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

Optimal. Leaf size=62 \[ \frac {\text {Ci}(b x) \sin (b x)}{b^2}-\frac {\text {Si}(2 b x)}{2 b^2}+\frac {\sin (b x) \cos (b x)}{2 b^2}-\frac {x \text {Ci}(b x) \cos (b x)}{b}+\frac {x}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6520, 12, 2635, 8, 6512, 4406, 3299} \[ \frac {\text {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {\text {Si}(2 b x)}{2 b^2}+\frac {\sin (b x) \cos (b x)}{2 b^2}-\frac {x \text {CosIntegral}(b x) \cos (b x)}{b}+\frac {x}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 6512

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Sin[a + b*x]*CosIntegral[c + d
*x])/b, x] - Dist[d/b, Int[(Sin[a + b*x]*Cos[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 6520

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

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 44, normalized size = 0.71 \[ \frac {\text {Ci}(b x) (4 \sin (b x)-4 b x \cos (b x))-2 \text {Si}(2 b x)+2 b x+\sin (2 b x)}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x + CosIntegral[b*x]*(-4*b*x*Cos[b*x] + 4*Sin[b*x]) + Sin[2*b*x] - 2*SinIntegral[2*b*x])/(4*b^2)

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fricas [F]  time = 2.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Ci}\left (b x\right ) \sin \left (b x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*cos_integral(b*x)*sin(b*x), x)

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giac [C]  time = 0.33, size = 126, normalized size = 2.03 \[ -{\left (\frac {x \cos \left (b x\right )}{b} - \frac {\sin \left (b x\right )}{b^{2}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {2 \, b x \tan \left (b x\right )^{2} - \Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 2 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} + 2 \, b x - \Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) + \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) - 2 \, \operatorname {Si}\left (2 \, b x\right ) + 2 \, \tan \left (b x\right )}{4 \, {\left (b^{2} \tan \left (b x\right )^{2} + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*cos(b*x)/b - sin(b*x)/b^2)*cos_integral(b*x) + 1/4*(2*b*x*tan(b*x)^2 - imag_part(cos_integral(2*b*x))*tan(
b*x)^2 + imag_part(cos_integral(-2*b*x))*tan(b*x)^2 - 2*sin_integral(2*b*x)*tan(b*x)^2 + 2*b*x - imag_part(cos
_integral(2*b*x)) + imag_part(cos_integral(-2*b*x)) - 2*sin_integral(2*b*x) + 2*tan(b*x))/(b^2*tan(b*x)^2 + b^
2)

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maple [A]  time = 0.03, size = 45, normalized size = 0.73 \[ \frac {\Ci \left (b x \right ) \left (\sin \left (b x \right )-b x \cos \left (b x \right )\right )-\frac {\Si \left (2 b x \right )}{2}+\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Ci(b*x)*sin(b*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*cosint(b*x)*sin(b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sin(b*x)*Ci(b*x), x)

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