Integrand size = 16, antiderivative size = 218 \[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cos (2 a+2 b x)}{2 b^3}-\frac {\operatorname {CosIntegral}(2 a+2 b x)}{b^3}+\frac {a^2 \operatorname {CosIntegral}(2 a+2 b x)}{2 b^3}+\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}-\frac {\sin ^2(a+b x)}{4 b^3}+\frac {2 x \cos (a+b x) \text {Si}(a+b x)}{b^2}-\frac {2 \sin (a+b x) \text {Si}(a+b x)}{b^3}+\frac {x^2 \sin (a+b x) \text {Si}(a+b x)}{b}+\frac {a \text {Si}(2 a+2 b x)}{b^3} \]
1/2*a*x/b^2-1/4*x^2/b-Ci(2*b*x+2*a)/b^3+1/2*a^2*Ci(2*b*x+2*a)/b^3+1/2*cos( 2*b*x+2*a)/b^3+ln(b*x+a)/b^3-1/2*a^2*ln(b*x+a)/b^3+2*x*cos(b*x+a)*Si(b*x+a )/b^2+a*Si(2*b*x+2*a)/b^3-1/2*a*cos(b*x+a)*sin(b*x+a)/b^3+1/2*x*cos(b*x+a) *sin(b*x+a)/b^2-2*Si(b*x+a)*sin(b*x+a)/b^3+x^2*Si(b*x+a)*sin(b*x+a)/b-1/4* sin(b*x+a)^2/b^3
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.61 \[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx=\frac {4 a b x-2 b^2 x^2+5 \cos (2 (a+b x))+4 \left (-2+a^2\right ) \operatorname {CosIntegral}(2 (a+b x))+8 \log (a+b x)-4 a^2 \log (a+b x)-2 a \sin (2 (a+b x))+2 b x \sin (2 (a+b x))+8 \left (2 b x \cos (a+b x)+\left (-2+b^2 x^2\right ) \sin (a+b x)\right ) \text {Si}(a+b x)+8 a \text {Si}(2 (a+b x))}{8 b^3} \]
(4*a*b*x - 2*b^2*x^2 + 5*Cos[2*(a + b*x)] + 4*(-2 + a^2)*CosIntegral[2*(a + b*x)] + 8*Log[a + b*x] - 4*a^2*Log[a + b*x] - 2*a*Sin[2*(a + b*x)] + 2*b *x*Sin[2*(a + b*x)] + 8*(2*b*x*Cos[a + b*x] + (-2 + b^2*x^2)*Sin[a + b*x]) *SinIntegral[a + b*x] + 8*a*SinIntegral[2*(a + b*x)])/(8*b^3)
Time = 2.00 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {7073, 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.
| \(\displaystyle \int x^2 \text {Si}(a+b x) \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 7073 |
| \(\displaystyle -\frac {2 \int x \sin (a+b x) \text {Si}(a+b x)dx}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7067 |
| \(\displaystyle -\frac {2 \left (\frac {\int \cos (a+b x) \text {Si}(a+b x)dx}{b}+\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 5084 |
| \(\displaystyle -\frac {2 \left (\frac {\int \cos (a+b x) \text {Si}(a+b x)dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7071 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin ^2(a+b x)}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\int \frac {\sin (a+b x)^2}{a+b x}dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {2 \left (\frac {\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right )dx}{b}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x}dx+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \int \frac {x \sin (2 a+2 b x)}{a+b x}dx+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}-\int \frac {x^2 \sin ^2(a+b x)}{a+b x}dx+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {x \sin ^2(a+b x)}{b}+\frac {a^2 \sin ^2(a+b x)}{b^2 (a+b x)}-\frac {a \sin ^2(a+b x)}{b^2}\right )dx-\frac {2 \left (\frac {1}{2} \int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right )dx+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \operatorname {CosIntegral}(2 a+2 b x)}{2 b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {\sin ^2(a+b x)}{4 b^3}-\frac {a \sin (a+b x) \cos (a+b x)}{2 b^3}-\frac {2 \left (\frac {1}{2} \left (-\frac {a \text {Si}(2 a+2 b x)}{b^2}-\frac {\cos (2 a+2 b x)}{2 b^2}\right )+\frac {\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b}\right )}{b}+\frac {a x}{2 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac {x^2 \text {Si}(a+b x) \sin (a+b x)}{b}-\frac {x^2}{4 b}\) |
(a*x)/(2*b^2) - x^2/(4*b) + (a^2*CosIntegral[2*a + 2*b*x])/(2*b^3) - (a^2* Log[a + b*x])/(2*b^3) - (a*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + (x*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) - Sin[a + b*x]^2/(4*b^3) + (x^2*Sin[a + b*x]*S inIntegral[a + b*x])/b - (2*(-((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]*SinIn tegral[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))/b
3.1.59.3.1 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]
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 = 4.36 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (a^{2} \sin \left (b x +a \right )-2 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}+\frac {a^{2} \operatorname {Ci}\left (2 b x +2 a \right )}{2}-\cos \left (b x +a \right ) \sin \left (b x +a \right ) a +\left (b x +a \right ) a -\left (b x +a \right ) \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right )^{2}}{4}-\frac {\sin \left (b x +a \right )^{2}}{4}+a \,\operatorname {Si}\left (2 b x +2 a \right )+\cos \left (b x +a \right )^{2}+\ln \left (b x +a \right )-\operatorname {Ci}\left (2 b x +2 a \right )}{b^{3}}\) | \(212\) |
