3.20.29 \(\int \frac {e^{\frac {2 (-7+e^{10}-4 x)}{x}} (14-2 e^{10})+x^2}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 5+e^{-8-\frac {2 \left (7-e^{10}\right )}{x}}+x \]

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Rubi [A]  time = 0.06, antiderivative size = 18, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {14, 2209} \begin {gather*} x+e^{-\frac {2 \left (7-e^{10}\right )}{x}-8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*(-7 + E^10 - 4*x))/x)*(14 - 2*E^10) + x^2)/x^2,x]

[Out]

E^(-8 - (2*(7 - E^10))/x) + 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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {2 e^{-8+\frac {-14+2 e^{10}}{x}} \left (7-e^{10}\right )}{x^2}\right ) \, dx\\ &=x+\left (2 \left (7-e^{10}\right )\right ) \int \frac {e^{-8+\frac {-14+2 e^{10}}{x}}}{x^2} \, dx\\ &=e^{-8-\frac {2 \left (7-e^{10}\right )}{x}}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 17, normalized size = 0.89 \begin {gather*} e^{\frac {2 \left (-7+e^{10}-4 x\right )}{x}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(-7 + E^10 - 4*x))/x)*(14 - 2*E^10) + x^2)/x^2,x]

[Out]

E^((2*(-7 + E^10 - 4*x))/x) + x

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fricas [A]  time = 0.87, size = 17, normalized size = 0.89 \begin {gather*} x + e^{\left (-\frac {2 \, {\left (4 \, x - e^{10} + 7\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(5)^2+14)*exp((exp(5)^2-4*x-7)/x)^2+x^2)/x^2,x, algorithm="fricas")

[Out]

x + e^(-2*(4*x - e^10 + 7)/x)

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giac [B]  time = 0.38, size = 129, normalized size = 6.79 \begin {gather*} \frac {\frac {{\left (4 \, x - e^{10} + 7\right )} e^{\left (-\frac {2 \, {\left (4 \, x - e^{10} + 7\right )}}{x} + 10\right )}}{x} - \frac {7 \, {\left (4 \, x - e^{10} + 7\right )} e^{\left (-\frac {2 \, {\left (4 \, x - e^{10} + 7\right )}}{x}\right )}}{x} - e^{20} + 14 \, e^{10} - 4 \, e^{\left (-\frac {2 \, {\left (4 \, x - e^{10} + 7\right )}}{x} + 10\right )} + 28 \, e^{\left (-\frac {2 \, {\left (4 \, x - e^{10} + 7\right )}}{x}\right )} - 49}{{\left (\frac {4 \, x - e^{10} + 7}{x} - 4\right )} {\left (e^{10} - 7\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(5)^2+14)*exp((exp(5)^2-4*x-7)/x)^2+x^2)/x^2,x, algorithm="giac")

[Out]

((4*x - e^10 + 7)*e^(-2*(4*x - e^10 + 7)/x + 10)/x - 7*(4*x - e^10 + 7)*e^(-2*(4*x - e^10 + 7)/x)/x - e^20 + 1
4*e^10 - 4*e^(-2*(4*x - e^10 + 7)/x + 10) + 28*e^(-2*(4*x - e^10 + 7)/x) - 49)/(((4*x - e^10 + 7)/x - 4)*(e^10
- 7))

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maple [A]  time = 0.06, size = 16, normalized size = 0.84

method	result	size
risch	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{10}-8 x -14}{x}}+x\)	\(16\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{10}-8 x -14}{x}}}{x}\)	\(27\)
derivativedivides	\(x +{\mathrm e}^{\frac {2 \,{\mathrm e}^{10}}{x}-8-\frac {14}{x}}\)	\(39\)
default	\(x +{\mathrm e}^{\frac {2 \,{\mathrm e}^{10}}{x}-8-\frac {14}{x}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(5)^2+14)*exp((exp(5)^2-4*x-7)/x)^2+x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(2*(exp(10)-4*x-7)/x)+x

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maxima [B]  time = 0.64, size = 47, normalized size = 2.47 \begin {gather*} x + \frac {e^{\left (\frac {2 \, e^{10}}{x} - \frac {14}{x} + 2\right )}}{e^{10} - 7} - \frac {7 \, e^{\left (\frac {2 \, e^{10}}{x} - \frac {14}{x} - 8\right )}}{e^{10} - 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(5)^2+14)*exp((exp(5)^2-4*x-7)/x)^2+x^2)/x^2,x, algorithm="maxima")

[Out]

x + e^(2*e^10/x - 14/x + 2)/(e^10 - 7) - 7*e^(2*e^10/x - 14/x - 8)/(e^10 - 7)

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mupad [B]  time = 1.36, size = 19, normalized size = 1.00 \begin {gather*} x+{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{10}}{x}}\,{\mathrm {e}}^{-8}\,{\mathrm {e}}^{-\frac {14}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*(4*x - exp(10) + 7))/x)*(2*exp(10) - 14) - x^2)/x^2,x)

[Out]

x + exp((2*exp(10))/x)*exp(-8)*exp(-14/x)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.74 \begin {gather*} x + e^{\frac {2 \left (- 4 x - 7 + e^{10}\right )}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(5)**2+14)*exp((exp(5)**2-4*x-7)/x)**2+x**2)/x**2,x)

[Out]

x + exp(2*(-4*x - 7 + exp(10))/x)

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