3.81.85 \(\int \frac {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+(-2-27 x^2+11 x^3-x^4+x^2 \log (x)) \log (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2})}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx\) [8085]

Optimal. Leaf size=24 \[ x+x \log \left (-2-(5-x)^2-\frac {2}{x^2}+x+\log (x)\right ) \]

[Out]

x+ln(-2+x-2/x^2+ln(x)-(5-x)^2)*x

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Rubi [F]
time = 0.93, 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 {2-26 x^2+22 x^3-3 x^4+x^2 \log (x)+\left (-2-27 x^2+11 x^3-x^4+x^2 \log (x)\right ) \log \left (\frac {-2-27 x^2+11 x^3-x^4+x^2 \log (x)}{x^2}\right )}{-2-27 x^2+11 x^3-x^4+x^2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 - 26*x^2 + 22*x^3 - 3*x^4 + x^2*Log[x] + (-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x])*Log[(-2 - 27*x^2 + 1
1*x^3 - x^4 + x^2*Log[x])/x^2])/(-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x]),x]

[Out]

x - 4*Defer[Int][(2 + 27*x^2 - 11*x^3 + x^4 - x^2*Log[x])^(-1), x] - Defer[Int][x^2/(2 + 27*x^2 - 11*x^3 + x^4
- x^2*Log[x]), x] - 11*Defer[Int][x^3/(2 + 27*x^2 - 11*x^3 + x^4 - x^2*Log[x]), x] + 2*Defer[Int][x^4/(2 + 27
*x^2 - 11*x^3 + x^4 - x^2*Log[x]), x] + Defer[Int][Log[-27 - 2/x^2 + 11*x - x^2 + Log[x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}+\frac {26 x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}-\frac {22 x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}+\frac {3 x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}-\frac {x^2 \log (x)}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}+\log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right )\right ) \, dx\\ &=-\left (2 \int \frac {1}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx\right )+3 \int \frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-22 \int \frac {x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+26 \int \frac {x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-\int \frac {x^2 \log (x)}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+\int \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \, dx\\ &=-\left (2 \int \frac {1}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx\right )+3 \int \frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-22 \int \frac {x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+26 \int \frac {x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-\int \left (-1+\frac {2+27 x^2-11 x^3+x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}\right ) \, dx+\int \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \, dx\\ &=x-2 \int \frac {1}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+3 \int \frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-22 \int \frac {x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+26 \int \frac {x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-\int \frac {2+27 x^2-11 x^3+x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+\int \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \, dx\\ &=x-2 \int \frac {1}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+3 \int \frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-22 \int \frac {x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+26 \int \frac {x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-\int \left (\frac {2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}+\frac {27 x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}-\frac {11 x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}+\frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)}\right ) \, dx+\int \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \, dx\\ &=x-2 \left (2 \int \frac {1}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx\right )+3 \int \frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+11 \int \frac {x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-22 \int \frac {x^3}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+26 \int \frac {x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-27 \int \frac {x^2}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx-\int \frac {x^4}{2+27 x^2-11 x^3+x^4-x^2 \log (x)} \, dx+\int \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.06, size = 22, normalized size = 0.92 \begin {gather*} x+x \log \left (-27-\frac {2}{x^2}+11 x-x^2+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 26*x^2 + 22*x^3 - 3*x^4 + x^2*Log[x] + (-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x])*Log[(-2 - 27*x
^2 + 11*x^3 - x^4 + x^2*Log[x])/x^2])/(-2 - 27*x^2 + 11*x^3 - x^4 + x^2*Log[x]),x]

[Out]

x + x*Log[-27 - 2/x^2 + 11*x - x^2 + Log[x]]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.68, size = 339, normalized size = 14.12

method	result	size
risch	\(x \ln \left (2+x^{4}-11 x^{3}+\left (-\ln \left (x \right )+27\right ) x^{2}\right )-2 x \ln \left (x \right )+\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}-i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )^{3}}{2}-i \pi x \mathrm {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )^{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (-2-x^{4}+11 x^{3}-\left (-\ln \left (x \right )+27\right ) x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )}{2}+i x \pi +x\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2*ln(x)-x^4+11*x^3-27*x^2-2)*ln((x^2*ln(x)-x^4+11*x^3-27*x^2-2)/x^2)+x^2*ln(x)-3*x^4+22*x^3-26*x^2+2)/
(x^2*ln(x)-x^4+11*x^3-27*x^2-2),x,method=_RETURNVERBOSE)

