}{4} + \frac {x^{2} \operatorname {Ci}{\left (b x \right )}}{2} - \frac {x \sin {\left (b x \right )}}{2 b} - \frac {\cos {\left (b x \right )}}{2 b^{2}} \]

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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