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

Optimal. Leaf size=30 \[ 5-x+x \left (-e^x-x+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \]

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Rubi [F]  time = 2.47, 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 {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

E^x - x - x^2 - E^x*(1 + x) + 3*Defer[Int][1/((x + Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]]^2), x] + 3*D
efer[Int][x/((x + Log[x])*Log[x + Log[x]]*Log[3/Log[x + Log[x]]]^2), x] + 3*Defer[Int][Log[3/Log[x + Log[x]]]^
(-1), x]

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 26, normalized size = 0.87 \begin {gather*} x \left (-1-e^x-x+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*(-1 - E^x - x + 3/Log[3/Log[x + Log[x]]])

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fricas [A]  time = 1.79, size = 38, normalized size = 1.27 \begin {gather*} -\frac {{\left (x^{2} + x e^{x} + x\right )} \log \left (\frac {3}{\log \left (x + \log \relax (x)\right )}\right ) - 3 \, x}{\log \left (\frac {3}{\log \left (x + \log \relax (x)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.66, size = 66, normalized size = 2.20 \begin {gather*} -\frac {x^{2} \log \relax (3) + x e^{x} \log \relax (3) - x^{2} \log \left (\log \left (x + \log \relax (x)\right )\right ) - x e^{x} \log \left (\log \left (x + \log \relax (x)\right )\right ) + x \log \relax (3) - x \log \left (\log \left (x + \log \relax (x)\right )\right ) - 3 \, x}{\log \relax (3) - \log \left (\log \left (x + \log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.43, size = 36, normalized size = 1.20

method	result	size
risch	\(-x^{2}-{\mathrm e}^{x} x -x +\frac {6 i x}{2 i \ln \relax (3)-2 i \ln \left (\ln \left (x +\ln \relax (x )\right )\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2-exp(x)*x-x+6*I*x/(2*I*ln(3)-2*I*ln(ln(x+ln(x))))

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maxima [A]  time = 0.50, size = 51, normalized size = 1.70 \begin {gather*} -\frac {x^{2} \log \relax (3) + x e^{x} \log \relax (3) + x {\left (\log \relax (3) - 3\right )} - {\left (x^{2} + x e^{x} + x\right )} \log \left (\log \left (x + \log \relax (x)\right )\right )}{\log \relax (3) - \log \left (\log \left (x + \log \relax (x)\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.25, size = 204, normalized size = 6.80 \begin {gather*} \frac {3\,x+\frac {3\,x\,\ln \left (x+\ln \relax (x)\right )\,\ln \left (\frac {3}{\ln \left (x+\ln \relax (x)\right )}\right )\,\left (x+\ln \relax (x)\right )}{x+1}}{\ln \left (\frac {3}{\ln \left (x+\ln \relax (x)\right )}\right )}-\ln \left (x+\ln \relax (x)\right )\,\left (\ln \relax (x)\,\left (\frac {3\,\left (x^3+2\,x^2+x\right )}{x\,{\left (x+1\right )}^2}-\frac {3\,x^2+3\,x}{x\,{\left (x+1\right )}^2}\right )-\frac {3\,x^4+12\,x^3+12\,x^2+3\,x}{x\,{\left (x+1\right )}^2}+\frac {6\,x^4+12\,x^3+6\,x^2}{x\,{\left (x+1\right )}^2}+\frac {3\,\left (x^3+3\,x^2+2\,x\right )}{x\,{\left (x+1\right )}^2}-\frac {3\,x^2+3\,x}{x\,{\left (x+1\right )}^2}\right )-x-x\,{\mathrm {e}}^x-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x + (3*x*log(x + log(x))*log(3/log(x + log(x)))*(x + log(x)))/(x + 1))/log(3/log(x + log(x))) - log(x + log
(x))*(log(x)*((3*(x + 2*x^2 + x^3))/(x*(x + 1)^2) - (3*x + 3*x^2)/(x*(x + 1)^2)) - (3*x + 12*x^2 + 12*x^3 + 3*
x^4)/(x*(x + 1)^2) + (6*x^2 + 12*x^3 + 6*x^4)/(x*(x + 1)^2) + (3*(2*x + 3*x^2 + x^3))/(x*(x + 1)^2) - (3*x + 3
*x^2)/(x*(x + 1)^2)) - x - x*exp(x) - x^2

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sympy [A]  time = 3.08, size = 22, normalized size = 0.73 \begin {gather*} - x^{2} - x e^{x} - x + \frac {3 x}{\log {\left (\frac {3}{\log {\left (x + \log {\relax (x )} \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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