Integrand size = 8, antiderivative size = 74 \[ \int x \text {Si}(b x)^2 \, dx=-\frac {\operatorname {CosIntegral}(2 b x)}{2 b^2}+\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \cos (b x) \text {Si}(b x)}{b}-\frac {\sin (b x) \text {Si}(b x)}{b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2 \]
-1/2*Ci(2*b*x)/b^2+1/2*ln(x)/b^2+x*cos(b*x)*Si(b*x)/b+1/2*x^2*Si(b*x)^2-Si (b*x)*sin(b*x)/b^2-1/2*sin(b*x)^2/b^2
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int x \text {Si}(b x)^2 \, dx=\frac {\cos (2 b x)-2 \operatorname {CosIntegral}(2 b x)+2 \log (x)+4 (b x \cos (b x)-\sin (b x)) \text {Si}(b x)+2 b^2 x^2 \text {Si}(b x)^2}{4 b^2} \]
(Cos[2*b*x] - 2*CosIntegral[2*b*x] + 2*Log[x] + 4*(b*x*Cos[b*x] - Sin[b*x] )*SinIntegral[b*x] + 2*b^2*x^2*SinIntegral[b*x]^2)/(4*b^2)
Time = 0.53 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {7061, 7067, 27, 3042, 3044, 15, 7071, 27, 3042, 3793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \text {Si}(b x)^2 \, dx\) |
\(\Big \downarrow \) 7061 |
\(\displaystyle \frac {1}{2} x^2 \text {Si}(b x)^2-\int x \sin (b x) \text {Si}(b x)dx\) |
\(\Big \downarrow \) 7067 |
\(\displaystyle -\frac {\int \cos (b x) \text {Si}(b x)dx}{b}-\int \frac {\cos (b x) \sin (b x)}{b}dx+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \cos (b x) \text {Si}(b x)dx}{b}-\frac {\int \cos (b x) \sin (b x)dx}{b}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \cos (b x) \text {Si}(b x)dx}{b}-\frac {\int \cos (b x) \sin (b x)dx}{b}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle -\frac {\int \sin (b x)d\sin (b x)}{b^2}-\frac {\int \cos (b x) \text {Si}(b x)dx}{b}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\int \cos (b x) \text {Si}(b x)dx}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 7071 |
\(\displaystyle -\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\int \frac {\sin ^2(b x)}{b x}dx}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin ^2(b x)}{x}dx}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (b x)^2}{x}dx}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right )dx}{b}}{b}-\frac {\sin ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sin ^2(b x)}{2 b^2}-\frac {\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\frac {\log (x)}{2}-\frac {\operatorname {CosIntegral}(2 b x)}{2}}{b}}{b}+\frac {1}{2} x^2 \text {Si}(b x)^2+\frac {x \text {Si}(b x) \cos (b x)}{b}\) |
-1/2*Sin[b*x]^2/b^2 + (x*Cos[b*x]*SinIntegral[b*x])/b + (x^2*SinIntegral[b *x]^2)/2 - (-((-1/2*CosIntegral[2*b*x] + Log[x]/2)/b) + (Sin[b*x]*SinInteg ral[b*x])/b)/b
3.1.12.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[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]))
Int[(x_)^(m_.)*SinIntegral[(b_.)*(x_)]^2, x_Symbol] :> Simp[x^(m + 1)*(SinI ntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1) Int[x^m*Sin[b*x]*SinIntegral [b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
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] + (Simp[d/b Int[(e + f*x)^m*Cos[a + b*x]*(Sin[c + d*x]/(c + d*x)), x], x] + Simp[f*(m/b) Int[(e + f*x)^(m - 1)*Cos[a + b*x]*SinIntegr al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
Int[Cos[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[Sin[a + b*x]*(SinIntegral[c + d*x]/b), x] - Simp[d/b Int[Sin[a + b*x] *(Sin[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
Time = 0.48 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {b^{2} x^{2} \operatorname {Si}\left (b x \right )^{2}}{2}-2 \,\operatorname {Si}\left (b x \right ) \left (\frac {\sin \left (b x \right )}{2}-\frac {b x \cos \left (b x \right )}{2}\right )+\frac {\cos \left (b x \right )^{2}}{2}+\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ci}\left (2 b x \right )}{2}}{b^{2}}\) | \(62\) |
default | \(\frac {\frac {b^{2} x^{2} \operatorname {Si}\left (b x \right )^{2}}{2}-2 \,\operatorname {Si}\left (b x \right ) \left (\frac {\sin \left (b x \right )}{2}-\frac {b x \cos \left (b x \right )}{2}\right )+\frac {\cos \left (b x \right )^{2}}{2}+\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ci}\left (2 b x \right )}{2}}{b^{2}}\) | \(62\) |
1/b^2*(1/2*b^2*x^2*Si(b*x)^2-2*Si(b*x)*(1/2*sin(b*x)-1/2*b*x*cos(b*x))+1/2 *cos(b*x)^2+1/2*ln(b*x)-1/2*Ci(2*b*x))
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.76 \[ \int x \text {Si}(b x)^2 \, dx=\frac {b^{2} x^{2} \operatorname {Si}\left (b x\right )^{2} + 2 \, b x \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) + \cos \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) + \log \left (x\right )}{2 \, b^{2}} \]
1/2*(b^2*x^2*sin_integral(b*x)^2 + 2*b*x*cos(b*x)*sin_integral(b*x) + cos( b*x)^2 - 2*sin(b*x)*sin_integral(b*x) - cos_integral(2*b*x) + log(x))/b^2
\[ \int x \text {Si}(b x)^2 \, dx=\int x \operatorname {Si}^{2}{\left (b x \right )}\, dx \]
\[ \int x \text {Si}(b x)^2 \, dx=\int { x \operatorname {Si}\left (b x\right )^{2} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int x \text {Si}(b x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {Si}\left (b x\right )^{2} + {\left (\frac {x \cos \left (b x\right )}{b} - \frac {\sin \left (b x\right )}{b^{2}}\right )} \operatorname {Si}\left (b x\right ) + \frac {\cos \left (2 \, b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) - \operatorname {Ci}\left (-2 \, b x\right ) + 2 \, \log \left (x\right )}{4 \, b^{2}} \]
1/2*x^2*sin_integral(b*x)^2 + (x*cos(b*x)/b - sin(b*x)/b^2)*sin_integral(b *x) + 1/4*(cos(2*b*x) - cos_integral(2*b*x) - cos_integral(-2*b*x) + 2*log (x))/b^2
Timed out. \[ \int x \text {Si}(b x)^2 \, dx=\int x\,{\mathrm {sinint}\left (b\,x\right )}^2 \,d x \]