Integrand size = 10, antiderivative size = 118 \[ \int x^2 \text {Si}(a+b x) \, dx=-\frac {2 \cos (a+b x)}{3 b^3}+\frac {a^2 \cos (a+b x)}{3 b^3}-\frac {a x \cos (a+b x)}{3 b^2}+\frac {x^2 \cos (a+b x)}{3 b}+\frac {a \sin (a+b x)}{3 b^3}-\frac {2 x \sin (a+b x)}{3 b^2}+\frac {a^3 \text {Si}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Si}(a+b x) \] Output:
-2/3*cos(b*x+a)/b^3+1/3*a^2*cos(b*x+a)/b^3-1/3*a*x*cos(b*x+a)/b^2+1/3*x^2* cos(b*x+a)/b+1/3*a*sin(b*x+a)/b^3-2/3*x*sin(b*x+a)/b^2+1/3*a^3*Si(b*x+a)/b ^3+1/3*x^3*Si(b*x+a)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.53 \[ \int x^2 \text {Si}(a+b x) \, dx=\frac {\left (-2+a^2-a b x+b^2 x^2\right ) \cos (a+b x)+(a-2 b x) \sin (a+b x)+\left (a^3+b^3 x^3\right ) \text {Si}(a+b x)}{3 b^3} \] Input:
Integrate[x^2*SinIntegral[a + b*x],x]
Output:
((-2 + a^2 - a*b*x + b^2*x^2)*Cos[a + b*x] + (a - 2*b*x)*Sin[a + b*x] + (a ^3 + b^3*x^3)*SinIntegral[a + b*x])/(3*b^3)
Time = 0.52 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7057, 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.
\(\displaystyle \int x^2 \text {Si}(a+b x) \, dx\) |
\(\Big \downarrow \) 7057 |
\(\displaystyle \frac {1}{3} x^3 \text {Si}(a+b x)-\frac {1}{3} b \int \frac {x^3 \sin (a+b x)}{a+b x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} x^3 \text {Si}(a+b x)-\frac {1}{3} b \int \left (-\frac {\sin (a+b x) a^3}{b^3 (a+b x)}+\frac {\sin (a+b x) a^2}{b^3}-\frac {x \sin (a+b x) a}{b^2}+\frac {x^2 \sin (a+b x)}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \text {Si}(a+b x)-\frac {1}{3} b \left (-\frac {a^3 \text {Si}(a+b x)}{b^4}-\frac {a^2 \cos (a+b x)}{b^4}-\frac {a \sin (a+b x)}{b^4}+\frac {2 \cos (a+b x)}{b^4}+\frac {2 x \sin (a+b x)}{b^3}+\frac {a x \cos (a+b x)}{b^3}-\frac {x^2 \cos (a+b x)}{b^2}\right )\) |
Input:
Int[x^2*SinIntegral[a + b*x],x]
Output:
(x^3*SinIntegral[a + b*x])/3 - (b*((2*Cos[a + b*x])/b^4 - (a^2*Cos[a + b*x ])/b^4 + (a*x*Cos[a + b*x])/b^3 - (x^2*Cos[a + b*x])/b^2 - (a*Sin[a + b*x] )/b^4 + (2*x*Sin[a + b*x])/b^3 - (a^3*SinIntegral[a + b*x])/b^4))/3
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 = 0.85 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Si}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Si}\left (b x +a \right )}{3}+a^{2} \cos \left (b x +a \right )+a \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )+\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )}{3}-\frac {2 \cos \left (b x +a \right )}{3}-\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )}{3}}{b^{3}}\) | \(99\) |
default | \(\frac {\frac {\operatorname {Si}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Si}\left (b x +a \right )}{3}+a^{2} \cos \left (b x +a \right )+a \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )+\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )}{3}-\frac {2 \cos \left (b x +a \right )}{3}-\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )}{3}}{b^{3}}\) | \(99\) |
parts | \(\frac {x^{3} \operatorname {Si}\left (b x +a \right )}{3}-\frac {-a^{3} \operatorname {Si}\left (b x +a \right )-3 a^{2} \cos \left (b x +a \right )-3 a \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )}{3 b^{3}}\) | \(100\) |
orering | \(\frac {\left (b^{5} x^{5}+a^{3} b^{2} x^{2}+8 b^{3} x^{3}+4 a \,b^{2} x^{2}-4 a^{2} b x +6 a^{3}-8 b x -12 a \right ) \operatorname {Si}\left (b x +a \right )}{3 b^{5} x^{2}}-\frac {\left (5 b^{3} x^{3}+2 a \,b^{2} x^{2}-2 a^{2} b x +4 a^{3}-6 b x -8 a \right ) \left (2 x \,\operatorname {Si}\left (b x +a \right )+\frac {x^{2} \sin \left (b x +a \right ) b}{b x +a}\right )}{3 b^{5} x^{3}}+\frac {\left (b^{2} x^{2}-b x a +a^{2}-2\right ) \left (b x +a \right ) \left (2 \,\operatorname {Si}\left (b x +a \right )+\frac {4 x \sin \left (b x +a \right ) b}{b x +a}+\frac {x^{2} b^{2} \cos \left (b x +a \right )}{b x +a}-\frac {x^{2} \sin \left (b x +a \right ) b^{2}}{\left (b x +a \right )^{2}}\right )}{3 b^{5} x^{2}}\) | \(240\) |
Input:
int(x^2*Si(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b^3*(1/3*Si(b*x+a)*b^3*x^3+1/3*a^3*Si(b*x+a)+a^2*cos(b*x+a)+a*(sin(b*x+a )-(b*x+a)*cos(b*x+a))+1/3*(b*x+a)^2*cos(b*x+a)-2/3*cos(b*x+a)-2/3*(b*x+a)* sin(b*x+a))
