\(\int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx\) [9]

Optimal result

Integrand size = 10, antiderivative size = 105 \[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=-\frac {5 e^{2 b x}}{6 b^3}+\frac {e^{2 b x} x}{3 b^2}-\frac {4 e^{b x} \operatorname {ExpIntegralEi}(b x)}{3 b^3}+\frac {4 e^{b x} x \operatorname {ExpIntegralEi}(b x)}{3 b^2}-\frac {2 e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)}{3 b}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2+\frac {4 \operatorname {ExpIntegralEi}(2 b x)}{3 b^3} \] Output:

-5/6*exp(2*b*x)/b^3+1/3*exp(2*b*x)*x/b^2-4/3*exp(b*x)*Ei(b*x)/b^3+4/3*exp( 
b*x)*x*Ei(b*x)/b^2-2/3*exp(b*x)*x^2*Ei(b*x)/b+1/3*x^3*Ei(b*x)^2+4/3*Ei(2*b 
*x)/b^3
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\frac {e^{2 b x} (-5+2 b x)-4 e^{b x} \left (2-2 b x+b^2 x^2\right ) \operatorname {ExpIntegralEi}(b x)+2 b^3 x^3 \operatorname {ExpIntegralEi}(b x)^2+8 \operatorname {ExpIntegralEi}(2 b x)}{6 b^3} \] Input:

Integrate[x^2*ExpIntegralEi[b*x]^2,x]
 

Output:

(E^(2*b*x)*(-5 + 2*b*x) - 4*E^(b*x)*(2 - 2*b*x + b^2*x^2)*ExpIntegralEi[b* 
x] + 2*b^3*x^3*ExpIntegralEi[b*x]^2 + 8*ExpIntegralEi[2*b*x])/(6*b^3)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {7041, 7044, 27, 2607, 2624, 7044, 27, 2624, 7043, 27, 2609}

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.

\(\displaystyle \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx\)

\(\Big \downarrow \) 7041

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \int e^{b x} x^2 \operatorname {ExpIntegralEi}(b x)dx\)

\(\Big \downarrow \) 7044

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\int \frac {e^{2 b x} x}{b}dx+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\int e^{2 b x} xdx}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {\int e^{2 b x}dx}{2 b}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 2624

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \int e^{b x} x \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 7044

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \left (-\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\int \frac {e^{2 b x}}{b}dx+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \left (-\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\int e^{2 b x}dx}{b}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 2624

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \left (-\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 7043

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\int \frac {e^{2 b x}}{b x}dx}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \left (-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\int \frac {e^{2 b x}}{x}dx}{b}}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(b x)^2-\frac {2}{3} \left (-\frac {2 \left (-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 b x)}{b}}{b}\right )}{b}-\frac {\frac {x e^{2 b x}}{2 b}-\frac {e^{2 b x}}{4 b^2}}{b}+\frac {x^2 e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\right )\)

Input:

Int[x^2*ExpIntegralEi[b*x]^2,x]
 

Output:

(x^3*ExpIntegralEi[b*x]^2)/3 - (2*(-((-1/4*E^(2*b*x)/b^2 + (E^(2*b*x)*x)/( 
2*b))/b) + (E^(b*x)*x^2*ExpIntegralEi[b*x])/b - (2*(-1/2*E^(2*b*x)/b^2 + ( 
E^(b*x)*x*ExpIntegralEi[b*x])/b - ((E^(b*x)*ExpIntegralEi[b*x])/b - ExpInt 
egralEi[2*b*x]/b)/b))/b))/3
 

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 2607

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), 
x] - Simp[d*(m/(f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ 
n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* 
m] &&  !TrueQ[$UseGamma]
 

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2624

Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; 
FreeQ[{F, n}, x] && LinearQ[v, x]
 

rule 7041

Int[ExpIntegralEi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Ex 
pIntegralEi[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*E^(b*x)*ExpInteg 
ralEi[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

rule 7043

Int[E^((a_.) + (b_.)*(x_))*ExpIntegralEi[(c_.) + (d_.)*(x_)], x_Symbol] :> 
Simp[E^(a + b*x)*(ExpIntegralEi[c + d*x]/b), x] - Simp[d/b   Int[E^(a + c + 
 (b + d)*x)/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]
 

