3.4 \(\int x \text {Si}(b x) \, dx\)

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

[Out]

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

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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 = {6503, 12, 3296, 2637} \[ -\frac {\sin (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)+\frac {x \cos (b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x*SinIntegral[b*x],x]

[Out]

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

Rule 12

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 6503

Int[((c_.) + (d_.)*(x_))^(m_.)*SinIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*SinIntegr
al[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Sin[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 {Si}(b x) \, dx &=\frac {1}{2} x^2 \text {Si}(b x)-\frac {1}{2} b \int \frac {x \sin (b x)}{b} \, dx\\ &=\frac {1}{2} x^2 \text {Si}(b x)-\frac {1}{2} \int x \sin (b x) \, dx\\ &=\frac {x \cos (b x)}{2 b}+\frac {1}{2} x^2 \text {Si}(b x)-\frac {\int \cos (b x) \, dx}{2 b}\\ &=\frac {x \cos (b x)}{2 b}-\frac {\sin (b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[x*SinIntegral[b*x],x]

[Out]

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

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fricas [F]  time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Si}\left (b x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*sin_integral(b*x), x)

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giac [A]  time = 0.73, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} \operatorname {Si}\left (b x\right ) + \frac {x \cos \left (b x\right )}{2 \, b} - \frac {\sin \left (b x\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Si(b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Si(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinint(b*x),x)

[Out]

(x^2*sinint(b*x))/2 - (sin(b*x) - b*x*cos(b*x))/(2*b^2)

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sympy [A]  time = 0.68, size = 29, normalized size = 0.83 \[ \frac {x^{2} \operatorname {Si}{\left (b x \right )}}{2} + \frac {x \cos {\left (b x \right )}}{2 b} - \frac {\sin {\left (b x \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Si(b*x),x)

[Out]

x**2*Si(b*x)/2 + x*cos(b*x)/(2*b) - sin(b*x)/(2*b**2)

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