3.168 \(\int \frac {e^{\frac {1}{x}} (1+x)}{x^4} \, dx\)

Optimal. Leaf size=27 \[ -\frac {e^{\frac {1}{x}}}{x^2}-e^{\frac {1}{x}}+\frac {e^{\frac {1}{x}}}{x} \]

[Out]

-exp(1/x)-exp(1/x)/x^2+exp(1/x)/x

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6742, 2212, 2209} \[ -\frac {e^{\frac {1}{x}}}{x^2}-e^{\frac {1}{x}}+\frac {e^{\frac {1}{x}}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(E^x^(-1)*(1 + x))/x^4,x]

[Out]

-E^x^(-1) - E^x^(-1)/x^2 + E^x^(-1)/x

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {e^{\frac {1}{x}} (1+x)}{x^4} \, dx &=\int \left (\frac {e^{\frac {1}{x}}}{x^4}+\frac {e^{\frac {1}{x}}}{x^3}\right ) \, dx\\ &=\int \frac {e^{\frac {1}{x}}}{x^4} \, dx+\int \frac {e^{\frac {1}{x}}}{x^3} \, dx\\ &=-\frac {e^{\frac {1}{x}}}{x^2}-\frac {e^{\frac {1}{x}}}{x}-2 \int \frac {e^{\frac {1}{x}}}{x^3} \, dx-\int \frac {e^{\frac {1}{x}}}{x^2} \, dx\\ &=e^{\frac {1}{x}}-\frac {e^{\frac {1}{x}}}{x^2}+\frac {e^{\frac {1}{x}}}{x}+2 \int \frac {e^{\frac {1}{x}}}{x^2} \, dx\\ &=-e^{\frac {1}{x}}-\frac {e^{\frac {1}{x}}}{x^2}+\frac {e^{\frac {1}{x}}}{x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 0.63 \[ \frac {e^{\frac {1}{x}} \left (-x^2+x-1\right )}{x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x^(-1)*(1 + x))/x^4,x]

[Out]

(E^x^(-1)*(-1 + x - x^2))/x^2

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 17, normalized size = 0.63 \[ -\frac {{\left (x^{2} - x + 1\right )} e^{\frac {1}{x}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/x)*(1+x)/x^4,x, algorithm="fricas")

[Out]

-(x^2 - x + 1)*e^(1/x)/x^2

________________________________________________________________________________________

giac [A]  time = 1.01, size = 24, normalized size = 0.89 \[ \frac {e^{\frac {1}{x}}}{x} - \frac {e^{\frac {1}{x}}}{x^{2}} - e^{\frac {1}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/x)*(1+x)/x^4,x, algorithm="giac")

[Out]

e^(1/x)/x - e^(1/x)/x^2 - e^(1/x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 18, normalized size = 0.67 \[ -\frac {\left (x^{2}-x +1\right ) {\mathrm e}^{\frac {1}{x}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/x)*(x+1)/x^4,x)

[Out]

-(x^2-x+1)*exp(1/x)/x^2

________________________________________________________________________________________

maxima [C]  time = 0.53, size = 17, normalized size = 0.63 \[ -\Gamma \left (3, -\frac {1}{x}\right ) + \Gamma \left (2, -\frac {1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/x)*(1+x)/x^4,x, algorithm="maxima")

[Out]

-gamma(3, -1/x) + gamma(2, -1/x)

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 17, normalized size = 0.63 \[ -\frac {{\mathrm {e}}^{1/x}\,\left (x^2-x+1\right )}{x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/x)*(x + 1))/x^4,x)

[Out]

-(exp(1/x)*(x^2 - x + 1))/x^2

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 14, normalized size = 0.52 \[ \frac {\left (- x^{2} + x - 1\right ) e^{\frac {1}{x}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/x)*(1+x)/x**4,x)

[Out]

(-x**2 + x - 1)*exp(1/x)/x**2

________________________________________________________________________________________