3.80.11 \(\int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+(-9+45 x^2+18 x^3+x^4) \log (x)+(-x+9 x^3+6 x^4+x^5+(-1+9 x^2+6 x^3+x^4) \log (x)) \log (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8})}{-x+9 x^3+6 x^4+x^5+(-1+9 x^2+6 x^3+x^4) \log (x)} \, dx\)

Optimal. Leaf size=25 \[ x \left (5+\log \left (\frac {(x+\log (x))^2}{\left (-\frac {1}{x^2}+(3+x)^2\right )^2}\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 11.19, 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 {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 18*x^3 + x^4)*Log[x] + (-x + 9*x^3 + 6*x^4
+ x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3 + 79*x
^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x]),x]

[Out]

5*x + x*Log[(x^4*(x + Log[x])^2)/(1 - 9*x^2 - 6*x^3 - x^4)^2] + 2*Defer[Int][(-x - Log[x])^(-1), x] + 2*Defer[
Int][(x + Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+11 x-18 x^2-75 x^3-32 x^4-3 x^5-\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)-\left (-1+9 x^2+6 x^3+x^4\right ) (x+\log (x)) \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right )}{\left (1-9 x^2-6 x^3-x^4\right ) (x+\log (x))} \, dx\\ &=\int \left (-\frac {2}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}-\frac {11 x}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {18 x^2}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {75 x^3}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {32 x^4}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {3 x^5}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\frac {\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}+\log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right )\right ) \, dx\\ &=-\left (2 \int \frac {1}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx\right )+3 \int \frac {x^5}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx-11 \int \frac {x}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+18 \int \frac {x^2}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+32 \int \frac {x^4}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+75 \int \frac {x^3}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \frac {\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right ) \, dx\\ &=x \log \left (\frac {x^4 (x+\log (x))^2}{\left (1-9 x^2-6 x^3-x^4\right )^2}\right )-2 \int \left (\frac {1}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {1}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+3 \int \left (-\frac {6}{x+\log (x)}+\frac {x}{x+\log (x)}+\frac {-33+109 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {-21-55 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx-11 \int \left (\frac {x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}-\frac {x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+18 \int \left (\frac {1-3 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {1+3 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+32 \int \left (\frac {1}{x+\log (x)}+\frac {10-33 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {8+21 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx+75 \int \left (\frac {-3+10 x}{2 \left (-1+3 x+x^2\right ) (x+\log (x))}+\frac {-3-8 x}{2 \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx-\int -\frac {2 \left (1+3 x-9 x^2-15 x^3-x^4+x^5+2 \left (1+3 x^3+x^4\right ) \log (x)\right )}{\left (-1+9 x^2+6 x^3+x^4\right ) (x+\log (x))} \, dx+\int \left (\frac {-9+45 x^2+18 x^3+x^4}{-1+9 x^2+6 x^3+x^4}-\frac {x \left (-9+45 x^2+18 x^3+x^4\right )}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))}\right ) \, dx\\ &=x \log \left (\frac {x^4 (x+\log (x))^2}{\left (1-9 x^2-6 x^3-x^4\right )^2}\right )+\frac {3}{2} \int \frac {-33+109 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {3}{2} \int \frac {-21-55 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+2 \int \frac {1+3 x-9 x^2-15 x^3-x^4+x^5+2 \left (1+3 x^3+x^4\right ) \log (x)}{\left (-1+9 x^2+6 x^3+x^4\right ) (x+\log (x))} \, dx+3 \int \frac {x}{x+\log (x)} \, dx-\frac {11}{2} \int \frac {x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {11}{2} \int \frac {x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+9 \int \frac {1-3 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+9 \int \frac {1+3 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+16 \int \frac {10-33 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+16 \int \frac {8+21 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-18 \int \frac {1}{x+\log (x)} \, dx+32 \int \frac {1}{x+\log (x)} \, dx+\frac {75}{2} \int \frac {-3+10 x}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\frac {75}{2} \int \frac {-3-8 x}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \frac {-9+45 x^2+18 x^3+x^4}{-1+9 x^2+6 x^3+x^4} \, dx-\int \frac {1}{\left (-1+3 x+x^2\right ) (x+\log (x))} \, dx+\int \frac {1}{\left (1+3 x+x^2\right ) (x+\log (x))} \, dx-\int \frac {x \left (-9+45 x^2+18 x^3+x^4\right )}{\left (-1+3 x+x^2\right ) \left (1+3 x+x^2\right ) (x+\log (x))} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 34, normalized size = 1.36 \begin {gather*} 5 x+x \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 18*x^3 + x^4)*Log[x] + (-x + 9*x^3 +
6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3
+ 79*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log[x]),x
]

