Integrand size = 10, antiderivative size = 61 \[ \int x \cos (b x) \text {Si}(b x) \, dx=-\frac {x}{2 b}+\frac {\cos (b x) \sin (b x)}{2 b^2}+\frac {\cos (b x) \text {Si}(b x)}{b^2}+\frac {x \sin (b x) \text {Si}(b x)}{b}-\frac {\text {Si}(2 b x)}{2 b^2} \] Output:
-1/2*x/b+1/2*cos(b*x)*sin(b*x)/b^2+cos(b*x)*Si(b*x)/b^2+x*sin(b*x)*Si(b*x) /b-1/2*Si(2*b*x)/b^2
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int x \cos (b x) \text {Si}(b x) \, dx=\frac {-2 b x+\sin (2 b x)+4 (\cos (b x)+b x \sin (b x)) \text {Si}(b x)-2 \text {Si}(2 b x)}{4 b^2} \] Input:
Integrate[x*Cos[b*x]*SinIntegral[b*x],x]
Output:
(-2*b*x + Sin[2*b*x] + 4*(Cos[b*x] + b*x*Sin[b*x])*SinIntegral[b*x] - 2*Si nIntegral[2*b*x])/(4*b^2)
Time = 0.47 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.16, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {7073, 27, 3042, 3115, 24, 7065, 27, 4906, 27, 3042, 3780}
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.
\(\displaystyle \int x \text {Si}(b x) \cos (b x) \, dx\) |
\(\Big \downarrow \) 7073 |
\(\displaystyle -\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\int \frac {\sin ^2(b x)}{b}dx+\frac {x \text {Si}(b x) \sin (b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int \sin ^2(b x)dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\int \sin (b x)^2dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\int \sin (b x) \text {Si}(b x)dx}{b}-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\int \sin (b x) \text {Si}(b x)dx}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 7065 |
\(\displaystyle -\frac {\int \frac {\cos (b x) \sin (b x)}{b x}dx-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {\cos (b x) \sin (b x)}{x}dx}{b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {\frac {\int \frac {\sin (2 b x)}{2 x}dx}{b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}+\frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {x \text {Si}(b x) \sin (b x)}{b}-\frac {\frac {\text {Si}(2 b x)}{2 b}-\frac {\text {Si}(b x) \cos (b x)}{b}}{b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{b}\) |
Input:
Int[x*Cos[b*x]*SinIntegral[b*x],x]
Output:
-((x/2 - (Cos[b*x]*Sin[b*x])/(2*b))/b) + (x*Sin[b*x]*SinIntegral[b*x])/b - (-((Cos[b*x]*SinIntegral[b*x])/b) + SinIntegral[2*b*x]/(2*b))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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 = 4.57 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {\operatorname {Si}\left (b x \right ) \left (\cos \left (b x \right )+b x \sin \left (b x \right )\right )-\frac {\operatorname {Si}\left (2 b x \right )}{2}+\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}-\frac {b x}{2}}{b^{2}}\) | \(44\) |
default | \(\frac {\operatorname {Si}\left (b x \right ) \left (\cos \left (b x \right )+b x \sin \left (b x \right )\right )-\frac {\operatorname {Si}\left (2 b x \right )}{2}+\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}-\frac {b x}{2}}{b^{2}}\) | \(44\) |
Input:
int(x*cos(b*x)*Si(b*x),x,method=_RETURNVERBOSE)
Output:
1/b^2*(Si(b*x)*(cos(b*x)+b*x*sin(b*x))-1/2*Si(2*b*x)+1/2*sin(b*x)*cos(b*x) -1/2*b*x)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70 \[ \int x \cos (b x) \text {Si}(b x) \, dx=-\frac {b x - {\left (2 \, b x \operatorname {Si}\left (b x\right ) + \cos \left (b x\right )\right )} \sin \left (b x\right ) - 2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) + \operatorname {Si}\left (2 \, b x\right )}{2 \, b^{2}} \] Input:
integrate(x*cos(b*x)*sin_integral(b*x),x, algorithm="fricas")
Output:
-1/2*(b*x - (2*b*x*sin_integral(b*x) + cos(b*x))*sin(b*x) - 2*cos(b*x)*sin _integral(b*x) + sin_integral(2*b*x))/b^2
\[ \int x \cos (b x) \text {Si}(b x) \, dx=\int x \cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \] Input:
integrate(x*cos(b*x)*Si(b*x),x)
Output:
Integral(x*cos(b*x)*Si(b*x), x)
\[ \int x \cos (b x) \text {Si}(b x) \, dx=\int { x \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) \,d x } \] Input:
integrate(x*cos(b*x)*sin_integral(b*x),x, algorithm="maxima")
Output:
integrate(x*cos(b*x)*sin_integral(b*x), x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.03 \[ \int x \cos (b x) \text {Si}(b x) \, dx={\left (\frac {x \sin \left (b x\right )}{b} + \frac {\cos \left (b x\right )}{b^{2}}\right )} \operatorname {Si}\left (b x\right ) - \frac {2 \, b x \tan \left (b x\right )^{2} + \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 2 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} + 2 \, b x + \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right ) - 2 \, \tan \left (b x\right )}{4 \, {\left (b^{2} \tan \left (b x\right )^{2} + b^{2}\right )}} \] Input:
integrate(x*cos(b*x)*sin_integral(b*x),x, algorithm="giac")
Output:
(x*sin(b*x)/b + cos(b*x)/b^2)*sin_integral(b*x) - 1/4*(2*b*x*tan(b*x)^2 + imag_part(cos_integral(2*b*x))*tan(b*x)^2 - imag_part(cos_integral(-2*b*x) )*tan(b*x)^2 + 2*sin_integral(2*b*x)*tan(b*x)^2 + 2*b*x + imag_part(cos_in tegral(2*b*x)) - imag_part(cos_integral(-2*b*x)) + 2*sin_integral(2*b*x) - 2*tan(b*x))/(b^2*tan(b*x)^2 + b^2)
Timed out. \[ \int x \cos (b x) \text {Si}(b x) \, dx=\int x\,\mathrm {sinint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \] Input:
int(x*sinint(b*x)*cos(b*x),x)
Output:
int(x*sinint(b*x)*cos(b*x), x)
\[ \int x \cos (b x) \text {Si}(b x) \, dx=\int \cos \left (b x \right ) \mathit {si} \left (b x \right ) x d x \] Input:
int(x*cos(b*x)*Si(b*x),x)
Output:
int(cos(b*x)*si(b*x)*x,x)