3.1.56 \(\int x \sin (a+b x) \text {Si}(a+b x) \, dx\) [56]

Optimal. Leaf size=97 \[ -\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\text {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2} \]

[Out]

1/2*Ci(2*b*x+2*a)/b^2-1/4*cos(2*b*x+2*a)/b^2-1/2*ln(b*x+a)/b^2-x*cos(b*x+a)*Si(b*x+a)/b-1/2*a*Si(2*b*x+2*a)/b^
2+Si(b*x+a)*sin(b*x+a)/b^2

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Rubi [A]
time = 0.20, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6648, 4669, 6873, 6874, 2718, 3380, 6652, 3393, 3383} \begin {gather*} \frac {\text {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {\cos (2 a+2 b x)}{4 b^2}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sin[a + b*x]*SinIntegral[a + b*x],x]

[Out]

-1/4*Cos[2*a + 2*b*x]/b^2 + CosIntegral[2*a + 2*b*x]/(2*b^2) - Log[a + b*x]/(2*b^2) - (x*Cos[a + b*x]*SinInteg
ral[a + b*x])/b + (Sin[a + b*x]*SinIntegral[a + b*x])/b^2 - (a*SinIntegral[2*a + 2*b*x])/(2*b^2)

Rule 2718

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

Rule 3380

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 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 4669

Int[Cos[w_]^(p_.)*(u_.)*Sin[v_]^(p_.), x_Symbol] :> Dist[1/2^p, Int[u*Sin[2*v]^p, x], x] /; EqQ[w, v] && Integ
erQ[p]

Rule 6648

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

Rule 6652

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

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x \sin (a+b x) \text {Si}(a+b x) \, dx &=-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\int \cos (a+b x) \text {Si}(a+b x) \, dx}{b}+\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx\\ &=-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x} \, dx-\frac {\int \frac {\sin ^2(a+b x)}{a+b x} \, dx}{b}\\ &=-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \frac {x \sin (2 a+2 b x)}{a+b x} \, dx-\frac {\int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx+\frac {\int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=\frac {\text {Ci}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {\int \sin (2 a+2 b x) \, dx}{2 b}+\frac {a \int \frac {\sin (2 a+2 b x)}{-a-b x} \, dx}{2 b}\\ &=-\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\text {Ci}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 71, normalized size = 0.73 \begin {gather*} -\frac {\cos (2 (a+b x))-2 \text {CosIntegral}(2 (a+b x))+2 \log (a+b x)+4 (b x \cos (a+b x)-\sin (a+b x)) \text {Si}(a+b x)+2 a \text {Si}(2 (a+b x))}{4 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sin[a + b*x]*SinIntegral[a + b*x],x]

[Out]

-1/4*(Cos[2*(a + b*x)] - 2*CosIntegral[2*(a + b*x)] + 2*Log[a + b*x] + 4*(b*x*Cos[a + b*x] - Sin[a + b*x])*Sin
Integral[a + b*x] + 2*a*SinIntegral[2*(a + b*x)])/b^2

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Maple [A]
time = 0.52, size = 82, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {\sinIntegral \left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a \sinIntegral \left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\cosineIntegral \left (2 b x +2 a \right )}{2}-\frac {\left (\cos ^{2}\left (b x +a \right )\right )}{2}}{b^{2}}\)	\(82\)
default	\(\frac {\sinIntegral \left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a \sinIntegral \left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\cosineIntegral \left (2 b x +2 a \right )}{2}-\frac {\left (\cos ^{2}\left (b x +a \right )\right )}{2}}{b^{2}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*Si(b*x+a)*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin_integral(b*x+a)*sin(b*x+a),x, algorithm="maxima")

[Out]

integrate(x*sin(b*x + a)*sin_integral(b*x + a), x)

