Integrand size = 14, antiderivative size = 108 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=-\frac {x}{2 b}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b^2} \]
-1/2*x/b-1/2*a*Ci(2*b*x+2*a)/b^2+1/2*a*ln(b*x+a)/b^2+cos(b*x+a)*Si(b*x+a)/ b^2-1/2*Si(2*b*x+2*a)/b^2+1/2*cos(b*x+a)*sin(b*x+a)/b^2+x*Si(b*x+a)*sin(b* x+a)/b
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\frac {-2 b x-2 a \operatorname {CosIntegral}(2 (a+b x))+2 a \log (a+b x)+\sin (2 (a+b x))+4 (\cos (a+b x)+b x \sin (a+b x)) \text {Si}(a+b x)-2 \text {Si}(2 (a+b x))}{4 b^2} \]
(-2*b*x - 2*a*CosIntegral[2*(a + b*x)] + 2*a*Log[a + b*x] + Sin[2*(a + b*x )] + 4*(Cos[a + b*x] + b*x*Sin[a + b*x])*SinIntegral[a + b*x] - 2*SinInteg ral[2*(a + b*x)])/(4*b^2)
Time = 0.89 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {7073, 7065, 4906, 27, 3042, 3780, 7293, 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.
| \(\displaystyle \int x \text {Si}(a+b x) \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 7073 |
| \(\displaystyle -\frac {\int \sin (a+b x) \text {Si}(a+b x)dx}{b}-\int \frac {x \sin ^2(a+b x)}{a+b x}dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7065 |
| \(\displaystyle -\frac {\int \frac {\cos (a+b x) \sin (a+b x)}{a+b x}dx-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}-\int \frac {x \sin ^2(a+b x)}{a+b x}dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle -\frac {\int \frac {\sin (2 a+2 b x)}{2 (a+b x)}dx-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}-\int \frac {x \sin ^2(a+b x)}{a+b x}dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{a+b x}dx-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}-\int \frac {x \sin ^2(a+b x)}{a+b x}dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {\sin (2 a+2 b x)}{a+b x}dx-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}-\int \frac {x \sin ^2(a+b x)}{a+b x}dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\int \frac {x \sin ^2(a+b x)}{a+b x}dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\frac {\text {Si}(2 a+2 b x)}{2 b}-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {\sin ^2(a+b x)}{b}-\frac {a \sin ^2(a+b x)}{b (a+b x)}\right )dx+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\frac {\text {Si}(2 a+2 b x)}{2 b}-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\frac {\text {Si}(2 a+2 b x)}{2 b}-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}-\frac {x}{2 b}\) |
-1/2*x/b - (a*CosIntegral[2*a + 2*b*x])/(2*b^2) + (a*Log[a + b*x])/(2*b^2) + (Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (x*Sin[a + b*x]*SinIntegral[a + b *x])/b - (-((Cos[a + b*x]*SinIntegral[a + b*x])/b) + SinIntegral[2*a + 2*b *x]/(2*b))/b
3.1.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
Int[Sin[(a_.) + (b_.)*(x_)]*SinIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[(-Cos[a + b*x])*(SinIntegral[c + d*x]/b), x] + Simp[d/b Int[Cos[a + b *x]*(Sin[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
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] + (-Simp[d/b Int[(e + f*x)^m*Sin[a + b*x]*(Sin[c + d*x]/(c + d* x)), x], x] - Simp[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]
Time = 2.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+a \left (\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}\right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}}{b^{2}}\) | \(95\) |
| default | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (-a \sin \left (b x +a \right )+\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+a \left (\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}\right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}}{b^{2}}\) | \(95\) |
1/b^2*(Si(b*x+a)*(-a*sin(b*x+a)+cos(b*x+a)+(b*x+a)*sin(b*x+a))+a*(1/2*ln(b *x+a)-1/2*Ci(2*b*x+2*a))-1/2*Si(2*b*x+2*a)+1/2*sin(b*x+a)*cos(b*x+a)-1/2*b *x-1/2*a)