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.54 \[ \int x^2 \text {Si}(a+b x) \, dx=\frac {{\left (b^{2} x^{2} - a b x + a^{2} - 2\right )} \cos \left (b x + a\right ) - {\left (2 \, b x - a\right )} \sin \left (b x + a\right ) + {\left (b^{3} x^{3} + a^{3}\right )} \operatorname {Si}\left (b x + a\right )}{3 \, b^{3}} \] Input:
integrate(x^2*sin_integral(b*x+a),x, algorithm="fricas")
Output:
1/3*((b^2*x^2 - a*b*x + a^2 - 2)*cos(b*x + a) - (2*b*x - a)*sin(b*x + a) + (b^3*x^3 + a^3)*sin_integral(b*x + a))/b^3
\[ \int x^2 \text {Si}(a+b x) \, dx=\int x^{2} \operatorname {Si}{\left (a + b x \right )}\, dx \] Input:
integrate(x**2*Si(b*x+a),x)
Output:
Integral(x**2*Si(a + b*x), x)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int x^2 \text {Si}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {Si}\left (b x + a\right ) - \frac {a^{3} {\left (i \, {\rm Ei}\left (i \, b x + i \, a\right ) - i \, {\rm Ei}\left (-i \, b x - i \, a\right )\right )} - 2 \, {\left ({\left (b x + a\right )}^{2} - 3 \, {\left (b x + a\right )} a + 3 \, a^{2} - 2\right )} \cos \left (b x + a\right ) + 2 \, {\left (2 \, b x - a\right )} \sin \left (b x + a\right )}{6 \, b^{3}} \] Input:
integrate(x^2*sin_integral(b*x+a),x, algorithm="maxima")
Output:
1/3*x^3*sin_integral(b*x + a) - 1/6*(a^3*(I*Ei(I*b*x + I*a) - I*Ei(-I*b*x - I*a)) - 2*((b*x + a)^2 - 3*(b*x + a)*a + 3*a^2 - 2)*cos(b*x + a) + 2*(2* b*x - a)*sin(b*x + a))/b^3
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.14 \[ \int x^2 \text {Si}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {Si}\left (b x + a\right ) - \frac {{\left (2 \, b^{2} x^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - a^{3} \Im \left ( \operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + a^{3} \Im \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 2 \, a^{3} \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 2 \, a b x \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - 2 \, b^{2} x^{2} - a^{3} \Im \left ( \operatorname {Ci}\left (b x + a\right ) \right ) + a^{3} \Im \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) - 2 \, a^{3} \operatorname {Si}\left (b x + a\right ) + 2 \, a^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 2 \, a b x + 8 \, b x \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 2 \, a^{2} - 4 \, a \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - 4 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 4\right )} b}{6 \, {\left (b^{4} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + b^{4}\right )}} \] Input:
integrate(x^2*sin_integral(b*x+a),x, algorithm="giac")
Output:
1/3*x^3*sin_integral(b*x + a) - 1/6*(2*b^2*x^2*tan(1/2*b*x + 1/2*a)^2 - a^ 3*imag_part(cos_integral(b*x + a))*tan(1/2*b*x + 1/2*a)^2 + a^3*imag_part( cos_integral(-b*x - a))*tan(1/2*b*x + 1/2*a)^2 - 2*a^3*sin_integral(b*x + a)*tan(1/2*b*x + 1/2*a)^2 - 2*a*b*x*tan(1/2*b*x + 1/2*a)^2 - 2*b^2*x^2 - a ^3*imag_part(cos_integral(b*x + a)) + a^3*imag_part(cos_integral(-b*x - a) ) - 2*a^3*sin_integral(b*x + a) + 2*a^2*tan(1/2*b*x + 1/2*a)^2 + 2*a*b*x + 8*b*x*tan(1/2*b*x + 1/2*a) - 2*a^2 - 4*a*tan(1/2*b*x + 1/2*a) - 4*tan(1/2 *b*x + 1/2*a)^2 + 4)*b/(b^4*tan(1/2*b*x + 1/2*a)^2 + b^4)
Timed out. \[ \int x^2 \text {Si}(a+b x) \, dx=\int x^2\,\mathrm {sinint}\left (a+b\,x\right ) \,d x \] Input:
int(x^2*sinint(a + b*x),x)
Output:
int(x^2*sinint(a + b*x), x)
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int x^2 \text {Si}(a+b x) \, dx=\frac {\cos \left (b x +a \right ) a^{2}-\cos \left (b x +a \right ) a b x +\cos \left (b x +a \right ) b^{2} x^{2}-2 \cos \left (b x +a \right )+\mathit {si} \left (b x +a \right ) a^{3}+\mathit {si} \left (b x +a \right ) b^{3} x^{3}+\sin \left (b x +a \right ) a -2 \sin \left (b x +a \right ) b x}{3 b^{3}} \] Input:
int(x^2*Si(b*x+a),x)
Output:
(cos(a + b*x)*a**2 - cos(a + b*x)*a*b*x + cos(a + b*x)*b**2*x**2 - 2*cos(a + b*x) + si(a + b*x)*a**3 + si(a + b*x)*b**3*x**3 + sin(a + b*x)*a - 2*si n(a + b*x)*b*x)/(3*b**3)