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Fricas [A]
time = 0.37, size = 88, normalized size = 0.91 \begin {gather*} -\frac {4 \, b x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + 2 \, \cos \left (b x + a\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) - \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) - \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) + 2 \, \log \left (b x + a\right )}{4 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin_integral(b*x+a)*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/4*(4*b*x*cos(b*x + a)*sin_integral(b*x + a) + 2*cos(b*x + a)^2 + 2*a*sin_integral(2*b*x + 2*a) - 4*sin(b*x
+ a)*sin_integral(b*x + a) - cos_integral(2*b*x + 2*a) - cos_integral(-2*b*x - 2*a) + 2*log(b*x + a))/b^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sin {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*Si(b*x+a)*sin(b*x+a),x)

[Out]

Integral(x*sin(a + b*x)*Si(a + b*x), x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.42, size = 507, normalized size = 5.23 \begin {gather*} -{\left (\frac {x \cos \left (b x + a\right )}{b} - \frac {\sin \left (b x + a\right )}{b^{2}}\right )} \operatorname {Si}\left (b x + a\right ) - \frac {a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} + a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (a\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (a\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (a\right )^{2} + \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} - \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (a\right )^{2} + a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - \tan \left (b x\right )^{2} - 4 \, \tan \left (b x\right ) \tan \left (a\right ) - \tan \left (a\right )^{2} + 2 \, \log \left ({\left | b x + a \right |}\right ) - \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 1}{4 \, {\left (b^{2} \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + b^{2} \tan \left (b x\right )^{2} + b^{2} \tan \left (a\right )^{2} + b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sin_integral(b*x+a)*sin(b*x+a),x, algorithm="giac")

[Out]

-(x*cos(b*x + a)/b - sin(b*x + a)/b^2)*sin_integral(b*x + a) - 1/4*(a*imag_part(cos_integral(2*b*x + 2*a))*tan
(b*x)^2*tan(a)^2 - a*imag_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2*tan(a)^2 + 2*a*sin_integral(2*b*x + 2*a)
*tan(b*x)^2*tan(a)^2 + 2*log(abs(b*x + a))*tan(b*x)^2*tan(a)^2 - real_part(cos_integral(2*b*x + 2*a))*tan(b*x)
^2*tan(a)^2 - real_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2*tan(a)^2 + a*imag_part(cos_integral(2*b*x + 2*a
))*tan(b*x)^2 - a*imag_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2 + 2*a*sin_integral(2*b*x + 2*a)*tan(b*x)^2
+ a*imag_part(cos_integral(2*b*x + 2*a))*tan(a)^2 - a*imag_part(cos_integral(-2*b*x - 2*a))*tan(a)^2 + 2*a*sin
_integral(2*b*x + 2*a)*tan(a)^2 + tan(b*x)^2*tan(a)^2 + 2*log(abs(b*x + a))*tan(b*x)^2 - real_part(cos_integra
l(2*b*x + 2*a))*tan(b*x)^2 - real_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2 + 2*log(abs(b*x + a))*tan(a)^2 -
real_part(cos_integral(2*b*x + 2*a))*tan(a)^2 - real_part(cos_integral(-2*b*x - 2*a))*tan(a)^2 + a*imag_part(
cos_integral(2*b*x + 2*a)) - a*imag_part(cos_integral(-2*b*x - 2*a)) + 2*a*sin_integral(2*b*x + 2*a) - tan(b*x
)^2 - 4*tan(b*x)*tan(a) - tan(a)^2 + 2*log(abs(b*x + a)) - real_part(cos_integral(2*b*x + 2*a)) - real_part(co
s_integral(-2*b*x - 2*a)) + 1)/(b^2*tan(b*x)^2*tan(a)^2 + b^2*tan(b*x)^2 + b^2*tan(a)^2 + b^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\mathrm {sinint}\left (a+b\,x\right )\,\sin \left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinint(a + b*x)*sin(a + b*x),x)

[Out]

int(x*sinint(a + b*x)*sin(a + b*x), x)

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