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=-\frac {b x + a \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) - a \log \left (b x + a\right ) - {\left (2 \, b x \operatorname {Si}\left (b x + a\right ) + \cos \left (b x + a\right )\right )} \sin \left (b x + a\right ) - 2 \, \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + \operatorname {Si}\left (2 \, b x + 2 \, a\right )}{2 \, b^{2}} \]
-1/2*(b*x + a*cos_integral(2*b*x + 2*a) - a*log(b*x + a) - (2*b*x*sin_inte gral(b*x + a) + cos(b*x + a))*sin(b*x + a) - 2*cos(b*x + a)*sin_integral(b *x + a) + sin_integral(2*b*x + 2*a))/b^2
\[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int x \cos {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \]
\[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) \,d x } \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.89 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx={\left (\frac {x \sin \left (b x + a\right )}{b} + \frac {\cos \left (b x + a\right )}{b^{2}}\right )} \operatorname {Si}\left (b x + a\right ) - \frac {2 \, b x \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - 2 \, a \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + a \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + a \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, b x \tan \left (b x\right )^{2} - 2 \, a \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} + a \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + a \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, b x \tan \left (a\right )^{2} - 2 \, a \log \left ({\left | b x + a \right |}\right ) \tan \left (a\right )^{2} + a \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (a\right )^{2} + a \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (a\right )^{2} + \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} + 2 \, \tan \left (b x\right )^{2} \tan \left (a\right ) + \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (a\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (a\right )^{2} + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (a\right )^{2} + 2 \, \tan \left (b x\right ) \tan \left (a\right )^{2} + 2 \, b x - 2 \, a \log \left ({\left | b x + a \right |}\right ) + a \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + a \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - 2 \, \tan \left (b x\right ) - 2 \, \tan \left (a\right )}{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 )}} \]
(x*sin(b*x + a)/b + cos(b*x + a)/b^2)*sin_integral(b*x + a) - 1/4*(2*b*x*t an(b*x)^2*tan(a)^2 - 2*a*log(abs(b*x + a))*tan(b*x)^2*tan(a)^2 + a*real_pa rt(cos_integral(2*b*x + 2*a))*tan(b*x)^2*tan(a)^2 + a*real_part(cos_integr al(-2*b*x - 2*a))*tan(b*x)^2*tan(a)^2 + imag_part(cos_integral(2*b*x + 2*a ))*tan(b*x)^2*tan(a)^2 - imag_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2* tan(a)^2 + 2*sin_integral(2*b*x + 2*a)*tan(b*x)^2*tan(a)^2 + 2*b*x*tan(b*x )^2 - 2*a*log(abs(b*x + a))*tan(b*x)^2 + a*real_part(cos_integral(2*b*x + 2*a))*tan(b*x)^2 + a*real_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2 + 2* b*x*tan(a)^2 - 2*a*log(abs(b*x + a))*tan(a)^2 + a*real_part(cos_integral(2 *b*x + 2*a))*tan(a)^2 + a*real_part(cos_integral(-2*b*x - 2*a))*tan(a)^2 + imag_part(cos_integral(2*b*x + 2*a))*tan(b*x)^2 - imag_part(cos_integral( -2*b*x - 2*a))*tan(b*x)^2 + 2*sin_integral(2*b*x + 2*a)*tan(b*x)^2 + 2*tan (b*x)^2*tan(a) + imag_part(cos_integral(2*b*x + 2*a))*tan(a)^2 - imag_part (cos_integral(-2*b*x - 2*a))*tan(a)^2 + 2*sin_integral(2*b*x + 2*a)*tan(a) ^2 + 2*tan(b*x)*tan(a)^2 + 2*b*x - 2*a*log(abs(b*x + a)) + a*real_part(cos _integral(2*b*x + 2*a)) + a*real_part(cos_integral(-2*b*x - 2*a)) + imag_p art(cos_integral(2*b*x + 2*a)) - imag_part(cos_integral(-2*b*x - 2*a)) + 2 *sin_integral(2*b*x + 2*a) - 2*tan(b*x) - 2*tan(a))/(b^2*tan(b*x)^2*tan(a) ^2 + b^2*tan(b*x)^2 + b^2*tan(a)^2 + b^2)
Timed out. \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int x\,\mathrm {sinint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]