Integrand size = 8, antiderivative size = 49 \[ \int x^2 \text {Si}(b x) \, dx=-\frac {2 \cos (b x)}{3 b^3}+\frac {x^2 \cos (b x)}{3 b}-\frac {2 x \sin (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x) \] Output:
-2/3*cos(b*x)/b^3+1/3*x^2*cos(b*x)/b-2/3*x*sin(b*x)/b^2+1/3*x^3*Si(b*x)
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int x^2 \text {Si}(b x) \, dx=\frac {\left (-2+b^2 x^2\right ) \cos (b x)-2 b x \sin (b x)+b^3 x^3 \text {Si}(b x)}{3 b^3} \] Input:
Integrate[x^2*SinIntegral[b*x],x]
Output:
((-2 + b^2*x^2)*Cos[b*x] - 2*b*x*Sin[b*x] + b^3*x^3*SinIntegral[b*x])/(3*b ^3)
Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {7057, 27, 3042, 3777, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int x^2 \text {Si}(b x) \, dx\) |
\(\Big \downarrow \) 7057 |
\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)-\frac {1}{3} b \int \frac {x^2 \sin (b x)}{b}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)-\frac {1}{3} \int x^2 \sin (b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \text {Si}(b x)-\frac {1}{3} \int x^2 \sin (b x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} \left (\frac {x^2 \cos (b x)}{b}-\frac {2 \int x \cos (b x)dx}{b}\right )+\frac {1}{3} x^3 \text {Si}(b x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {x^2 \cos (b x)}{b}-\frac {2 \int x \sin \left (b x+\frac {\pi }{2}\right )dx}{b}\right )+\frac {1}{3} x^3 \text {Si}(b x)\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} \left (\frac {x^2 \cos (b x)}{b}-\frac {2 \left (\frac {\int -\sin (b x)dx}{b}+\frac {x \sin (b x)}{b}\right )}{b}\right )+\frac {1}{3} x^3 \text {Si}(b x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \left (\frac {x^2 \cos (b x)}{b}-\frac {2 \left (\frac {x \sin (b x)}{b}-\frac {\int \sin (b x)dx}{b}\right )}{b}\right )+\frac {1}{3} x^3 \text {Si}(b x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \left (\frac {x^2 \cos (b x)}{b}-\frac {2 \left (\frac {x \sin (b x)}{b}-\frac {\int \sin (b x)dx}{b}\right )}{b}\right )+\frac {1}{3} x^3 \text {Si}(b x)\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {1}{3} \left (\frac {x^2 \cos (b x)}{b}-\frac {2 \left (\frac {\cos (b x)}{b^2}+\frac {x \sin (b x)}{b}\right )}{b}\right )+\frac {1}{3} x^3 \text {Si}(b x)\) |
Input:
Int[x^2*SinIntegral[b*x],x]
Output:
((x^2*Cos[b*x])/b - (2*(Cos[b*x]/b^2 + (x*Sin[b*x])/b))/b)/3 + (x^3*SinInt egral[b*x])/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*SinIntegral[(a_.) + (b_.)*(x_)], x_Symbol] : > Simp[(c + d*x)^(m + 1)*(SinIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/(d *(m + 1)) Int[(c + d*x)^(m + 1)*(Sin[a + b*x]/(a + b*x)), x], x] /; FreeQ [{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.66 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88
method | result | size |
parts | \(\frac {x^{3} \operatorname {Si}\left (b x \right )}{3}-\frac {-b^{2} x^{2} \cos \left (b x \right )+2 \cos \left (b x \right )+2 b x \sin \left (b x \right )}{3 b^{3}}\) | \(43\) |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )}{3}+\frac {b^{2} x^{2} \cos \left (b x \right )}{3}-\frac {2 \cos \left (b x \right )}{3}-\frac {2 b x \sin \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
