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

Optimal. Leaf size=26 \[ -5-x+\log (2)-x \log \left (x-\left (-4+e^{3+x} x\right ) \log (x)\right ) \]

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Rubi [F]  time = 1.26, 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 {4+2 x-e^{3+x} x+\left (4+e^{3+x} \left (-2 x-x^2\right )\right ) \log (x)+\left (x+\left (4-e^{3+x} x\right ) \log (x)\right ) \log \left (x+\left (4-e^{3+x} x\right ) \log (x)\right )}{-x+\left (-4+e^{3+x} x\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 23, normalized size = 0.88 \begin {gather*} -x-x \log \left (x+\left (4-e^{3+x} x\right ) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x - x*Log[x + (4 - E^(3 + x)*x)*Log[x]]

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fricas [A]  time = 0.90, size = 22, normalized size = 0.85 \begin {gather*} -x \log \left (-{\left (x e^{\left (x + 3\right )} - 4\right )} \log \relax (x) + x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.43, size = 23, normalized size = 0.88 \begin {gather*} -x \log \left (-x e^{\left (x + 3\right )} \log \relax (x) + x + 4 \, \log \relax (x)\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 23, normalized size = 0.88

method	result	size
risch	\(-x \ln \left (\left (-{\mathrm e}^{3+x} x +4\right ) \ln \relax (x )+x \right )-x\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.76, size = 22, normalized size = 0.85 \begin {gather*} -x \log \left (-{\left (x e^{\left (x + 3\right )} - 4\right )} \log \relax (x) + x\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.90, size = 20, normalized size = 0.77 \begin {gather*} -x\,\left (\ln \left (x-\ln \relax (x)\,\left (x\,{\mathrm {e}}^3\,{\mathrm {e}}^x-4\right )\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(log(x - log(x)*(x*exp(3)*exp(x) - 4)) + 1)

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sympy [A]  time = 1.00, size = 19, normalized size = 0.73 \begin {gather*} - x \log {\left (x + \left (- x e^{x + 3} + 4\right ) \log {\relax (x )} \right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*log(x + (-x*exp(x + 3) + 4)*log(x)) - x

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