\(\int x \cos (a+b x) \text {Si}(a+b x) \, dx\) [60]

Optimal result

Integrand size = 14, antiderivative size = 108 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=-\frac {x}{2 b}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {\cos (a+b x) \text {Si}(a+b x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(a+b x)}{b}-\frac {\text {Si}(2 a+2 b x)}{2 b^2} \] Output:

-1/2*x/b-1/2*a*Ci(2*b*x+2*a)/b^2+1/2*a*ln(b*x+a)/b^2+1/2*cos(b*x+a)*sin(b* 
x+a)/b^2+cos(b*x+a)*Si(b*x+a)/b^2+x*sin(b*x+a)*Si(b*x+a)/b-1/2*Si(2*b*x+2* 
a)/b^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\frac {-2 b x-2 a \operatorname {CosIntegral}(2 (a+b x))+2 a \log (a+b x)+\sin (2 (a+b x))+4 (\cos (a+b x)+b x \sin (a+b x)) \text {Si}(a+b x)-2 \text {Si}(2 (a+b x))}{4 b^2} \] Input:

Integrate[x*Cos[a + b*x]*SinIntegral[a + b*x],x]
 

Output:

(-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)
 

Rubi [A] (verified)

Time = 0.88 (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

\(\displaystyle -\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {x \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\frac {\text {Si}(2 a+2 b x)}{2 b}-\frac {\text {Si}(a+b x) \cos (a+b x)}{b}}{b}-\frac {x}{2 b}\)

Input:

Int[x*Cos[a + b*x]*SinIntegral[a + b*x],x]
 

Output:

-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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

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]
 

rule 4906

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]
 

rule 7065

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]
 

rule 7073

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]
 

rule 7293

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

Maple [A] (verified)

Time = 8.98 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\operatorname {Si}\left (b x +a \right ) \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )-a \sin \left (b x +a \right )\right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}+a \left (\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}\right )}{b^{2}}\)	\(95\)
default	\(\frac {\operatorname {Si}\left (b x +a \right ) \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )-a \sin \left (b x +a \right )\right )-\frac {\operatorname {Si}\left (2 b x +2 a \right )}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}+a \left (\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}\right )}{b^{2}}\)	\(95\)

Input:

int(x*cos(b*x+a)*Si(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/b^2*(Si(b*x+a)*(cos(b*x+a)+(b*x+a)*sin(b*x+a)-a*sin(b*x+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+a*(1/2*ln(b*x+a)-1/2*Ci(2*b* 
x+2*a)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=-\frac {b x + a \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) - a \log \left (b x + a\right ) - {\left (2 \, b x \operatorname {Si}\left (b x + a\right ) + \cos \left (b x + a\right )\right )} \sin \left (b x + a\right ) - 2 \, \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + \operatorname {Si}\left (2 \, b x + 2 \, a\right )}{2 \, b^{2}} \] Input:

integrate(x*cos(b*x+a)*sin_integral(b*x+a),x, algorithm="fricas")
 

Output:

-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
 

Sympy [F]

\[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int x \cos {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \] Input:

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

Output:

Integral(x*cos(a + b*x)*Si(a + b*x), x)
 

Maxima [F]

\[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) \,d x } \] Input:

integrate(x*cos(b*x+a)*sin_integral(b*x+a),x, algorithm="maxima")
 

Output:

integrate(x*cos(b*x + a)*sin_integral(b*x + a), x)
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.89 \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx =\text {Too large to display} \] Input:

integrate(x*cos(b*x+a)*sin_integral(b*x+a),x, algorithm="giac")
 

Output:

(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)
 

Mupad [F(-1)]

Timed out. \[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int x\,\mathrm {sinint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \] Input:

int(x*sinint(a + b*x)*cos(a + b*x),x)
 

Output:

int(x*sinint(a + b*x)*cos(a + b*x), x)
 

Reduce [F]

\[ \int x \cos (a+b x) \text {Si}(a+b x) \, dx=\int \cos \left (b x +a \right ) \mathit {si} \left (b x +a \right ) x d x \] Input:

int(x*cos(b*x+a)*Si(b*x+a),x)
 

Output:

int(cos(a + b*x)*si(a + b*x)*x,x)