default | \(\frac {\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )}{3}+\frac {b^{2} x^{2} \cos \left (b x \right )}{3}-\frac {2 \cos \left (b x \right )}{3}-\frac {2 b x \sin \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
meijerg | \(\frac {2 \sqrt {\pi }\, \left (\frac {1}{3 \sqrt {\pi }}-\frac {\left (-\frac {b^{2} x^{2}}{2}+1\right ) \cos \left (b x \right )}{3 \sqrt {\pi }}-\frac {b x \sin \left (b x \right )}{3 \sqrt {\pi }}+\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}\) | \(60\) |
orering | \(\frac {\left (b^{4} x^{4}+8 b^{2} x^{2}-8\right ) \operatorname {Si}\left (b x \right )}{3 b^{4} x}-\frac {\left (5 b^{2} x^{2}-6\right ) \left (2 x \,\operatorname {Si}\left (b x \right )+x \sin \left (b x \right )\right )}{3 b^{4} x^{2}}+\frac {\left (b^{2} x^{2}-2\right ) \left (2 \,\operatorname {Si}\left (b x \right )+3 \sin \left (b x \right )+b x \cos \left (b x \right )\right )}{3 b^{4} x}\) | \(100\) |
Input:
int(x^2*Si(b*x),x,method=_RETURNVERBOSE)
Output:
1/3*x^3*Si(b*x)-1/3/b^3*(-b^2*x^2*cos(b*x)+2*cos(b*x)+2*b*x*sin(b*x))
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int x^2 \text {Si}(b x) \, dx=\frac {b^{3} x^{3} \operatorname {Si}\left (b x\right ) - 2 \, b x \sin \left (b x\right ) + {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \] Input:
integrate(x^2*sin_integral(b*x),x, algorithm="fricas")
Output:
1/3*(b^3*x^3*sin_integral(b*x) - 2*b*x*sin(b*x) + (b^2*x^2 - 2)*cos(b*x))/ b^3
Time = 0.87 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int x^2 \text {Si}(b x) \, dx=\frac {x^{3} \operatorname {Si}{\left (b x \right )}}{3} + \frac {x^{2} \cos {\left (b x \right )}}{3 b} - \frac {2 x \sin {\left (b x \right )}}{3 b^{2}} - \frac {2 \cos {\left (b x \right )}}{3 b^{3}} \] Input:
integrate(x**2*Si(b*x),x)
Output:
x**3*Si(b*x)/3 + x**2*cos(b*x)/(3*b) - 2*x*sin(b*x)/(3*b**2) - 2*cos(b*x)/ (3*b**3)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int x^2 \text {Si}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right ) - \frac {2 \, b x \sin \left (b x\right ) - {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \] Input:
integrate(x^2*sin_integral(b*x),x, algorithm="maxima")
Output:
1/3*x^3*sin_integral(b*x) - 1/3*(2*b*x*sin(b*x) - (b^2*x^2 - 2)*cos(b*x))/ b^3
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78 \[ \int x^2 \text {Si}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right ) - \frac {2 \, x \sin \left (b x\right )}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \] Input:
integrate(x^2*sin_integral(b*x),x, algorithm="giac")
Output:
1/3*x^3*sin_integral(b*x) - 2/3*x*sin(b*x)/b^2 + 1/3*(b^2*x^2 - 2)*cos(b*x )/b^3
Timed out. \[ \int x^2 \text {Si}(b x) \, dx=\frac {x^3\,\mathrm {sinint}\left (b\,x\right )}{3}-\frac {\cos \left (b\,x\right )\,\left (\frac {2}{b^3}-\frac {x^2}{b}\right )}{3}-\frac {2\,x\,\sin \left (b\,x\right )}{3\,b^2} \] Input:
int(x^2*sinint(b*x),x)
Output:
(x^3*sinint(b*x))/3 - (cos(b*x)*(2/b^3 - x^2/b))/3 - (2*x*sin(b*x))/(3*b^2 )
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int x^2 \text {Si}(b x) \, dx=\frac {\cos \left (b x \right ) b^{2} x^{2}-2 \cos \left (b x \right )+\mathit {si} \left (b x \right ) b^{3} x^{3}-2 \sin \left (b x \right ) b x}{3 b^{3}} \] Input:
int(x^2*Si(b*x),x)
Output:
(cos(b*x)*b**2*x**2 - 2*cos(b*x) + si(b*x)*b**3*x**3 - 2*sin(b*x)*b*x)/(3* b**3)