rule 7044

Int[E^((a_.) + (b_.)*(x_))*ExpIntegralEi[(c_.) + (d_.)*(x_)]*(x_)^(m_.), x_ 
Symbol] :> Simp[x^m*E^(a + b*x)*(ExpIntegralEi[c + d*x]/b), x] + (-Simp[d/b 
   Int[x^m*(E^(a + c + (b + d)*x)/(c + d*x)), x], x] - Simp[m/b   Int[x^(m 
- 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x], x]) /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {expIntegral}\left (b x \right )^{2} b^{3} x^{3}}{3}-2 \,\operatorname {expIntegral}\left (b x \right ) \left (\frac {b^{2} x^{2} {\mathrm e}^{b x}}{3}-\frac {2 b x \,{\mathrm e}^{b x}}{3}+\frac {2 \,{\mathrm e}^{b x}}{3}\right )+\frac {{\mathrm e}^{2 b x} b x}{3}-\frac {5 \,{\mathrm e}^{2 b x}}{6}-\frac {4 \,\operatorname {expIntegral}_{1}\left (-2 b x \right )}{3}}{b^{3}}\)	\(79\)
default	\(\frac {\frac {\operatorname {expIntegral}\left (b x \right )^{2} b^{3} x^{3}}{3}-2 \,\operatorname {expIntegral}\left (b x \right ) \left (\frac {b^{2} x^{2} {\mathrm e}^{b x}}{3}-\frac {2 b x \,{\mathrm e}^{b x}}{3}+\frac {2 \,{\mathrm e}^{b x}}{3}\right )+\frac {{\mathrm e}^{2 b x} b x}{3}-\frac {5 \,{\mathrm e}^{2 b x}}{6}-\frac {4 \,\operatorname {expIntegral}_{1}\left (-2 b x \right )}{3}}{b^{3}}\)	\(79\)

Input:

int(x^2*Ei(b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b^3*(1/3*Ei(b*x)^2*b^3*x^3-2*Ei(b*x)*(1/3*b^2*x^2*exp(b*x)-2/3*b*x*exp(b 
*x)+2/3*exp(b*x))+1/3*exp(b*x)^2*b*x-5/6*exp(b*x)^2-4/3*Ei(1,-2*b*x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.59 \[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\frac {2 \, b^{3} x^{3} {\rm Ei}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} {\rm Ei}\left (b x\right ) e^{\left (b x\right )} + {\left (2 \, b x - 5\right )} e^{\left (2 \, b x\right )} + 8 \, {\rm Ei}\left (2 \, b x\right )}{6 \, b^{3}} \] Input:

integrate(x^2*Ei(b*x)^2,x, algorithm="fricas")
 

Output:

1/6*(2*b^3*x^3*Ei(b*x)^2 - 4*(b^2*x^2 - 2*b*x + 2)*Ei(b*x)*e^(b*x) + (2*b* 
x - 5)*e^(2*b*x) + 8*Ei(2*b*x))/b^3
 

Sympy [F]

\[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\int x^{2} \operatorname {Ei}^{2}{\left (b x \right )}\, dx \] Input:

integrate(x**2*Ei(b*x)**2,x)
 

Output:

Integral(x**2*Ei(b*x)**2, x)
 

Maxima [F]

\[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\int { x^{2} {\rm Ei}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^2*Ei(b*x)^2,x, algorithm="maxima")
 

Output:

integrate(x^2*Ei(b*x)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\frac {1}{3} \, x^{3} {\rm Ei}\left (b x\right )^{2} - \frac {2 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} {\rm Ei}\left (b x\right ) e^{\left (b x\right )}}{3 \, b^{3}} + \frac {2 \, b x e^{\left (2 \, b x\right )} + 8 \, {\rm Ei}\left (2 \, b x\right ) - 5 \, e^{\left (2 \, b x\right )}}{6 \, b^{3}} \] Input:

integrate(x^2*Ei(b*x)^2,x, algorithm="giac")
 

Output:

1/3*x^3*Ei(b*x)^2 - 2/3*(b^2*x^2 - 2*b*x + 2)*Ei(b*x)*e^(b*x)/b^3 + 1/6*(2 
*b*x*e^(2*b*x) + 8*Ei(2*b*x) - 5*e^(2*b*x))/b^3
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\int x^2\,{\mathrm {ei}\left (b\,x\right )}^2 \,d x \] Input:

int(x^2*ei(b*x)^2,x)
 

Output:

int(x^2*ei(b*x)^2, x)
 

Reduce [F]

\[ \int x^2 \operatorname {ExpIntegralEi}(b x)^2 \, dx=\int \mathit {ei} \left (b x \right )^{2} x^{2}d x \] Input:

int(x^2*Ei(b*x)^2,x)
 

Output:

int(ei(b*x)**2*x**2,x)