3.2.68 \(\int \frac {e^{\frac {1}{x}} (1+x)}{x^4} \, dx\) [168]

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

[Out]

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

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Rubi [A]
time = 0.08, 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 = {6874, 2243, 2240} \begin {gather*} -\frac {e^{\frac {1}{x}}}{x^2}-e^{\frac {1}{x}}+\frac {e^{\frac {1}{x}}}{x} \end {gather*}

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 2240

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 2243

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 6874

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*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.59 \begin {gather*} e^{\frac {1}{x}} \left (-1-\frac {1}{x^2}+\frac {1}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^(-1)*(1 + x))/x^4,x]

[Out]

E^x^(-1)*(-1 - x^(-2) + x^(-1))

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Maple [A]
time = 0.02, size = 25, normalized size = 0.93

method	result	size
gosper	\(-\frac {\left (x^{2}-x +1\right ) {\mathrm e}^{\frac {1}{x}}}{x^{2}}\)	\(18\)
risch	\(-\frac {\left (x^{2}-x +1\right ) {\mathrm e}^{\frac {1}{x}}}{x^{2}}\)	\(18\)
derivativedivides	\(-{\mathrm e}^{\frac {1}{x}}-\frac {{\mathrm e}^{\frac {1}{x}}}{x^{2}}+\frac {{\mathrm e}^{\frac {1}{x}}}{x}\)	\(25\)
default	\(-{\mathrm e}^{\frac {1}{x}}-\frac {{\mathrm e}^{\frac {1}{x}}}{x^{2}}+\frac {{\mathrm e}^{\frac {1}{x}}}{x}\)	\(25\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {1}{x}}-x \,{\mathrm e}^{\frac {1}{x}}-x^{3} {\mathrm e}^{\frac {1}{x}}}{x^{3}}\)	\(30\)
meijerg	\(1-\frac {\left (\frac {3}{x^{2}}-\frac {6}{x}+6\right ) {\mathrm e}^{\frac {1}{x}}}{3}+\frac {\left (2-\frac {2}{x}\right ) {\mathrm e}^{\frac {1}{x}}}{2}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.79, size = 17, normalized size = 0.63 \begin {gather*} -\Gamma \left (3, -\frac {1}{x}\right ) + \Gamma \left (2, -\frac {1}{x}\right ) \end {gather*}

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)

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Fricas [A]
time = 0.45, size = 17, normalized size = 0.63 \begin {gather*} -\frac {{\left (x^{2} - x + 1\right )} e^{\frac {1}{x}}}{x^{2}} \end {gather*}

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

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Sympy [A]
time = 0.03, size = 14, normalized size = 0.52 \begin {gather*} \frac {\left (- x^{2} + x - 1\right ) e^{\frac {1}{x}}}{x^{2}} \end {gather*}

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

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Giac [A]
time = 0.81, size = 24, normalized size = 0.89 \begin {gather*} \frac {e^{\frac {1}{x}}}{x} - \frac {e^{\frac {1}{x}}}{x^{2}} - e^{\frac {1}{x}} \end {gather*}

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)

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

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

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