3.86.87 \(\int \frac {-3+(-x-e^x x+6 e^{2 x} x-2 x^2) \log (x)}{x \log (x)} \, dx\) [8587]

Optimal. Leaf size=27 \[ -e^x-x (4+x)+3 \left (1+e^{2 x}+x-\log (\log (x))\right ) \]

[Out]

3*exp(x)^2-3*ln(ln(x))+3+3*x-exp(x)-(4+x)*x

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Rubi [A]
time = 0.14, antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6820, 2225, 2339, 29} \begin {gather*} -x^2-x-e^x+3 e^{2 x}-3 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + (-x - E^x*x + 6*E^(2*x)*x - 2*x^2)*Log[x])/(x*Log[x]),x]

[Out]

-E^x + 3*E^(2*x) - x - x^2 - 3*Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-e^x+6 e^{2 x}-2 x-\frac {3}{x \log (x)}\right ) \, dx\\ &=-x-x^2-3 \int \frac {1}{x \log (x)} \, dx+6 \int e^{2 x} \, dx-\int e^x \, dx\\ &=-e^x+3 e^{2 x}-x-x^2-3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-e^x+3 e^{2 x}-x-x^2-3 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 26, normalized size = 0.96 \begin {gather*} -e^x+3 e^{2 x}-x-x^2-3 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + (-x - E^x*x + 6*E^(2*x)*x - 2*x^2)*Log[x])/(x*Log[x]),x]

[Out]

-E^x + 3*E^(2*x) - x - x^2 - 3*Log[Log[x]]

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Maple [A]
time = 0.25, size = 25, normalized size = 0.93

method	result	size
default	\(-x^{2}-x +3 \,{\mathrm e}^{2 x}-3 \ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\)	\(25\)
norman	\(-x^{2}-x +3 \,{\mathrm e}^{2 x}-3 \ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\)	\(25\)
risch	\(-x^{2}-x +3 \,{\mathrm e}^{2 x}-3 \ln \left (\ln \left (x \right )\right )-{\mathrm e}^{x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x*exp(x)^2-exp(x)*x-2*x^2-x)*ln(x)-3)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

-x^2-x+3*exp(x)^2-3*ln(ln(x))-exp(x)

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Maxima [A]
time = 0.28, size = 24, normalized size = 0.89 \begin {gather*} -x^{2} - x + 3 \, e^{\left (2 \, x\right )} - e^{x} - 3 \, \log \left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(x)^2-exp(x)*x-2*x^2-x)*log(x)-3)/x/log(x),x, algorithm="maxima")

[Out]

-x^2 - x + 3*e^(2*x) - e^x - 3*log(log(x))

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Fricas [A]
time = 0.38, size = 24, normalized size = 0.89 \begin {gather*} -x^{2} - x + 3 \, e^{\left (2 \, x\right )} - e^{x} - 3 \, \log \left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(x)^2-exp(x)*x-2*x^2-x)*log(x)-3)/x/log(x),x, algorithm="fricas")

[Out]

-x^2 - x + 3*e^(2*x) - e^x - 3*log(log(x))

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Sympy [A]
time = 0.08, size = 20, normalized size = 0.74 \begin {gather*} - x^{2} - x + 3 e^{2 x} - e^{x} - 3 \log {\left (\log {\left (x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(x)**2-exp(x)*x-2*x**2-x)*ln(x)-3)/x/ln(x),x)

[Out]

-x**2 - x + 3*exp(2*x) - exp(x) - 3*log(log(x))

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Giac [A]
time = 0.42, size = 24, normalized size = 0.89 \begin {gather*} -x^{2} - x + 3 \, e^{\left (2 \, x\right )} - e^{x} - 3 \, \log \left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x*exp(x)^2-exp(x)*x-2*x^2-x)*log(x)-3)/x/log(x),x, algorithm="giac")

[Out]

-x^2 - x + 3*e^(2*x) - e^x - 3*log(log(x))

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Mupad [B]
time = 5.94, size = 24, normalized size = 0.89 \begin {gather*} 3\,{\mathrm {e}}^{2\,x}-x-3\,\ln \left (\ln \left (x\right )\right )-{\mathrm {e}}^x-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(x - 6*x*exp(2*x) + x*exp(x) + 2*x^2) + 3)/(x*log(x)),x)

[Out]

3*exp(2*x) - x - 3*log(log(x)) - exp(x) - x^2

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