3.46.56 \(\int \frac {-26+26 x^2+e^{\frac {1-3 x-x^2}{x}} (-1-x^2)}{26 x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{26} e^{-3+\frac {1}{x}-x}+\frac {(1-x)^2}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 19, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {12, 14, 6706} \begin {gather*} x+\frac {1}{26} e^{-x+\frac {1}{x}-3}+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-26 + 26*x^2 + E^((1 - 3*x - x^2)/x)*(-1 - x^2))/(26*x^2),x]

[Out]

E^(-3 + x^(-1) - x)/26 + x^(-1) + x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{26} \int \frac {-26+26 x^2+e^{\frac {1-3 x-x^2}{x}} \left (-1-x^2\right )}{x^2} \, dx\\ &=\frac {1}{26} \int \left (\frac {26 \left (-1+x^2\right )}{x^2}-\frac {e^{-3+\frac {1}{x}-x} \left (1+x^2\right )}{x^2}\right ) \, dx\\ &=-\left (\frac {1}{26} \int \frac {e^{-3+\frac {1}{x}-x} \left (1+x^2\right )}{x^2} \, dx\right )+\int \frac {-1+x^2}{x^2} \, dx\\ &=\frac {1}{26} e^{-3+\frac {1}{x}-x}+\int \left (1-\frac {1}{x^2}\right ) \, dx\\ &=\frac {1}{26} e^{-3+\frac {1}{x}-x}+\frac {1}{x}+x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 19, normalized size = 0.73 \begin {gather*} \frac {1}{26} e^{-3+\frac {1}{x}-x}+\frac {1}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-26 + 26*x^2 + E^((1 - 3*x - x^2)/x)*(-1 - x^2))/(26*x^2),x]

[Out]

E^(-3 + x^(-1) - x)/26 + x^(-1) + x

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 28, normalized size = 1.08 \begin {gather*} \frac {26 \, x^{2} + x e^{\left (-\frac {x^{2} + 3 \, x - 1}{x}\right )} + 26}{26 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/26*((-x^2-1)*exp((-x^2-3*x+1)/x)+26*x^2-26)/x^2,x, algorithm="fricas")

[Out]

1/26*(26*x^2 + x*e^(-(x^2 + 3*x - 1)/x) + 26)/x

________________________________________________________________________________________

giac [A]  time = 0.15, size = 28, normalized size = 1.08 \begin {gather*} \frac {26 \, x^{2} + x e^{\left (-\frac {x^{2} + 3 \, x - 1}{x}\right )} + 26}{26 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/26*((-x^2-1)*exp((-x^2-3*x+1)/x)+26*x^2-26)/x^2,x, algorithm="giac")

[Out]

1/26*(26*x^2 + x*e^(-(x^2 + 3*x - 1)/x) + 26)/x

________________________________________________________________________________________

maple [A]  time = 0.11, size = 22, normalized size = 0.85

method	result	size
risch	\(x +\frac {1}{x}+\frac {{\mathrm e}^{-\frac {x^{2}+3 x -1}{x}}}{26}\)	\(22\)
norman	\(\frac {1+x^{2}+\frac {x \,{\mathrm e}^{\frac {-x^{2}-3 x +1}{x}}}{26}}{x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/26*((-x^2-1)*exp((-x^2-3*x+1)/x)+26*x^2-26)/x^2,x,method=_RETURNVERBOSE)

[Out]

x+1/x+1/26*exp(-(x^2+3*x-1)/x)

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 16, normalized size = 0.62 \begin {gather*} x + \frac {1}{x} + \frac {1}{26} \, e^{\left (-x + \frac {1}{x} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/26*((-x^2-1)*exp((-x^2-3*x+1)/x)+26*x^2-26)/x^2,x, algorithm="maxima")

[Out]

x + 1/x + 1/26*e^(-x + 1/x - 3)

________________________________________________________________________________________

mupad [B]  time = 3.34, size = 17, normalized size = 0.65 \begin {gather*} x+\frac {1}{x}+\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-3}}{26} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(-(3*x + x^2 - 1)/x)*(x^2 + 1))/26 - x^2 + 1)/x^2,x)

[Out]

x + 1/x + (exp(-x)*exp(1/x)*exp(-3))/26

________________________________________________________________________________________

sympy [A]  time = 0.27, size = 17, normalized size = 0.65 \begin {gather*} x + \frac {e^{\frac {- x^{2} - 3 x + 1}{x}}}{26} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/26*((-x**2-1)*exp((-x**2-3*x+1)/x)+26*x**2-26)/x**2,x)

[Out]

x + exp((-x**2 - 3*x + 1)/x)/26 + 1/x

________________________________________________________________________________________