3.9.100 \(\int \frac {3 x+e^{\frac {1}{3} (x-e^x \log (x))} (e^x-x+e^x x \log (x))}{3 x} \, dx\)

Optimal. Leaf size=24 \[ 5-e^{x+\frac {1}{3} \left (-2 x-e^x \log (x)\right )}+x \]

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Rubi [A]  time = 0.09, antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {12, 14, 2288} \begin {gather*} x-e^{x/3} x^{\frac {1}{3} \left (-e^x-3\right )+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - E^(x/3)*x^(1 + (-3 - E^x)/3)

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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 20, normalized size = 0.83 \begin {gather*} x-e^{x/3} x^{-\frac {e^x}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - E^(x/3)/x^(E^x/3)

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fricas [A]  time = 0.61, size = 15, normalized size = 0.62 \begin {gather*} x - e^{\left (-\frac {1}{3} \, e^{x} \log \relax (x) + \frac {1}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - e^(-1/3*e^x*log(x) + 1/3*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x e^{x} \log \relax (x) - x + e^{x}\right )} e^{\left (-\frac {1}{3} \, e^{x} \log \relax (x) + \frac {1}{3} \, x\right )} + 3 \, x}{3 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/3*((x*e^x*log(x) - x + e^x)*e^(-1/3*e^x*log(x) + 1/3*x) + 3*x)/x, x)

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maple [A]  time = 0.08, size = 15, normalized size = 0.62

method	result	size
risch	\(x -x^{-\frac {{\mathrm e}^{x}}{3}} {\mathrm e}^{\frac {x}{3}}\)	\(15\)
norman	\(x -{\mathrm e}^{-\frac {{\mathrm e}^{x} \ln \relax (x )}{3}+\frac {x}{3}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((x*exp(x)*ln(x)+exp(x)-x)*exp(-1/3*exp(x)*ln(x)+1/3*x)+3*x)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x + \frac {1}{3} \, \int \frac {{\left (x \log \relax (x) + 1\right )} e^{\left (\frac {4}{3} \, x\right )} - x e^{\left (\frac {1}{3} \, x\right )}}{x x^{\frac {1}{3} \, e^{x}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/3*integrate(((x*log(x) + 1)*e^(4/3*x) - x*e^(1/3*x))/(x*x^(1/3*e^x)), x)

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mupad [B]  time = 0.69, size = 15, normalized size = 0.62 \begin {gather*} x-{\mathrm {e}}^{\frac {x}{3}-\frac {{\mathrm {e}}^x\,\ln \relax (x)}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + (exp(x/3 - (exp(x)*log(x))/3)*(exp(x) - x + x*exp(x)*log(x)))/3)/x,x)

[Out]

x - exp(x/3 - (exp(x)*log(x))/3)

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sympy [A]  time = 0.44, size = 14, normalized size = 0.58 \begin {gather*} x - e^{\frac {x}{3} - \frac {e^{x} \log {\relax (x )}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - exp(x/3 - exp(x)*log(x)/3)

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