3.95.82 \(\int \frac {-1-x-\frac {e^x x}{4}}{e^2 x+\frac {1}{16} e^{2 x} x+2 e x^2+x^3+(2 e x+2 x^2) \log (x)+x \log ^2(x)+\frac {e^x (2 e x^2+2 x^3+2 x^2 \log (x))}{4 x}} \, dx\)

Optimal. Leaf size=14 \[ \frac {1}{e+\frac {e^x}{4}+x+\log (x)} \]

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Rubi [A]  time = 0.31, antiderivative size = 18, normalized size of antiderivative = 1.29, number of steps used = 3, number of rules used = 3, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6688, 12, 6686} \begin {gather*} \frac {4}{4 x+e^x+4 \log (x)+4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - x - (E^x*x)/4)/(E^2*x + (E^(2*x)*x)/16 + 2*E*x^2 + x^3 + (2*E*x + 2*x^2)*Log[x] + x*Log[x]^2 + (E^x*
(2*E*x^2 + 2*x^3 + 2*x^2*Log[x]))/(4*x)),x]

[Out]

4/(4*E + E^x + 4*x + 4*Log[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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.29 \begin {gather*} \frac {4}{4 e+e^x+4 x+4 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - x - (E^x*x)/4)/(E^2*x + (E^(2*x)*x)/16 + 2*E*x^2 + x^3 + (2*E*x + 2*x^2)*Log[x] + x*Log[x]^2 +
(E^x*(2*E*x^2 + 2*x^3 + 2*x^2*Log[x]))/(4*x)),x]

[Out]

4/(4*E + E^x + 4*x + 4*Log[x])

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fricas [A]  time = 0.51, size = 21, normalized size = 1.50 \begin {gather*} \frac {1}{x e^{\left (x - 2 \, \log \relax (2) - \log \relax (x)\right )} + x + e + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(x*e^(x - 2*log(2) - log(x)) + x + e + log(x))

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giac [A]  time = 0.13, size = 18, normalized size = 1.29 \begin {gather*} \frac {4}{4 \, x + 4 \, e + e^{x} + 4 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(4*x + 4*e + e^x + 4*log(x))

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maple [A]  time = 0.06, size = 13, normalized size = 0.93

method	result	size
risch	\(\frac {1}{\frac {{\mathrm e}^{x}}{4}+{\mathrm e}+x +\ln \relax (x )}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(1/4*exp(x)+exp(1)+x+ln(x))

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maxima [A]  time = 0.39, size = 18, normalized size = 1.29 \begin {gather*} \frac {4}{4 \, x + 4 \, e + e^{x} + 4 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(4*x + 4*e + e^x + 4*log(x))

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mupad [B]  time = 9.05, size = 18, normalized size = 1.29 \begin {gather*} \frac {4}{4\,x+4\,\mathrm {e}+{\mathrm {e}}^x+4\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + x^2*exp(x - log(4*x)) + 1)/(log(x)*(2*x*exp(1) + 2*x^2) + x*log(x)^2 + x*exp(2) + x^3*exp(2*x - 2*lo
g(4*x)) + exp(x - log(4*x))*(2*x^2*log(x) + 2*x^2*exp(1) + 2*x^3) + 2*x^2*exp(1) + x^3),x)

[Out]

4/(4*x + 4*exp(1) + exp(x) + 4*log(x))

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sympy [A]  time = 0.27, size = 17, normalized size = 1.21 \begin {gather*} \frac {4}{4 x + e^{x} + 4 \log {\relax (x )} + 4 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(4*x + exp(x) + 4*log(x) + 4*E)

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