Integrand size = 10, antiderivative size = 82 \[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=-\frac {b e^{a+b x}}{2 a x}-\frac {b^2 e^a \operatorname {ExpIntegralEi}(b x)}{2 a^2}+\frac {b^2 e^a \operatorname {ExpIntegralEi}(b x)}{2 a}+\frac {b^2 \operatorname {ExpIntegralEi}(a+b x)}{2 a^2}-\frac {\operatorname {ExpIntegralEi}(a+b x)}{2 x^2} \] Output:
-1/2*b*exp(b*x+a)/a/x-1/2*b^2*exp(a)*Ei(b*x)/a^2+1/2*b^2*exp(a)*Ei(b*x)/a+ 1/2*b^2*Ei(b*x+a)/a^2-1/2*Ei(b*x+a)/x^2
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.73 \[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\frac {-a b e^{a+b x} x+(-1+a) b^2 e^a x^2 \operatorname {ExpIntegralEi}(b x)+\left (-a^2+b^2 x^2\right ) \operatorname {ExpIntegralEi}(a+b x)}{2 a^2 x^2} \] Input:
Integrate[ExpIntegralEi[a + b*x]/x^3,x]
Output:
(-(a*b*E^(a + b*x)*x) + (-1 + a)*b^2*E^a*x^2*ExpIntegralEi[b*x] + (-a^2 + b^2*x^2)*ExpIntegralEi[a + b*x])/(2*a^2*x^2)
Time = 0.41 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7039, 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.
\(\displaystyle \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 7039 |
\(\displaystyle \frac {1}{2} b \int \frac {e^{a+b x}}{x^2 (a+b x)}dx-\frac {\operatorname {ExpIntegralEi}(a+b x)}{2 x^2}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} b \int \left (\frac {e^{a+b x} b^2}{a^2 (a+b x)}-\frac {e^{a+b x} b}{a^2 x}+\frac {e^{a+b x}}{a x^2}\right )dx-\frac {\operatorname {ExpIntegralEi}(a+b x)}{2 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} b \left (-\frac {e^a b \operatorname {ExpIntegralEi}(b x)}{a^2}+\frac {b \operatorname {ExpIntegralEi}(a+b x)}{a^2}+\frac {e^a b \operatorname {ExpIntegralEi}(b x)}{a}-\frac {e^{a+b x}}{a x}\right )-\frac {\operatorname {ExpIntegralEi}(a+b x)}{2 x^2}\) |
Input:
Int[ExpIntegralEi[a + b*x]/x^3,x]
Output:
-1/2*ExpIntegralEi[a + b*x]/x^2 + (b*(-(E^(a + b*x)/(a*x)) - (b*E^a*ExpInt egralEi[b*x])/a^2 + (b*E^a*ExpIntegralEi[b*x])/a + (b*ExpIntegralEi[a + b* x])/a^2))/2
Int[ExpIntegralEi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(ExpIntegralEi[a + b*x]/(d*(m + 1))), x] - Simp[ b/(d*(m + 1)) Int[(c + d*x)^(m + 1)*(E^(a + b*x)/(a + b*x)), x], x] /; Fr eeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91
method | result | size |
parts | \(-\frac {\operatorname {expIntegral}\left (b x +a \right )}{2 x^{2}}+\frac {b^{2} \left (\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{a^{2}}+\frac {-\frac {{\mathrm e}^{b x +a}}{b x}-{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{a}-\frac {\operatorname {expIntegral}_{1}\left (-b x -a \right )}{a^{2}}\right )}{2}\) | \(75\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {expIntegral}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{2 a^{2}}+\frac {-\frac {{\mathrm e}^{b x +a}}{b x}-{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{2 a}-\frac {\operatorname {expIntegral}_{1}\left (-b x -a \right )}{2 a^{2}}\right )\) | \(78\) |
default | \(b^{2} \left (-\frac {\operatorname {expIntegral}\left (b x +a \right )}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{2 a^{2}}+\frac {-\frac {{\mathrm e}^{b x +a}}{b x}-{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-b x \right )}{2 a}-\frac {\operatorname {expIntegral}_{1}\left (-b x -a \right )}{2 a^{2}}\right )\) | \(78\) |
Input:
int(Ei(b*x+a)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*Ei(b*x+a)/x^2+1/2*b^2*(1/a^2*exp(a)*Ei(1,-b*x)+1/a*(-exp(b*x+a)/b/x-e xp(a)*Ei(1,-b*x))-1/a^2*Ei(1,-b*x-a))
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\frac {{\left (a - 1\right )} b^{2} x^{2} {\rm Ei}\left (b x\right ) e^{a} - a b x e^{\left (b x + a\right )} + {\left (b^{2} x^{2} - a^{2}\right )} {\rm Ei}\left (b x + a\right )}{2 \, a^{2} x^{2}} \] Input:
integrate(Ei(b*x+a)/x^3,x, algorithm="fricas")
Output:
1/2*((a - 1)*b^2*x^2*Ei(b*x)*e^a - a*b*x*e^(b*x + a) + (b^2*x^2 - a^2)*Ei( b*x + a))/(a^2*x^2)
\[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {Ei}{\left (a + b x \right )}}{x^{3}}\, dx \] Input:
integrate(Ei(b*x+a)/x**3,x)
Output:
Integral(Ei(a + b*x)/x**3, x)
\[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\int { \frac {{\rm Ei}\left (b x + a\right )}{x^{3}} \,d x } \] Input:
integrate(Ei(b*x+a)/x^3,x, algorithm="maxima")
Output:
integrate(Ei(b*x + a)/x^3, x)
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.73 \[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\frac {{\left (a b x {\rm Ei}\left (b x\right ) e^{a} - b x {\rm Ei}\left (b x\right ) e^{a} + b x {\rm Ei}\left (b x + a\right ) - a e^{\left (b x + a\right )}\right )} b}{2 \, a^{2} x} - \frac {{\rm Ei}\left (b x + a\right )}{2 \, x^{2}} \] Input:
integrate(Ei(b*x+a)/x^3,x, algorithm="giac")
Output:
1/2*(a*b*x*Ei(b*x)*e^a - b*x*Ei(b*x)*e^a + b*x*Ei(b*x + a) - a*e^(b*x + a) )*b/(a^2*x) - 1/2*Ei(b*x + a)/x^2
Timed out. \[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {ei}\left (a+b\,x\right )}{x^3} \,d x \] Input:
int(ei(a + b*x)/x^3,x)
Output:
int(ei(a + b*x)/x^3, x)
\[ \int \frac {\operatorname {ExpIntegralEi}(a+b x)}{x^3} \, dx=\int \frac {\mathit {ei} \left (b x +a \right )}{x^{3}}d x \] Input:
int(Ei(b*x+a)/x^3,x)
Output:
int(ei(a + b*x)/x**3,x)