3.57.6 \(\int \frac {e^{\frac {1+x}{3 x}}+9 x^2}{9 x^2} \, dx\)

Optimal. Leaf size=27 \[ -4-\frac {1}{3} e^{\frac {1+x}{3 x}}+x+\log \left (\frac {8}{x}\right )+\log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 14, 2209} \begin {gather*} x-\frac {1}{3} e^{\frac {1}{3 x}+\frac {1}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((1 + x)/(3*x)) + 9*x^2)/(9*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{3} e^{\frac {1}{3}+\frac {1}{3 x}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 + x)/(3*x)) + 9*x^2)/(9*x^2),x]

[Out]

-1/3*E^(1/3 + 1/(3*x)) + x

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fricas [A]  time = 0.70, size = 13, normalized size = 0.48 \begin {gather*} x - \frac {1}{3} \, e^{\left (\frac {x + 1}{3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(exp(1/3*(x+1)/x)+9*x^2)/x^2,x, algorithm="fricas")

[Out]

x - 1/3*e^(1/3*(x + 1)/x)

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giac [A]  time = 0.15, size = 42, normalized size = 1.56 \begin {gather*} -\frac {\frac {{\left (x + 1\right )} e^{\left (\frac {x + 1}{3 \, x}\right )}}{x} - e^{\left (\frac {x + 1}{3 \, x}\right )} - 3}{3 \, {\left (\frac {x + 1}{x} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(exp(1/3*(x+1)/x)+9*x^2)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 13, normalized size = 0.48

method	result	size
derivativedivides	\(x -\frac {{\mathrm e}^{\frac {1}{3}+\frac {1}{3 x}}}{3}\)	\(13\)
default	\(x -\frac {{\mathrm e}^{\frac {1}{3}+\frac {1}{3 x}}}{3}\)	\(13\)
risch	\(x -\frac {{\mathrm e}^{\frac {x +1}{3 x}}}{3}\)	\(14\)
norman	\(\frac {x^{2}-\frac {x \,{\mathrm e}^{\frac {x +1}{3 x}}}{3}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(exp(1/3*(x+1)/x)+9*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.44, size = 12, normalized size = 0.44 \begin {gather*} x - \frac {1}{3} \, e^{\left (\frac {1}{3 \, x} + \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(exp(1/3*(x+1)/x)+9*x^2)/x^2,x, algorithm="maxima")

[Out]

x - 1/3*e^(1/3/x + 1/3)

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mupad [B]  time = 3.72, size = 12, normalized size = 0.44 \begin {gather*} x-\frac {{\mathrm {e}}^{\frac {1}{3\,x}+\frac {1}{3}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x/3 + 1/3)/x)/9 + x^2)/x^2,x)

[Out]

x - exp(1/(3*x) + 1/3)/3

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sympy [A]  time = 0.10, size = 12, normalized size = 0.44 \begin {gather*} x - \frac {e^{\frac {\frac {x}{3} + \frac {1}{3}}{x}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(exp(1/3*(x+1)/x)+9*x**2)/x**2,x)

[Out]

x - exp((x/3 + 1/3)/x)/3

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