3.31.63 \(\int \frac {-2 x-x^2+(-2-x-2 x^2-2 x^3-x^4) \log (x)+(-2 x-4 x^2-2 x^3) \log ^2(x)+(-2 x-x^2) \log ^3(x)+(2+3 x+x^2) \log (x) \log ((8+8 x+2 x^2) \log (x))}{(2 x^3+x^4) \log (x)+(4 x^2+2 x^3) \log ^2(x)+(2 x+x^2) \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ 13-x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \]

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Rubi [F]  time = 3.29, 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 {-2 x-x^2+\left (-2-x-2 x^2-2 x^3-x^4\right ) \log (x)+\left (-2 x-4 x^2-2 x^3\right ) \log ^2(x)+\left (-2 x-x^2\right ) \log ^3(x)+\left (2+3 x+x^2\right ) \log (x) \log \left (\left (8+8 x+2 x^2\right ) \log (x)\right )}{\left (2 x^3+x^4\right ) \log (x)+\left (4 x^2+2 x^3\right ) \log ^2(x)+\left (2 x+x^2\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-x - ExpIntegralEi[-Log[x]] + Defer[Int][1/(x^2*(x + Log[x])), x] - 2*Defer[Int][1/((2 + x)*(x + Log[x])), x]
+ Defer[Int][Log[2*(2 + x)^2*Log[x]]/(x + Log[x])^2, x] + Defer[Int][Log[2*(2 + x)^2*Log[x]]/(x*(x + Log[x])^2
), x]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 22, normalized size = 0.96 \begin {gather*} -x-\frac {\log \left (2 (2+x)^2 \log (x)\right )}{x+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - x^2 + (-2 - x - 2*x^2 - 2*x^3 - x^4)*Log[x] + (-2*x - 4*x^2 - 2*x^3)*Log[x]^2 + (-2*x - x^2)
*Log[x]^3 + (2 + 3*x + x^2)*Log[x]*Log[(8 + 8*x + 2*x^2)*Log[x]])/((2*x^3 + x^4)*Log[x] + (4*x^2 + 2*x^3)*Log[
x]^2 + (2*x + x^2)*Log[x]^3),x]

[Out]

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

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fricas [A]  time = 0.49, size = 29, normalized size = 1.26 \begin {gather*} -\frac {x^{2} + x \log \relax (x) + \log \left (2 \, {\left (x^{2} + 4 \, x + 4\right )} \log \relax (x)\right )}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+3*x+2)*log(x)*log((2*x^2+8*x+8)*log(x))+(-x^2-2*x)*log(x)^3+(-2*x^3-4*x^2-2*x)*log(x)^2+(-x^4-
2*x^3-2*x^2-x-2)*log(x)-x^2-2*x)/((x^2+2*x)*log(x)^3+(2*x^3+4*x^2)*log(x)^2+(x^4+2*x^3)*log(x)),x, algorithm="
fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (x^{2} + 2 \, x\right )} \log \relax (x)^{3} - {\left (x^{2} + 3 \, x + 2\right )} \log \left (2 \, {\left (x^{2} + 4 \, x + 4\right )} \log \relax (x)\right ) \log \relax (x) + 2 \, {\left (x^{3} + 2 \, x^{2} + x\right )} \log \relax (x)^{2} + x^{2} + {\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + x + 2\right )} \log \relax (x) + 2 \, x}{{\left (x^{2} + 2 \, x\right )} \log \relax (x)^{3} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (x)^{2} + {\left (x^{4} + 2 \, x^{3}\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+3*x+2)*log(x)*log((2*x^2+8*x+8)*log(x))+(-x^2-2*x)*log(x)^3+(-2*x^3-4*x^2-2*x)*log(x)^2+(-x^4-
2*x^3-2*x^2-x-2)*log(x)-x^2-2*x)/((x^2+2*x)*log(x)^3+(2*x^3+4*x^2)*log(x)^2+(x^4+2*x^3)*log(x)),x, algorithm="
giac")

[Out]

