3.27.38 \(\int \frac {e^{3+\frac {e^3-3 x}{4 x}}+4 x^2}{4 x^2} \, dx\)

Optimal. Leaf size=20 \[ -e^{-1+\frac {e^3+x}{4 x}}+x \]

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 14, 2209} \begin {gather*} x-e^{\frac {e^3}{4 x}-\frac {3}{4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3 + (E^3 - 3*x)/(4*x)) + 4*x^2)/(4*x^2),x]

[Out]

-E^(-3/4 + E^3/(4*x)) + 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 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} &=\frac {1}{4} \int \frac {e^{3+\frac {e^3-3 x}{4 x}}+4 x^2}{x^2} \, dx\\ &=\frac {1}{4} \int \left (4+\frac {e^{\frac {9}{4}+\frac {e^3}{4 x}}}{x^2}\right ) \, dx\\ &=x+\frac {1}{4} \int \frac {e^{\frac {9}{4}+\frac {e^3}{4 x}}}{x^2} \, dx\\ &=-e^{-\frac {3}{4}+\frac {e^3}{4 x}}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 20, normalized size = 1.00 \begin {gather*} -e^{-\frac {3}{4}+\frac {e^3}{4 x}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 + (E^3 - 3*x)/(4*x)) + 4*x^2)/(4*x^2),x]

[Out]

-E^(-3/4 + E^3/(4*x)) + x

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fricas [A]  time = 0.55, size = 22, normalized size = 1.10 \begin {gather*} {\left (x e^{3} - e^{\left (\frac {9 \, x + e^{3}}{4 \, x}\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(3)*exp(1/4*(exp(3)-3*x)/x)+4*x^2)/x^2,x, algorithm="fricas")

[Out]

(x*e^3 - e^(1/4*(9*x + e^3)/x))*e^(-3)

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giac [B]  time = 0.23, size = 59, normalized size = 2.95 \begin {gather*} -\frac {{\left (\frac {{\left (9 \, x + e^{3}\right )} e^{\left (\frac {9 \, x + e^{3}}{4 \, x}\right )}}{x} - e^{6} - 9 \, e^{\left (\frac {9 \, x + e^{3}}{4 \, x}\right )}\right )} e^{\left (-3\right )}}{\frac {9 \, x + e^{3}}{x} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(3)*exp(1/4*(exp(3)-3*x)/x)+4*x^2)/x^2,x, algorithm="giac")

[Out]

-((9*x + e^3)*e^(1/4*(9*x + e^3)/x)/x - e^6 - 9*e^(1/4*(9*x + e^3)/x))*e^(-3)/((9*x + e^3)/x - 9)

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maple [A]  time = 0.04, size = 15, normalized size = 0.75

method	result	size
derivativedivides	\(x -{\mathrm e}^{-\frac {3}{4}+\frac {{\mathrm e}^{3}}{4 x}}\)	\(15\)
default	\(x -{\mathrm e}^{-\frac {3}{4}+\frac {{\mathrm e}^{3}}{4 x}}\)	\(15\)
risch	\(x -{\mathrm e}^{\frac {{\mathrm e}^{3}-3 x}{4 x}}\)	\(17\)
norman	\(\frac {x^{2}-x \,{\mathrm e}^{\frac {{\mathrm e}^{3}-3 x}{4 x}}}{x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(exp(3)*exp(1/4*(exp(3)-3*x)/x)+4*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x-exp(-3/4+1/4*exp(3)/x)

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maxima [A]  time = 0.35, size = 14, normalized size = 0.70 \begin {gather*} x - e^{\left (\frac {e^{3}}{4 \, x} - \frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(3)*exp(1/4*(exp(3)-3*x)/x)+4*x^2)/x^2,x, algorithm="maxima")

[Out]

x - e^(1/4*e^3/x - 3/4)

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mupad [B]  time = 1.51, size = 14, normalized size = 0.70 \begin {gather*} x-{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{4\,x}-\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(3)*exp(-((3*x)/4 - exp(3)/4)/x))/4 + x^2)/x^2,x)

[Out]

x - exp(exp(3)/(4*x) - 3/4)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.70 \begin {gather*} x - e^{\frac {- \frac {3 x}{4} + \frac {e^{3}}{4}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(exp(3)*exp(1/4*(exp(3)-3*x)/x)+4*x**2)/x**2,x)

[Out]

x - exp((-3*x/4 + exp(3)/4)/x)

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