Integrand size = 8, antiderivative size = 63 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=-\frac {e^{b x}}{9 x^3}-\frac {b e^{b x}}{18 x^2}-\frac {b^2 e^{b x}}{18 x}+\frac {1}{18} b^3 \operatorname {ExpIntegralEi}(b x)-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3} \] Output:
-1/9*exp(b*x)/x^3-1/18*b*exp(b*x)/x^2-1/18*b^2*exp(b*x)/x+1/18*b^3*Ei(b*x) -1/3*Ei(b*x)/x^3
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {-e^{b x} \left (2+b x+b^2 x^2\right )+\left (-6+b^3 x^3\right ) \operatorname {ExpIntegralEi}(b x)}{18 x^3} \] Input:
Integrate[ExpIntegralEi[b*x]/x^4,x]
Output:
(-(E^(b*x)*(2 + b*x + b^2*x^2)) + (-6 + b^3*x^3)*ExpIntegralEi[b*x])/(18*x ^3)
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7039, 27, 2608, 2608, 2608, 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.
\(\displaystyle \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx\) |
\(\Big \downarrow \) 7039 |
\(\displaystyle \frac {1}{3} b \int \frac {e^{b x}}{b x^4}dx-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {e^{b x}}{x^4}dx-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} b \int \frac {e^{b x}}{x^3}dx-\frac {e^{b x}}{3 x^3}\right )-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} b \left (\frac {1}{2} b \int \frac {e^{b x}}{x^2}dx-\frac {e^{b x}}{2 x^2}\right )-\frac {e^{b x}}{3 x^3}\right )-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} b \left (\frac {1}{2} b \left (b \int \frac {e^{b x}}{x}dx-\frac {e^{b x}}{x}\right )-\frac {e^{b x}}{2 x^2}\right )-\frac {e^{b x}}{3 x^3}\right )-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} b \left (\frac {1}{2} b \left (b \operatorname {ExpIntegralEi}(b x)-\frac {e^{b x}}{x}\right )-\frac {e^{b x}}{2 x^2}\right )-\frac {e^{b x}}{3 x^3}\right )-\frac {\operatorname {ExpIntegralEi}(b x)}{3 x^3}\) |
Input:
Int[ExpIntegralEi[b*x]/x^4,x]
Output:
-1/3*ExpIntegralEi[b*x]/x^3 + (-1/3*E^(b*x)/x^3 + (b*(-1/2*E^(b*x)/x^2 + ( b*(-(E^(b*x)/x) + b*ExpIntegralEi[b*x]))/2))/3)/3
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[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 = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {b^{3} x^{3} \operatorname {expIntegral}\left (b x \right )-b^{2} x^{2} {\mathrm e}^{b x}-b x \,{\mathrm e}^{b x}-6 \,\operatorname {expIntegral}\left (b x \right )-2 \,{\mathrm e}^{b x}}{18 x^{3}}\) | \(50\) |
parts | \(-\frac {\operatorname {expIntegral}\left (b x \right )}{3 x^{3}}+\frac {b^{3} \left (-\frac {{\mathrm e}^{b x}}{3 b^{3} x^{3}}-\frac {{\mathrm e}^{b x}}{6 b^{2} x^{2}}-\frac {{\mathrm e}^{b x}}{6 b x}-\frac {\operatorname {expIntegral}_{1}\left (-b x \right )}{6}\right )}{3}\) | \(61\) |
derivativedivides | \(b^{3} \left (-\frac {\operatorname {expIntegral}\left (b x \right )}{3 b^{3} x^{3}}-\frac {{\mathrm e}^{b x}}{9 b^{3} x^{3}}-\frac {{\mathrm e}^{b x}}{18 b^{2} x^{2}}-\frac {{\mathrm e}^{b x}}{18 b x}-\frac {\operatorname {expIntegral}_{1}\left (-b x \right )}{18}\right )\) | \(62\) |
