3.26 \(\int x^2 \text{Si}(a+b x)^2 \, dx\)

Optimal. Leaf size=329 \[ \frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a^2 \text{Si}(2 a+2 b x)}{b^3}+\frac{2 a^2 \text{Si}(a+b x) \cos (a+b x)}{3 b^3}+\frac{a \text{CosIntegral}(2 a+2 b x)}{b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{2 \text{Si}(2 a+2 b x)}{3 b^3}+\frac{2 a \text{Si}(a+b x) \sin (a+b x)}{3 b^3}-\frac{4 x \text{Si}(a+b x) \sin (a+b x)}{3 b^2}-\frac{2 a x \text{Si}(a+b x) \cos (a+b x)}{3 b^2}-\frac{4 \text{Si}(a+b x) \cos (a+b x)}{3 b^3}-\frac{a \log (a+b x)}{b^3}-\frac{\sin (2 a+2 b x)}{12 b^3}-\frac{a \cos (2 a+2 b x)}{3 b^3}+\frac{x \cos (2 a+2 b x)}{6 b^2}-\frac{2 \sin (a+b x) \cos (a+b x)}{3 b^3}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}+\frac{2 x^2 \text{Si}(a+b x) \cos (a+b x)}{3 b}+\frac{2 x}{3 b^2} \]

[Out]

(2*x)/(3*b^2) - (a*Cos[2*a + 2*b*x])/(3*b^3) + (x*Cos[2*a + 2*b*x])/(6*b^2) + (a*CosIntegral[2*a + 2*b*x])/b^3
- (a*Log[a + b*x])/b^3 - (2*Cos[a + b*x]*Sin[a + b*x])/(3*b^3) - Sin[2*a + 2*b*x]/(12*b^3) - (4*Cos[a + b*x]*
SinIntegral[a + b*x])/(3*b^3) + (2*a^2*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b^3) - (2*a*x*Cos[a + b*x]*SinInt
egral[a + b*x])/(3*b^2) + (2*x^2*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b) + (2*a*Sin[a + b*x]*SinIntegral[a +
b*x])/(3*b^3) - (4*x*Sin[a + b*x]*SinIntegral[a + b*x])/(3*b^2) + (a^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b^
3) - (a*x*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b^2) + (x^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b) + (2*SinInt
egral[2*a + 2*b*x])/(3*b^3) - (a^2*SinIntegral[2*a + 2*b*x])/b^3

________________________________________________________________________________________

Rubi [A]  time = 1.49501, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 19, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.583, Rules used = {6509, 6513, 4573, 6741, 6742, 2638, 3296, 2637, 3299, 6519, 2635, 8, 3312, 3302, 6511, 4406, 12, 6517, 6505} \[ \frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a^2 \text{Si}(2 a+2 b x)}{b^3}+\frac{2 a^2 \text{Si}(a+b x) \cos (a+b x)}{3 b^3}+\frac{a \text{CosIntegral}(2 a+2 b x)}{b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{2 \text{Si}(2 a+2 b x)}{3 b^3}+\frac{2 a \text{Si}(a+b x) \sin (a+b x)}{3 b^3}-\frac{4 x \text{Si}(a+b x) \sin (a+b x)}{3 b^2}-\frac{2 a x \text{Si}(a+b x) \cos (a+b x)}{3 b^2}-\frac{4 \text{Si}(a+b x) \cos (a+b x)}{3 b^3}-\frac{a \log (a+b x)}{b^3}-\frac{\sin (2 a+2 b x)}{12 b^3}-\frac{a \cos (2 a+2 b x)}{3 b^3}+\frac{x \cos (2 a+2 b x)}{6 b^2}-\frac{2 \sin (a+b x) \cos (a+b x)}{3 b^3}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}+\frac{2 x^2 \text{Si}(a+b x) \cos (a+b x)}{3 b}+\frac{2 x}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*SinIntegral[a + b*x]^2,x]

[Out]

(2*x)/(3*b^2) - (a*Cos[2*a + 2*b*x])/(3*b^3) + (x*Cos[2*a + 2*b*x])/(6*b^2) + (a*CosIntegral[2*a + 2*b*x])/b^3
- (a*Log[a + b*x])/b^3 - (2*Cos[a + b*x]*Sin[a + b*x])/(3*b^3) - Sin[2*a + 2*b*x]/(12*b^3) - (4*Cos[a + b*x]*
SinIntegral[a + b*x])/(3*b^3) + (2*a^2*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b^3) - (2*a*x*Cos[a + b*x]*SinInt
egral[a + b*x])/(3*b^2) + (2*x^2*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b) + (2*a*Sin[a + b*x]*SinIntegral[a +
b*x])/(3*b^3) - (4*x*Sin[a + b*x]*SinIntegral[a + b*x])/(3*b^2) + (a^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b^
3) - (a*x*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b^2) + (x^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b) + (2*SinInt
egral[2*a + 2*b*x])/(3*b^3) - (a^2*SinIntegral[2*a + 2*b*x])/b^3

