3.30.85 \(\int \frac {e^{2 x} (2+x)+(e^x (4+2 x)+e^{2 x} (3 x+2 x^2)) \log (x)+(2-4 e^{2 x}+x-4 e^{3 x} x+e^x (6 x+4 x^2)) \log ^2(x)+(3 x+2 x^2) \log ^3(x)}{e^{4 x} x \log ^2(x)+2 e^{3 x} x \log ^3(x)+e^{2 x} x \log ^4(x)} \, dx\)

Optimal. Leaf size=25 \[ -\frac {e^{-2 x} (2+x)}{\log (x)}+\frac {4}{e^x+\log (x)} \]

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Rubi [F]  time = 5.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{2 x} (2+x)+\left (e^x (4+2 x)+e^{2 x} \left (3 x+2 x^2\right )\right ) \log (x)+\left (2-4 e^{2 x}+x-4 e^{3 x} x+e^x \left (6 x+4 x^2\right )\right ) \log ^2(x)+\left (3 x+2 x^2\right ) \log ^3(x)}{e^{4 x} x \log ^2(x)+2 e^{3 x} x \log ^3(x)+e^{2 x} x \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

4/E^x - (4*Log[x])/E^(2*x) + Defer[Int][1/(E^(2*x)*Log[x]^2), x] + 2*Defer[Int][1/(E^(2*x)*x*Log[x]^2), x] + 3
*Defer[Int][1/(E^(2*x)*Log[x]), x] + 2*Defer[Int][x/(E^(2*x)*Log[x]), x] - 4*Defer[Int][Log[x]^2/(E^(2*x)*x*(E
^x + Log[x])^2), x] + 4*Defer[Int][Log[x]^3/(E^(2*x)*(E^x + Log[x])^2), x] + 8*Defer[Int][Log[x]/(E^(2*x)*x*(E
^x + Log[x])), x] - 12*Defer[Int][Log[x]^2/(E^(2*x)*(E^x + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-2 x} \left (e^{2 x} (2+x)+\left (e^x (4+2 x)+e^{2 x} \left (3 x+2 x^2\right )\right ) \log (x)+\left (2-4 e^{2 x}+x-4 e^{3 x} x+e^x \left (6 x+4 x^2\right )\right ) \log ^2(x)+\left (3 x+2 x^2\right ) \log ^3(x)\right )}{x \log ^2(x) \left (e^x+\log (x)\right )^2} \, dx\\ &=\int \left (-4 e^{-x}+\frac {4 e^{-2 x} \log ^2(x) (-1+x \log (x))}{x \left (e^x+\log (x)\right )^2}-\frac {4 e^{-2 x} \log (x) (-2+3 x \log (x))}{x \left (e^x+\log (x)\right )}+\frac {e^{-2 x} \left (2+x+3 x \log (x)+2 x^2 \log (x)-4 \log ^2(x)+8 x \log ^3(x)\right )}{x \log ^2(x)}\right ) \, dx\\ &=-\left (4 \int e^{-x} \, dx\right )+4 \int \frac {e^{-2 x} \log ^2(x) (-1+x \log (x))}{x \left (e^x+\log (x)\right )^2} \, dx-4 \int \frac {e^{-2 x} \log (x) (-2+3 x \log (x))}{x \left (e^x+\log (x)\right )} \, dx+\int \frac {e^{-2 x} \left (2+x+3 x \log (x)+2 x^2 \log (x)-4 \log ^2(x)+8 x \log ^3(x)\right )}{x \log ^2(x)} \, dx\\ &=4 e^{-x}+4 \int \left (-\frac {e^{-2 x} \log ^2(x)}{x \left (e^x+\log (x)\right )^2}+\frac {e^{-2 x} \log ^3(x)}{\left (e^x+\log (x)\right )^2}\right ) \, dx-4 \int \left (-\frac {2 e^{-2 x} \log (x)}{x \left (e^x+\log (x)\right )}+\frac {3 e^{-2 x} \log ^2(x)}{e^x+\log (x)}\right ) \, dx+\int \left (-\frac {4 e^{-2 x}}{x}+\frac {e^{-2 x} (2+x)}{x \log ^2(x)}+\frac {e^{-2 x} (3+2 x)}{\log (x)}+8 e^{-2 x} \log (x)\right ) \, dx\\ &=4 e^{-x}-4 \int \frac {e^{-2 x}}{x} \, dx-4 \int \frac {e^{-2 x} \log ^2(x)}{x \left (e^x+\log (x)\right )^2} \, dx+4 \int \frac {e^{-2 x} \log ^3(x)}{\left (e^x+\log (x)\right )^2} \, dx+8 \int e^{-2 x} \log (x) \, dx+8 \int \frac {e^{-2 x} \log (x)}{x \left (e^x+\log (x)\right )} \, dx-12 \int \frac {e^{-2 x} \log ^2(x)}{e^x+\log (x)} \, dx+\int \frac {e^{-2 x} (2+x)}{x \log ^2(x)} \, dx+\int \frac {e^{-2 x} (3+2 x)}{\log (x)} \, dx\\ &=4 e^{-x}-4 \text {Ei}(-2 x)-4 e^{-2 x} \log (x)-4 \int \frac {e^{-2 x} \log ^2(x)}{x \left (e^x+\log (x)\right )^2} \, dx+4 \int \frac {e^{-2 x} \log ^3(x)}{\left (e^x+\log (x)\right )^2} \, dx-8 \int -\frac {e^{-2 x}}{2 x} \, dx+8 \int \frac {e^{-2 x} \log (x)}{x \left (e^x+\log (x)\right )} \, dx-12 \int \frac {e^{-2 x} \log ^2(x)}{e^x+\log (x)} \, dx+\int \left (\frac {e^{-2 x}}{\log ^2(x)}+\frac {2 e^{-2 x}}{x \log ^2(x)}\right ) \, dx+\int \left (\frac {3 e^{-2 x}}{\log (x)}+\frac {2 e^{-2 x} x}{\log (x)}\right ) \, dx\\ &=4 e^{-x}-4 \text {Ei}(-2 x)-4 e^{-2 x} \log (x)+2 \int \frac {e^{-2 x}}{x \log ^2(x)} \, dx+2 \int \frac {e^{-2 x} x}{\log (x)} \, dx+3 \int \frac {e^{-2 x}}{\log (x)} \, dx+4 \int \frac {e^{-2 x}}{x} \, dx-4 \int \frac {e^{-2 x} \log ^2(x)}{x \left (e^x+\log (x)\right )^2} \, dx+4 \int \frac {e^{-2 x} \log ^3(x)}{\left (e^x+\log (x)\right )^2} \, dx+8 \int \frac {e^{-2 x} \log (x)}{x \left (e^x+\log (x)\right )} \, dx-12 \int \frac {e^{-2 x} \log ^2(x)}{e^x+\log (x)} \, dx+\int \frac {e^{-2 x}}{\log ^2(x)} \, dx\\ &=4 e^{-x}-4 e^{-2 x} \log (x)+2 \int \frac {e^{-2 x}}{x \log ^2(x)} \, dx+2 \int \frac {e^{-2 x} x}{\log (x)} \, dx+3 \int \frac {e^{-2 x}}{\log (x)} \, dx-4 \int \frac {e^{-2 x} \log ^2(x)}{x \left (e^x+\log (x)\right )^2} \, dx+4 \int \frac {e^{-2 x} \log ^3(x)}{\left (e^x+\log (x)\right )^2} \, dx+8 \int \frac {e^{-2 x} \log (x)}{x \left (e^x+\log (x)\right )} \, dx-12 \int \frac {e^{-2 x} \log ^2(x)}{e^x+\log (x)} \, dx+\int \frac {e^{-2 x}}{\log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 25, normalized size = 1.00 \begin {gather*} -\frac {e^{-2 x} (2+x)}{\log (x)}+\frac {4}{e^x+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(2 + x) + (E^x*(4 + 2*x) + E^(2*x)*(3*x + 2*x^2))*Log[x] + (2 - 4*E^(2*x) + x - 4*E^(3*x)*x
+ E^x*(6*x + 4*x^2))*Log[x]^2 + (3*x + 2*x^2)*Log[x]^3)/(E^(4*x)*x*Log[x]^2 + 2*E^(3*x)*x*Log[x]^3 + E^(2*x)*
x*Log[x]^4),x]

[Out]

-((2 + x)/(E^(2*x)*Log[x])) + 4/(E^x + Log[x])

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fricas [A]  time = 1.17, size = 40, normalized size = 1.60 \begin {gather*} -\frac {{\left (x + 2\right )} e^{x} + {\left (x - 4 \, e^{\left (2 \, x\right )} + 2\right )} \log \relax (x)}{e^{\left (2 \, x\right )} \log \relax (x)^{2} + e^{\left (3 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+3*x)*log(x)^3+(-4*x*exp(x)^3-4*exp(x)^2+(4*x^2+6*x)*exp(x)+2+x)*log(x)^2+((2*x^2+3*x)*exp(x)
^2+(2*x+4)*exp(x))*log(x)+(2+x)*exp(x)^2)/(x*exp(x)^2*log(x)^4+2*x*exp(x)^3*log(x)^3+x*exp(x)^4*log(x)^2),x, a
lgorithm="fricas")

[Out]