| default | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (a^{2} \sin \left (b x +a \right )-2 a \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}+\frac {a^{2} \operatorname {Ci}\left (2 b x +2 a \right )}{2}-\cos \left (b x +a \right ) \sin \left (b x +a \right ) a +\left (b x +a \right ) a -\left (b x +a \right ) \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right )^{2}}{4}-\frac {\sin \left (b x +a \right )^{2}}{4}+a \,\operatorname {Si}\left (2 b x +2 a \right )+\cos \left (b x +a \right )^{2}+\ln \left (b x +a \right )-\operatorname {Ci}\left (2 b x +2 a \right )}{b^{3}}\) | \(212\) |
1/b^3*(Si(b*x+a)*(a^2*sin(b*x+a)-2*a*(cos(b*x+a)+(b*x+a)*sin(b*x+a))+(b*x+ a)^2*sin(b*x+a)-2*sin(b*x+a)+2*(b*x+a)*cos(b*x+a))-1/2*a^2*ln(b*x+a)+1/2*a ^2*Ci(2*b*x+2*a)-cos(b*x+a)*sin(b*x+a)*a+(b*x+a)*a-(b*x+a)*(-1/2*sin(b*x+a )*cos(b*x+a)+1/2*b*x+1/2*a)+1/4*(b*x+a)^2-1/4*sin(b*x+a)^2+a*Si(2*b*x+2*a) +cos(b*x+a)^2+ln(b*x+a)-Ci(2*b*x+2*a))
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.57 \[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx=-\frac {b^{2} x^{2} - 8 \, b x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) - 2 \, a b x - 5 \, \cos \left (b x + a\right )^{2} - 2 \, {\left (a^{2} - 2\right )} \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) + 2 \, {\left (a^{2} - 2\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (b x - a\right )} \cos \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} - 2\right )} \operatorname {Si}\left (b x + a\right )\right )} \sin \left (b x + a\right ) - 4 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right )}{4 \, b^{3}} \]
-1/4*(b^2*x^2 - 8*b*x*cos(b*x + a)*sin_integral(b*x + a) - 2*a*b*x - 5*cos (b*x + a)^2 - 2*(a^2 - 2)*cos_integral(2*b*x + 2*a) + 2*(a^2 - 2)*log(b*x + a) - 2*((b*x - a)*cos(b*x + a) + 2*(b^2*x^2 - 2)*sin_integral(b*x + a))* sin(b*x + a) - 4*a*sin_integral(2*b*x + 2*a))/b^3
\[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx=\int x^{2} \cos {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \]
\[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx=\int { x^{2} \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) \,d x } \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.98 \[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx={\left (\frac {2 \, x \cos \left (b x + a\right )}{b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \sin \left (b x + a\right )}{b^{3}}\right )} \operatorname {Si}\left (b x + a\right ) - \frac {2 \, b^{2} x^{2} \tan \left (b x + a\right )^{2} - 4 \, a b x \tan \left (b x + a\right )^{2} + 4 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} - 4 \, a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 4 \, a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 8 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )^{2} - 4 \, a b x + 4 \, a^{2} \log \left ({\left | b x + a \right |}\right ) - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - 2 \, a^{2} \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 4 \, b x \tan \left (b x + a\right ) - 8 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} + 4 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 4 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 4 \, a \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 4 \, a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 8 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) + 4 \, a \tan \left (b x + a\right ) + 5 \, \tan \left (b x + a\right )^{2} - 8 \, \log \left ({\left | b x + a \right |}\right ) + 4 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 4 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 5}{8 \, {\left (b^{3} \tan \left (b x + a\right )^{2} + b^{3}\right )}} \]
(2*x*cos(b*x + a)/b^2 + (b^2*x^2 - 2)*sin(b*x + a)/b^3)*sin_integral(b*x + a) - 1/8*(2*b^2*x^2*tan(b*x + a)^2 - 4*a*b*x*tan(b*x + a)^2 + 4*a^2*log(a bs(b*x + a))*tan(b*x + a)^2 - 2*a^2*real_part(cos_integral(2*b*x + 2*a))*t an(b*x + a)^2 - 2*a^2*real_part(cos_integral(-2*b*x - 2*a))*tan(b*x + a)^2 + 2*b^2*x^2 - 4*a*imag_part(cos_integral(2*b*x + 2*a))*tan(b*x + a)^2 + 4 *a*imag_part(cos_integral(-2*b*x - 2*a))*tan(b*x + a)^2 - 8*a*sin_integral (2*b*x + 2*a)*tan(b*x + a)^2 - 4*a*b*x + 4*a^2*log(abs(b*x + a)) - 2*a^2*r eal_part(cos_integral(2*b*x + 2*a)) - 2*a^2*real_part(cos_integral(-2*b*x - 2*a)) - 4*b*x*tan(b*x + a) - 8*log(abs(b*x + a))*tan(b*x + a)^2 + 4*real _part(cos_integral(2*b*x + 2*a))*tan(b*x + a)^2 + 4*real_part(cos_integral (-2*b*x - 2*a))*tan(b*x + a)^2 - 4*a*imag_part(cos_integral(2*b*x + 2*a)) + 4*a*imag_part(cos_integral(-2*b*x - 2*a)) - 8*a*sin_integral(2*b*x + 2*a ) + 4*a*tan(b*x + a) + 5*tan(b*x + a)^2 - 8*log(abs(b*x + a)) + 4*real_par t(cos_integral(2*b*x + 2*a)) + 4*real_part(cos_integral(-2*b*x - 2*a)) - 5 )/(b^3*tan(b*x + a)^2 + b^3)
Timed out. \[ \int x^2 \cos (a+b x) \text {Si}(a+b x) \, dx=\int x^2\,\mathrm {sinint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]