Integrand size = 13, antiderivative size = 45 \[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=-\frac {e^{2 b x}}{x}-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}+\frac {1}{2} b \operatorname {ExpIntegralEi}(b x)^2+2 b \operatorname {ExpIntegralEi}(2 b x) \] Output:
-exp(2*b*x)/x-exp(b*x)*Ei(b*x)/x+1/2*b*Ei(b*x)^2+2*b*Ei(2*b*x)
\[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx \] Input:
Integrate[(E^(b*x)*ExpIntegralEi[b*x])/x^2,x]
Output:
Integrate[(E^(b*x)*ExpIntegralEi[b*x])/x^2, x]
Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {7045, 27, 2608, 2609, 7237}
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.
\(\displaystyle \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 7045 |
\(\displaystyle b \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}dx+b \int \frac {e^{2 b x}}{b x^2}dx-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}dx+\int \frac {e^{2 b x}}{x^2}dx-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle b \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}dx+2 b \int \frac {e^{2 b x}}{x}dx-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}-\frac {e^{2 b x}}{x}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle b \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}dx-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}+2 b \operatorname {ExpIntegralEi}(2 b x)-\frac {e^{2 b x}}{x}\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {1}{2} b \operatorname {ExpIntegralEi}(b x)^2-\frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x}+2 b \operatorname {ExpIntegralEi}(2 b x)-\frac {e^{2 b x}}{x}\) |
Input:
Int[(E^(b*x)*ExpIntegralEi[b*x])/x^2,x]
Output:
-(E^(2*b*x)/x) - (E^(b*x)*ExpIntegralEi[b*x])/x + (b*ExpIntegralEi[b*x]^2) /2 + 2*b*ExpIntegralEi[2*b*x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))) , x] - Simp[f*g*n*(Log[F]/(d*(m + 1))) 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] && LtQ[m, -1] && In tegerQ[2*m] && !TrueQ[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[E^((a_.) + (b_.)*(x_))*ExpIntegralEi[(c_.) + (d_.)*(x_)]*(x_)^(m_), x_S ymbol] :> Simp[x^(m + 1)*E^(a + b*x)*(ExpIntegralEi[c + d*x]/(m + 1)), x] + (-Simp[b/(m + 1) Int[x^(m + 1)*E^(a + b*x)*ExpIntegralEi[c + d*x], x], x ] - Simp[d/(m + 1) Int[x^(m + 1)*(E^(a + c + (b + d)*x)/(c + d*x)), x], x ]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
\[\int \frac {{\mathrm e}^{b x} \operatorname {expIntegral}\left (b x \right )}{x^{2}}d x\]
Input:
int(exp(b*x)*Ei(b*x)/x^2,x)
Output:
int(exp(b*x)*Ei(b*x)/x^2,x)
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\frac {b x {\rm Ei}\left (b x\right )^{2} + 4 \, b x {\rm Ei}\left (2 \, b x\right ) - 2 \, {\rm Ei}\left (b x\right ) e^{\left (b x\right )} - 2 \, e^{\left (2 \, b x\right )}}{2 \, x} \] Input:
integrate(exp(b*x)*Ei(b*x)/x^2,x, algorithm="fricas")
Output:
1/2*(b*x*Ei(b*x)^2 + 4*b*x*Ei(2*b*x) - 2*Ei(b*x)*e^(b*x) - 2*e^(2*b*x))/x
\[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\int \frac {e^{b x} \operatorname {Ei}{\left (b x \right )}}{x^{2}}\, dx \] Input:
integrate(exp(b*x)*Ei(b*x)/x**2,x)
Output:
Integral(exp(b*x)*Ei(b*x)/x**2, x)
\[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\int { \frac {{\rm Ei}\left (b x\right ) e^{\left (b x\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b*x)*Ei(b*x)/x^2,x, algorithm="maxima")
Output:
integrate(Ei(b*x)*e^(b*x)/x^2, x)
\[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\int { \frac {{\rm Ei}\left (b x\right ) e^{\left (b x\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b*x)*Ei(b*x)/x^2,x, algorithm="giac")
Output:
integrate(Ei(b*x)*e^(b*x)/x^2, x)
Timed out. \[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{b\,x}\,\mathrm {ei}\left (b\,x\right )}{x^2} \,d x \] Input:
int((exp(b*x)*ei(b*x))/x^2,x)
Output:
int((exp(b*x)*ei(b*x))/x^2, x)
\[ \int \frac {e^{b x} \operatorname {ExpIntegralEi}(b x)}{x^2} \, dx=\int \frac {e^{b x} \mathit {ei} \left (b x \right )}{x^{2}}d x \] Input:
int(exp(b*x)*Ei(b*x)/x^2,x)
Output:
int((e**(b*x)*ei(b*x))/x**2,x)