default | \(b^{3} \left (-\frac {\operatorname {expIntegral}\left (b x \right )}{3 b^{3} x^{3}}-\frac {{\mathrm e}^{b x}}{9 b^{3} x^{3}}-\frac {{\mathrm e}^{b x}}{18 b^{2} x^{2}}-\frac {{\mathrm e}^{b x}}{18 b x}-\frac {\operatorname {expIntegral}_{1}\left (-b x \right )}{18}\right )\) | \(62\) |
meijerg | \(b^{3} \left (-\frac {\frac {1}{3}+\gamma +\ln \left (x \right )+\ln \left (-b \right )}{3 x^{3} b^{3}}-\frac {1}{2 x^{2} b^{2}}-\frac {1}{4 x b}-\frac {13}{108}+\frac {\gamma }{18}+\frac {\ln \left (x \right )}{18}+\frac {\ln \left (-b \right )}{18}+\frac {104 b^{3} x^{3}+216 b^{2} x^{2}+432 b x +96}{864 b^{3} x^{3}}-\frac {\left (16 b^{2} x^{2}+16 b x +32\right ) {\mathrm e}^{b x}}{288 b^{3} x^{3}}-\frac {\left (-16 b^{3} x^{3}+96\right ) \left (-\gamma -\ln \left (-b x \right )-\operatorname {expIntegral}_{1}\left (-b x \right )\right )}{288 b^{3} x^{3}}\right )\) | \(146\) |
Input:
int(Ei(b*x)/x^4,x,method=_RETURNVERBOSE)
Output:
1/18/x^3*(b^3*x^3*Ei(b*x)-b^2*x^2*exp(b*x)-b*x*exp(b*x)-6*Ei(b*x)-2*exp(b* x))
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {{\left (b^{3} x^{3} - 6\right )} {\rm Ei}\left (b x\right ) - {\left (b^{2} x^{2} + b x + 2\right )} e^{\left (b x\right )}}{18 \, x^{3}} \] Input:
integrate(Ei(b*x)/x^4,x, algorithm="fricas")
Output:
1/18*((b^3*x^3 - 6)*Ei(b*x) - (b^2*x^2 + b*x + 2)*e^(b*x))/x^3
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {b^{3} \operatorname {Ei}{\left (b x \right )}}{18} - \frac {b^{2} e^{b x}}{18 x} - \frac {b e^{b x}}{18 x^{2}} - \frac {e^{b x}}{9 x^{3}} - \frac {\operatorname {Ei}{\left (b x \right )}}{3 x^{3}} \] Input:
integrate(Ei(b*x)/x**4,x)
Output:
b**3*Ei(b*x)/18 - b**2*exp(b*x)/(18*x) - b*exp(b*x)/(18*x**2) - exp(b*x)/( 9*x**3) - Ei(b*x)/(3*x**3)
Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.33 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {1}{3} \, b^{3} \Gamma \left (-3, -b x\right ) - \frac {{\rm Ei}\left (b x\right )}{3 \, x^{3}} \] Input:
integrate(Ei(b*x)/x^4,x, algorithm="maxima")
Output:
1/3*b^3*gamma(-3, -b*x) - 1/3*Ei(b*x)/x^3
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {b^{3} x^{3} {\rm Ei}\left (b x\right ) - b^{2} x^{2} e^{\left (b x\right )} - b x e^{\left (b x\right )} - 2 \, e^{\left (b x\right )}}{18 \, x^{3}} - \frac {{\rm Ei}\left (b x\right )}{3 \, x^{3}} \] Input:
integrate(Ei(b*x)/x^4,x, algorithm="giac")
Output:
1/18*(b^3*x^3*Ei(b*x) - b^2*x^2*e^(b*x) - b*x*e^(b*x) - 2*e^(b*x))/x^3 - 1 /3*Ei(b*x)/x^3
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {b^3\,\mathrm {ei}\left (b\,x\right )}{18}-\frac {\frac {\mathrm {ei}\left (b\,x\right )}{3}+\frac {{\mathrm {e}}^{b\,x}}{9}+\frac {b^2\,x^2\,{\mathrm {e}}^{b\,x}}{18}+\frac {b\,x\,{\mathrm {e}}^{b\,x}}{18}}{x^3} \] Input:
int(ei(b*x)/x^4,x)
Output:
(b^3*ei(b*x))/18 - (ei(b*x)/3 + exp(b*x)/9 + (b^2*x^2*exp(b*x))/18 + (b*x* exp(b*x))/18)/x^3
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\operatorname {ExpIntegralEi}(b x)}{x^4} \, dx=\frac {\mathit {ei} \left (b x \right ) b^{3} x^{3}-6 \mathit {ei} \left (b x \right )-e^{b x} b^{2} x^{2}-e^{b x} b x -2 e^{b x}}{18 x^{3}} \] Input:
int(Ei(b*x)/x^4,x)
Output:
(ei(b*x)*b**3*x**3 - 6*ei(b*x) - e**(b*x)*b**2*x**2 - e**(b*x)*b*x - 2*e** (b*x))/(18*x**3)