Integrand size = 11, antiderivative size = 51 \[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=-\frac {e^{2 b x}}{2 b^2}-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b^2}+\frac {e^{b x} x \operatorname {ExpIntegralEi}(b x)}{b}+\frac {\operatorname {ExpIntegralEi}(2 b x)}{b^2} \] Output:
-1/2*exp(2*b*x)/b^2-exp(b*x)*Ei(b*x)/b^2+exp(b*x)*x*Ei(b*x)/b+Ei(2*b*x)/b^ 2
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=-\frac {e^{2 b x}}{2 b^2}+\frac {e^{b x} (-1+b x) \operatorname {ExpIntegralEi}(b x)}{b^2}+\frac {\operatorname {ExpIntegralEi}(2 b x)}{b^2} \] Input:
Integrate[E^(b*x)*x*ExpIntegralEi[b*x],x]
Output:
-1/2*E^(2*b*x)/b^2 + (E^(b*x)*(-1 + b*x)*ExpIntegralEi[b*x])/b^2 + ExpInte gralEi[2*b*x]/b^2
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {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.
\(\displaystyle \int x e^{b x} \operatorname {ExpIntegralEi}(b x) \, dx\) |
\(\Big \downarrow \) 7044 |
\(\displaystyle -\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\int \frac {e^{2 b x}}{b}dx+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {\int e^{2 b x}dx}{b}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle -\frac {\int e^{b x} \operatorname {ExpIntegralEi}(b x)dx}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\) |
\(\Big \downarrow \) 7043 |
\(\displaystyle -\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\int \frac {e^{2 b x}}{b x}dx}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\int \frac {e^{2 b x}}{x}dx}{b}}{b}-\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle -\frac {e^{2 b x}}{2 b^2}+\frac {x e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{b}-\frac {\operatorname {ExpIntegralEi}(2 b x)}{b}}{b}\) |
Input:
Int[E^(b*x)*x*ExpIntegralEi[b*x],x]
Output:
-1/2*E^(2*b*x)/b^2 + (E^(b*x)*x*ExpIntegralEi[b*x])/b - ((E^(b*x)*ExpInteg ralEi[b*x])/b - ExpIntegralEi[2*b*x]/b)/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
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]
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]
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\operatorname {expIntegral}\left (b x \right ) \left (b x \,{\mathrm e}^{b x}-{\mathrm e}^{b x}\right )-\frac {{\mathrm e}^{2 b x}}{2}-\operatorname {expIntegral}_{1}\left (-2 b x \right )}{b^{2}}\) | \(41\) |
default | \(\frac {\operatorname {expIntegral}\left (b x \right ) \left (b x \,{\mathrm e}^{b x}-{\mathrm e}^{b x}\right )-\frac {{\mathrm e}^{2 b x}}{2}-\operatorname {expIntegral}_{1}\left (-2 b x \right )}{b^{2}}\) | \(41\) |
Input:
int(exp(b*x)*x*Ei(b*x),x,method=_RETURNVERBOSE)
Output:
1/b^2*(Ei(b*x)*(b*x*exp(b*x)-exp(b*x))-1/2*exp(b*x)^2-Ei(1,-2*b*x))
Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.69 \[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=\frac {2 \, {\left (b x - 1\right )} {\rm Ei}\left (b x\right ) e^{\left (b x\right )} + 2 \, {\rm Ei}\left (2 \, b x\right ) - e^{\left (2 \, b x\right )}}{2 \, b^{2}} \] Input:
integrate(exp(b*x)*x*Ei(b*x),x, algorithm="fricas")
Output:
1/2*(2*(b*x - 1)*Ei(b*x)*e^(b*x) + 2*Ei(2*b*x) - e^(2*b*x))/b^2
\[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=\int x e^{b x} \operatorname {Ei}{\left (b x \right )}\, dx \] Input:
integrate(exp(b*x)*x*Ei(b*x),x)
Output:
Integral(x*exp(b*x)*Ei(b*x), x)
\[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=\int { x {\rm Ei}\left (b x\right ) e^{\left (b x\right )} \,d x } \] Input:
integrate(exp(b*x)*x*Ei(b*x),x, algorithm="maxima")
Output:
integrate(x*Ei(b*x)*e^(b*x), x)
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.75 \[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=\frac {{\left (b x - 1\right )} {\rm Ei}\left (b x\right ) e^{\left (b x\right )}}{b^{2}} + \frac {2 \, {\rm Ei}\left (2 \, b x\right ) - e^{\left (2 \, b x\right )}}{2 \, b^{2}} \] Input:
integrate(exp(b*x)*x*Ei(b*x),x, algorithm="giac")
Output:
(b*x - 1)*Ei(b*x)*e^(b*x)/b^2 + 1/2*(2*Ei(2*b*x) - e^(2*b*x))/b^2
Timed out. \[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=\int x\,{\mathrm {e}}^{b\,x}\,\mathrm {ei}\left (b\,x\right ) \,d x \] Input:
int(x*exp(b*x)*ei(b*x),x)
Output:
int(x*exp(b*x)*ei(b*x), x)
\[ \int e^{b x} x \operatorname {ExpIntegralEi}(b x) \, dx=\int e^{b x} \mathit {ei} \left (b x \right ) x d x \] Input:
int(exp(b*x)*x*Ei(b*x),x)
Output:
int(e**(b*x)*ei(b*x)*x,x)