-((x + 2)*e^x + (x - 4*e^(2*x) + 2)*log(x))/(e^(2*x)*log(x)^2 + e^(3*x)*log(x))

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giac [A]  time = 0.28, size = 46, normalized size = 1.84 \begin {gather*} -\frac {x e^{x} + x \log \relax (x) - 4 \, e^{\left (2 \, x\right )} \log \relax (x) + 2 \, e^{x} + 2 \, \log \relax (x)}{e^{\left (2 \, x\right )} \log \relax (x)^{2} + e^{\left (3 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+3*x)*log(x)^3+(-4*x*exp(x)^3-4*exp(x)^2+(4*x^2+6*x)*exp(x)+2+x)*log(x)^2+((2*x^2+3*x)*exp(x)
^2+(2*x+4)*exp(x))*log(x)+(2+x)*exp(x)^2)/(x*exp(x)^2*log(x)^4+2*x*exp(x)^3*log(x)^3+x*exp(x)^4*log(x)^2),x, a
lgorithm="giac")

[Out]

-(x*e^x + x*log(x) - 4*e^(2*x)*log(x) + 2*e^x + 2*log(x))/(e^(2*x)*log(x)^2 + e^(3*x)*log(x))

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maple [A]  time = 0.05, size = 43, normalized size = 1.72

method	result	size
risch	\(-\frac {\left (-4 \,{\mathrm e}^{2 x} \ln \relax (x )+{\mathrm e}^{x} x +x \ln \relax (x )+2 \,{\mathrm e}^{x}+2 \ln \relax (x )\right ) {\mathrm e}^{-2 x}}{\left (\ln \relax (x )+{\mathrm e}^{x}\right ) \ln \relax (x )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2+3*x)*ln(x)^3+(-4*x*exp(x)^3-4*exp(x)^2+(4*x^2+6*x)*exp(x)+2+x)*ln(x)^2+((2*x^2+3*x)*exp(x)^2+(2*x+
4)*exp(x))*ln(x)+(2+x)*exp(x)^2)/(x*exp(x)^2*ln(x)^4+2*x*exp(x)^3*ln(x)^3+x*exp(x)^4*ln(x)^2),x,method=_RETURN
VERBOSE)

[Out]

-(-4*exp(2*x)*ln(x)+exp(x)*x+x*ln(x)+2*exp(x)+2*ln(x))/(ln(x)+exp(x))*exp(-2*x)/ln(x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2+3*x)*log(x)^3+(-4*x*exp(x)^3-4*exp(x)^2+(4*x^2+6*x)*exp(x)+2+x)*log(x)^2+((2*x^2+3*x)*exp(x)
^2+(2*x+4)*exp(x))*log(x)+(2+x)*exp(x)^2)/(x*exp(x)^2*log(x)^4+2*x*exp(x)^3*log(x)^3+x*exp(x)^4*log(x)^2),x, a
lgorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 1.90, size = 42, normalized size = 1.68 \begin {gather*} -\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,{\mathrm {e}}^x+2\,\ln \relax (x)-4\,{\mathrm {e}}^{2\,x}\,\ln \relax (x)+x\,{\mathrm {e}}^x+x\,\ln \relax (x)\right )}{\ln \relax (x)\,\left ({\mathrm {e}}^x+\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^3*(3*x + 2*x^2) + log(x)*(exp(2*x)*(3*x + 2*x^2) + exp(x)*(2*x + 4)) + exp(2*x)*(x + 2) + log(x)^2
*(x - 4*exp(2*x) - 4*x*exp(3*x) + exp(x)*(6*x + 4*x^2) + 2))/(x*exp(2*x)*log(x)^4 + 2*x*exp(3*x)*log(x)^3 + x*
exp(4*x)*log(x)^2),x)

[Out]

-(exp(-2*x)*(2*exp(x) + 2*log(x) - 4*exp(2*x)*log(x) + x*exp(x) + x*log(x)))/(log(x)*(exp(x) + log(x)))

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sympy [A]  time = 0.36, size = 20, normalized size = 0.80 \begin {gather*} \frac {\left (- x - 2\right ) e^{- 2 x}}{\log {\relax (x )}} + \frac {4}{e^{x} + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2+3*x)*ln(x)**3+(-4*x*exp(x)**3-4*exp(x)**2+(4*x**2+6*x)*exp(x)+2+x)*ln(x)**2+((2*x**2+3*x)*e
xp(x)**2+(2*x+4)*exp(x))*ln(x)+(2+x)*exp(x)**2)/(x*exp(x)**2*ln(x)**4+2*x*exp(x)**3*ln(x)**3+x*exp(x)**4*ln(x)
**2),x)

[Out]

(-x - 2)*exp(-2*x)/log(x) + 4/(exp(x) + log(x))

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