[Out]

x*ln(2+x^4-11*x^3+(-ln(x)+27)*x^2)-2*x*ln(x)+1/2*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)-I*Pi*x*csgn(I*x)*csgn(I*x^2)^2
+1/2*I*Pi*x*csgn(I*x^2)^3-1/2*I*Pi*x*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)*x^2))*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)
*x^2)/x^2)^2-1/2*I*Pi*x*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)*x^2))*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)*x^2)/x^2)*cs
gn(I/x^2)-1/2*I*Pi*x*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)*x^2)/x^2)^3-I*Pi*x*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)*x^
2)/x^2)^2+1/2*I*Pi*x*csgn(I*(-2-x^4+11*x^3-(-ln(x)+27)*x^2)/x^2)^2*csgn(I/x^2)+I*x*Pi+x

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Maxima [A]
time = 0.29, size = 33, normalized size = 1.38 \begin {gather*} x \log \left (-x^{4} + 11 \, x^{3} + x^{2} \log \left (x\right ) - 27 \, x^{2} - 2\right ) - 2 \, x \log \left (x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27*x^2-2)/x^2)+x^2*log(x)-3*x^4+22*x^3-
26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^2-2),x, algorithm="maxima")

[Out]

x*log(-x^4 + 11*x^3 + x^2*log(x) - 27*x^2 - 2) - 2*x*log(x) + x

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Fricas [A]
time = 0.37, size = 32, normalized size = 1.33 \begin {gather*} x \log \left (-\frac {x^{4} - 11 \, x^{3} - x^{2} \log \left (x\right ) + 27 \, x^{2} + 2}{x^{2}}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27*x^2-2)/x^2)+x^2*log(x)-3*x^4+22*x^3-
26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^2-2),x, algorithm="fricas")

[Out]

x*log(-(x^4 - 11*x^3 - x^2*log(x) + 27*x^2 + 2)/x^2) + x

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Sympy [A]
time = 0.21, size = 29, normalized size = 1.21 \begin {gather*} x \log {\left (\frac {- x^{4} + 11 x^{3} + x^{2} \log {\left (x \right )} - 27 x^{2} - 2}{x^{2}} \right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2*ln(x)-x**4+11*x**3-27*x**2-2)*ln((x**2*ln(x)-x**4+11*x**3-27*x**2-2)/x**2)+x**2*ln(x)-3*x**4+
22*x**3-26*x**2+2)/(x**2*ln(x)-x**4+11*x**3-27*x**2-2),x)

[Out]

x*log((-x**4 + 11*x**3 + x**2*log(x) - 27*x**2 - 2)/x**2) + x

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Giac [A]
time = 0.44, size = 33, normalized size = 1.38 \begin {gather*} x \log \left (-x^{4} + 11 \, x^{3} + x^{2} \log \left (x\right ) - 27 \, x^{2} - 2\right ) - 2 \, x \log \left (x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2*log(x)-x^4+11*x^3-27*x^2-2)*log((x^2*log(x)-x^4+11*x^3-27*x^2-2)/x^2)+x^2*log(x)-3*x^4+22*x^3-
26*x^2+2)/(x^2*log(x)-x^4+11*x^3-27*x^2-2),x, algorithm="giac")

[Out]

x*log(-x^4 + 11*x^3 + x^2*log(x) - 27*x^2 - 2) - 2*x*log(x) + x

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Mupad [B]
time = 5.42, size = 32, normalized size = 1.33 \begin {gather*} x\,\left (\ln \left (-\frac {27\,x^2-x^2\,\ln \left (x\right )-11\,x^3+x^4+2}{x^2}\right )+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*log(x) - log(-(27*x^2 - x^2*log(x) - 11*x^3 + x^4 + 2)/x^2)*(27*x^2 - x^2*log(x) - 11*x^3 + x^4 + 2)
- 26*x^2 + 22*x^3 - 3*x^4 + 2)/(27*x^2 - x^2*log(x) - 11*x^3 + x^4 + 2),x)

[Out]

x*(log(-(27*x^2 - x^2*log(x) - 11*x^3 + x^4 + 2)/x^2) + 1)

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