Rule 6509

Int[((c_.) + (d_.)*(x_))^(m_.)*SinIntegral[(a_) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*(c + d*x)^m*SinI
ntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Sin[a + b*x]*SinIntegral[a + b*x], x],
x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*SinIntegral[a + b*x]^2, x], x]) /; FreeQ[{a, b,
c, d}, x] && IGtQ[m, 0]

Rule 6513

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 4573

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 6741

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

Rule 6742

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

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 6519

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

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 8

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

Rule 3312

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 3302

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 6511

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

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 12

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

Rule 6517

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 6505

Int[SinIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*SinIntegral[a + b*x]^2)/b, x] - Dist[2, In
t[Sin[a + b*x]*SinIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int x^2 \text{Si}(a+b x)^2 \, dx &=\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}-\frac{2}{3} \int x^2 \sin (a+b x) \text{Si}(a+b x) \, dx-\frac{(2 a) \int x \text{Si}(a+b x)^2 \, dx}{3 b}\\ &=\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}-\frac{2}{3} \int \frac{x^2 \cos (a+b x) \sin (a+b x)}{a+b x} \, dx+\frac{a^2 \int \text{Si}(a+b x)^2 \, dx}{3 b^2}-\frac{4 \int x \cos (a+b x) \text{Si}(a+b x) \, dx}{3 b}+\frac{(2 a) \int x \sin (a+b x) \text{Si}(a+b x) \, dx}{3 b}\\ &=-\frac{2 a x \cos (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}-\frac{4 x \sin (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}-\frac{1}{3} \int \frac{x^2 \sin (2 (a+b x))}{a+b x} \, dx+\frac{4 \int \sin (a+b x) \text{Si}(a+b x) \, dx}{3 b^2}+\frac{(2 a) \int \cos (a+b x) \text{Si}(a+b x) \, dx}{3 b^2}-\frac{\left (2 a^2\right ) \int \sin (a+b x) \text{Si}(a+b x) \, dx}{3 b^2}+\frac{4 \int \frac{x \sin ^2(a+b x)}{a+b x} \, dx}{3 b}+\frac{(2 a) \int \frac{x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{3 b}\\ &=-\frac{4 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}+\frac{2 a^2 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{2 a x \cos (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}+\frac{2 a \sin (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{4 x \sin (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}-\frac{1}{3} \int \frac{x^2 \sin (2 a+2 b x)}{a+b x} \, dx+\frac{4 \int \frac{\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{3 b^2}-\frac{(2 a) \int \frac{\sin ^2(a+b x)}{a+b x} \, dx}{3 b^2}-\frac{\left (2 a^2\right ) \int \frac{\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{3 b^2}+\frac{4 \int \left (\frac{\sin ^2(a+b x)}{b}-\frac{a \sin ^2(a+b x)}{b (a+b x)}\right ) \, dx}{3 b}+\frac{a \int \frac{x \sin (2 (a+b x))}{a+b x} \, dx}{3 b}\\ &=-\frac{4 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}+\frac{2 a^2 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{2 a x \cos (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}+\frac{2 a \sin (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{4 x \sin (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}-\frac{1}{3} \int \left (-\frac{a \sin (2 a+2 b x)}{b^2}+\frac{x \sin (2 a+2 b x)}{b}+\frac{a^2 \sin (2 a+2 b x)}{b^2 (a+b x)}\right ) \, dx+\frac{4 \int \sin ^2(a+b x) \, dx}{3 b^2}+\frac{4 \int \frac{\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}-\frac{(2 a) \int \left (\frac{1}{2 (a+b x)}-\frac{\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}-\frac{(4 a) \int \frac{\sin ^2(a+b x)}{a+b x} \, dx}{3 b^2}-\frac{\left (2 a^2\right ) \int \frac{\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}+\frac{a \int \frac{x \sin (2 a+2 b x)}{a+b x} \, dx}{3 b}\\ &=-\frac{a \log (a+b x)}{3 b^3}-\frac{2 \cos (a+b x) \sin (a+b x)}{3 b^3}-\frac{4 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}+\frac{2 a^2 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{2 a x \cos (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}+\frac{2 a \sin (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{4 x \sin (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}+\frac{2 \int 1 \, dx}{3 b^2}+\frac{2 \int \frac{\sin (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac{a \int \frac{\cos (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac{a \int \sin (2 a+2 b x) \, dx}{3 b^2}-\frac{(4 a) \int \left (\frac{1}{2 (a+b x)}-\frac{\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}-2 \frac{a^2 \int \frac{\sin (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac{\int x \sin (2 a+2 b x) \, dx}{3 b}+\frac{a \int \left (\frac{\sin (2 a+2 b x)}{b}+\frac{a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{3 b}\\ &=\frac{2 x}{3 b^2}-\frac{a \cos (2 a+2 b x)}{6 b^3}+\frac{x \cos (2 a+2 b x)}{6 b^2}+\frac{a \text{Ci}(2 a+2 b x)}{3 b^3}-\frac{a \log (a+b x)}{b^3}-\frac{2 \cos (a+b x) \sin (a+b x)}{3 b^3}-\frac{4 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}+\frac{2 a^2 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{2 a x \cos (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}+\frac{2 a \sin (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{4 x \sin (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}+\frac{2 \text{Si}(2 a+2 b x)}{3 b^3}-\frac{2 a^2 \text{Si}(2 a+2 b x)}{3 b^3}-\frac{\int \cos (2 a+2 b x) \, dx}{6 b^2}+\frac{a \int \sin (2 a+2 b x) \, dx}{3 b^2}+\frac{(2 a) \int \frac{\cos (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac{a^2 \int \frac{\sin (2 a+2 b x)}{-a-b x} \, dx}{3 b^2}\\ &=\frac{2 x}{3 b^2}-\frac{a \cos (2 a+2 b x)}{3 b^3}+\frac{x \cos (2 a+2 b x)}{6 b^2}+\frac{a \text{Ci}(2 a+2 b x)}{b^3}-\frac{a \log (a+b x)}{b^3}-\frac{2 \cos (a+b x) \sin (a+b x)}{3 b^3}-\frac{\sin (2 a+2 b x)}{12 b^3}-\frac{4 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}+\frac{2 a^2 \cos (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{2 a x \cos (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{2 x^2 \cos (a+b x) \text{Si}(a+b x)}{3 b}+\frac{2 a \sin (a+b x) \text{Si}(a+b x)}{3 b^3}-\frac{4 x \sin (a+b x) \text{Si}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Si}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Si}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Si}(a+b x)^2}{3 b}+\frac{2 \text{Si}(2 a+2 b x)}{3 b^3}-\frac{a^2 \text{Si}(2 a+2 b x)}{b^3}\\ \end{align*}