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

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maple [C]  time = 0.21, size = 197, normalized size = 8.57

method	result	size
risch	\(-\frac {2 \ln \left (2+x \right )}{x +\ln \relax (x )}-\frac {-i \pi \mathrm {csgn}\left (i \left (2+x \right )\right )^{2} \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \left (2+x \right )\right ) \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \left (2+x \right )^{2}\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (2+x \right )^{2}\right )+i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (2+x \right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (2+x \right )^{2}\right )^{3}+i \pi \,\mathrm {csgn}\left (i \left (2+x \right )^{2}\right ) \mathrm {csgn}\left (i \ln \relax (x ) \left (2+x \right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \ln \relax (x ) \left (2+x \right )^{2}\right )^{3}+2 x^{2}+2 x \ln \relax (x )+2 \ln \relax (2)+2 \ln \left (\ln \relax (x )\right )}{2 \left (x +\ln \relax (x )\right )}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+3*x+2)*ln(x)*ln((2*x^2+8*x+8)*ln(x))+(-x^2-2*x)*ln(x)^3+(-2*x^3-4*x^2-2*x)*ln(x)^2+(-x^4-2*x^3-2*x^2
-x-2)*ln(x)-x^2-2*x)/((x^2+2*x)*ln(x)^3+(2*x^3+4*x^2)*ln(x)^2+(x^4+2*x^3)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-2/(x+ln(x))*ln(2+x)-1/2*(-I*Pi*csgn(I*(2+x))^2*csgn(I*(2+x)^2)+2*I*Pi*csgn(I*(2+x))*csgn(I*(2+x)^2)^2-I*Pi*cs
gn(I*ln(x))*csgn(I*(2+x)^2)*csgn(I*ln(x)*(2+x)^2)+I*Pi*csgn(I*ln(x))*csgn(I*ln(x)*(2+x)^2)^2-I*Pi*csgn(I*(2+x)
^2)^3+I*Pi*csgn(I*(2+x)^2)*csgn(I*ln(x)*(2+x)^2)^2-I*Pi*csgn(I*ln(x)*(2+x)^2)^3+2*x^2+2*x*ln(x)+2*ln(2)+2*ln(l
n(x)))/(x+ln(x))

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maxima [A]  time = 0.58, size = 27, normalized size = 1.17 \begin {gather*} -\frac {x^{2} + x \log \relax (x) + \log \relax (2) + 2 \, \log \left (x + 2\right ) + \log \left (\log \relax (x)\right )}{x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+3*x+2)*log(x)*log((2*x^2+8*x+8)*log(x))+(-x^2-2*x)*log(x)^3+(-2*x^3-4*x^2-2*x)*log(x)^2+(-x^4-
2*x^3-2*x^2-x-2)*log(x)-x^2-2*x)/((x^2+2*x)*log(x)^3+(2*x^3+4*x^2)*log(x)^2+(x^4+2*x^3)*log(x)),x, algorithm="
maxima")

[Out]

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

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mupad [B]  time = 2.10, size = 26, normalized size = 1.13 \begin {gather*} -x-\frac {\ln \left (\ln \relax (x)\,\left (2\,x^2+8\,x+8\right )\right )}{x+\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + log(x)^2*(2*x + 4*x^2 + 2*x^3) + log(x)*(x + 2*x^2 + 2*x^3 + x^4 + 2) + log(x)^3*(2*x + x^2) + x^2
- log(log(x)*(8*x + 2*x^2 + 8))*log(x)*(3*x + x^2 + 2))/(log(x)^2*(4*x^2 + 2*x^3) + log(x)^3*(2*x + x^2) + lo
g(x)*(2*x^3 + x^4)),x)

[Out]

- x - log(log(x)*(8*x + 2*x^2 + 8))/(x + log(x))

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sympy [A]  time = 0.43, size = 22, normalized size = 0.96 \begin {gather*} - x - \frac {\log {\left (\left (2 x^{2} + 8 x + 8\right ) \log {\relax (x )} \right )}}{x + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+3*x+2)*ln(x)*ln((2*x**2+8*x+8)*ln(x))+(-x**2-2*x)*ln(x)**3+(-2*x**3-4*x**2-2*x)*ln(x)**2+(-x*
*4-2*x**3-2*x**2-x-2)*ln(x)-x**2-2*x)/((x**2+2*x)*ln(x)**3+(2*x**3+4*x**2)*ln(x)**2+(x**4+2*x**3)*ln(x)),x)

[Out]

-x - log((2*x**2 + 8*x + 8)*log(x))/(x + log(x))

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