Integrand size = 14, antiderivative size = 97 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2} \]
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
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {\cos (2 (a+b x))-2 \operatorname {CosIntegral}(2 (a+b x))+2 \log (a+b x)+4 (b x \cos (a+b x)-\sin (a+b x)) \text {Si}(a+b x)+2 a \text {Si}(2 (a+b x))}{4 b^2} \]
-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])*SinIntegral[a + b*x] + 2*a*SinIntegral[2* (a + b*x)])/b^2
Time = 0.93 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {7067, 5084, 7071, 3042, 3793, 2009, 7292, 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.
| \(\displaystyle \int x \text {Si}(a+b x) \sin (a+b x) \, dx\) |
| \(\Big \downarrow \) 7067 |
| \(\displaystyle \frac {\int \cos (a+b x) \text {Si}(a+b x)dx}{b}+\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 5084 |
| \(\displaystyle \frac {\int \cos (a+b x) \text {Si}(a+b x)dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 7071 |
| \(\displaystyle \frac {\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin ^2(a+b x)}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin (a+b x)^2}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right )dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{2} \int \frac {x \sin (2 a+2 b x)}{a+b x}dx+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right )dx+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \text {Si}(2 a+2 b x)}{b^2}-\frac {\cos (2 a+2 b x)}{2 b^2}\right )+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\) |
-((x*Cos[a + b*x]*SinIntegral[a + b*x])/b) + (CosIntegral[2*a + 2*b*x]/(2* b) - Log[a + b*x]/(2*b) + (Sin[a + b*x]*SinIntegral[a + b*x])/b)/b + (-1/2 *Cos[2*a + 2*b*x]/b^2 - (a*SinIntegral[2*a + 2*b*x])/b^2)/2
3.1.56.3.1 Defintions of rubi rules used
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[Cos[w_]^(p_.)*(u_.)*Sin[v_]^(p_.), x_Symbol] :> Simp[1/2^p Int[u*Sin[ 2*v]^p, x], x] /; EqQ[w, v] && IntegerQ[p]
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 = 1.58 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a \,\operatorname {Si}\left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}-\frac {\cos \left (b x +a \right )^{2}}{2}}{b^{2}}\) | \(82\) |
| default | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a \,\operatorname {Si}\left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}-\frac {\cos \left (b x +a \right )^{2}}{2}}{b^{2}}\) | \(82\) |
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)
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {2 \, b x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + \cos \left (b x + a\right )^{2} + a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - 2 \, \sin \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) - \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) + \log \left (b x + a\right )}{2 \, b^{2}} \]
-1/2*(2*b*x*cos(b*x + a)*sin_integral(b*x + a) + cos(b*x + a)^2 + a*sin_in tegral(2*b*x + 2*a) - 2*sin(b*x + a)*sin_integral(b*x + a) - cos_integral( 2*b*x + 2*a) + log(b*x + a))/b^2
\[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\int x \sin {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \]
\[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\int { x \sin \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) \,d x } \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 507, normalized size of antiderivative = 5.23 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-{\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 )}} \]
-(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_int egral(-2*b*x - 2*a))*tan(b*x)^2*tan(a)^2 + 2*a*sin_integral(2*b*x + 2*a)*t an(b*x)^2*tan(a)^2 + 2*log(abs(b*x + a))*tan(b*x)^2*tan(a)^2 - real_part(c os_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))*t an(b*x)^2 - a*imag_part(cos_integral(-2*b*x - 2*a))*tan(b*x)^2 + 2*a*sin_i ntegral(2*b*x + 2*a)*tan(b*x)^2 + a*imag_part(cos_integral(2*b*x + 2*a))*t an(a)^2 - a*imag_part(cos_integral(-2*b*x - 2*a))*tan(a)^2 + 2*a*sin_integ ral(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_integral(2*b*x + 2*a))*tan(b*x)^2 - real_part(cos_i ntegral(-2*b*x - 2*a))*tan(b*x)^2 + 2*log(abs(b*x + a))*tan(a)^2 - real_pa rt(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_i ntegral(-2*b*x - 2*a)) + 2*a*sin_integral(2*b*x + 2*a) - tan(b*x)^2 - 4*ta n(b*x)*tan(a) - tan(a)^2 + 2*log(abs(b*x + a)) - real_part(cos_integral(2* b*x + 2*a)) - real_part(cos_integral(-2*b*x - 2*a)) + 1)/(b^2*tan(b*x)^2*t an(a)^2 + b^2*tan(b*x)^2 + b^2*tan(a)^2 + b^2)
Timed out. \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\int x\,\mathrm {sinint}\left (a+b\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]