Mathematica [A]  time = 1.71065, size = 158, normalized size = 0.48 \[ \frac{4 \left (a^3+b^3 x^3\right ) \text{Si}(a+b x)^2+8 \text{Si}(a+b x) \left (\left (a^2-a b x+b^2 x^2-2\right ) \cos (a+b x)+(a-2 b x) \sin (a+b x)\right )-12 a^2 \text{Si}(2 (a+b x))+12 a \text{CosIntegral}(2 (a+b x))+8 \text{Si}(2 (a+b x))-12 a \log (a+b x)-5 \sin (2 (a+b x))-4 a \cos (2 (a+b x))+2 b x \cos (2 (a+b x))+8 a+8 b x}{12 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*SinIntegral[a + b*x]^2,x]

[Out]

(8*a + 8*b*x - 4*a*Cos[2*(a + b*x)] + 2*b*x*Cos[2*(a + b*x)] + 12*a*CosIntegral[2*(a + b*x)] - 12*a*Log[a + b*
x] - 5*Sin[2*(a + b*x)] + 8*((-2 + a^2 - a*b*x + b^2*x^2)*Cos[a + b*x] + (a - 2*b*x)*Sin[a + b*x])*SinIntegral
[a + b*x] + 4*(a^3 + b^3*x^3)*SinIntegral[a + b*x]^2 + 8*SinIntegral[2*(a + b*x)] - 12*a^2*SinIntegral[2*(a +
b*x)])/(12*b^3)

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\it Si} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*Si(b*x+a)^2,x)

[Out]

int(x^2*Si(b*x+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Si}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*Si(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate(x^2*Si(b*x + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{Si}\left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*Si(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(x^2*sin_integral(b*x + a)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{Si}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*Si(b*x+a)**2,x)

[Out]

Integral(x**2*Si(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Si}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*Si(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^2*Si(b*x + a)^2, x)