[Out]

5*x + x*Log[(x^4*(x + Log[x])^2)/(-1 + 9*x^2 + 6*x^3 + x^4)^2]

________________________________________________________________________________________

fricas [B]  time = 1.26, size = 64, normalized size = 2.56 \begin {gather*} x \log \left (\frac {x^{6} + 2 \, x^{5} \log \relax (x) + x^{4} \log \relax (x)^{2}}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 79 \, x^{4} - 12 \, x^{3} - 18 \, x^{2} + 1}\right ) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x
^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^
3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm="fricas")

[Out]

x*log((x^6 + 2*x^5*log(x) + x^4*log(x)^2)/(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 79*x^4 - 12*x^3 - 18*x^2 + 1)) +
5*x

________________________________________________________________________________________

giac [B]  time = 3.68, size = 64, normalized size = 2.56 \begin {gather*} -x \log \left (x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 79 \, x^{4} - 12 \, x^{3} - 18 \, x^{2} + 1\right ) + x \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}\right ) + 4 \, x \log \relax (x) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x
^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^
3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm="giac")

[Out]

-x*log(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 79*x^4 - 12*x^3 - 18*x^2 + 1) + x*log(x^2 + 2*x*log(x) + log(x)^2) +
4*x*log(x) + 5*x

________________________________________________________________________________________

maple [C]  time = 0.25, size = 844, normalized size = 33.76

method	result	size
risch	\(5 x +4 x \ln \relax (x )-\frac {i \pi x \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i x^{4} \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{3}\right )^{3}}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{4}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-2 x \ln \left (x^{4}+6 x^{3}+9 x^{2}-1\right )+2 x \ln \left (x +\ln \relax (x )\right )-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right ) \mathrm {csgn}\left (i x^{4}\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right ) \mathrm {csgn}\left (\frac {i}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )}{2}-\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 \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )}{2}+i \pi x \,\mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i x^{4} \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left (x^{4}+6 x^{3}+9 x^{2}-1\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}\right )}{2}-i \pi x \,\mathrm {csgn}\left (i \left (x^{4}+6 x^{3}+9 x^{2}-1\right )\right ) \mathrm {csgn}\left (i \left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x +\ln \relax (x )\right )^{2}}{\left (x^{4}+6 x^{3}+9 x^{2}-1\right )^{2}}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (x +\ln \relax (x )\right )^{2}\right )}{2}\)	\(844\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4+6*x^3+9*x^2-1)*ln(x)+x^5+6*x^4+9*x^3-x)*ln((x^4*ln(x)^2+2*x^5*ln(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5
+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*ln(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^3+9*x^2-1)*
ln(x)+x^5+6*x^4+9*x^3-x),x,method=_RETURNVERBOSE)

[Out]

5*x+4*x*ln(x)-1/2*I*Pi*x*csgn(I*(x+ln(x))^2)^3-1/2*I*Pi*x*csgn(I*(x+ln(x))^2/(x^4+6*x^3+9*x^2-1)^2)^3-1/2*I*Pi
*x*csgn(I*x^4/(x^4+6*x^3+9*x^2-1)^2*(x+ln(x))^2)^3-1/2*I*Pi*x*csgn(I*x^3)^3+1/2*I*Pi*x*csgn(I*(x^4+6*x^3+9*x^2
-1)^2)^3-1/2*I*Pi*x*csgn(I*x^4)^3-1/2*I*Pi*x*csgn(I*x^2)^3-2*x*ln(x^4+6*x^3+9*x^2-1)+2*x*ln(x+ln(x))+I*Pi*x*cs
gn(I*(x+ln(x)))*csgn(I*(x+ln(x))^2)^2+1/2*I*Pi*x*csgn(I*(x+ln(x))^2/(x^4+6*x^3+9*x^2-1)^2)*csgn(I*x^4/(x^4+6*x
^3+9*x^2-1)^2*(x+ln(x))^2)^2+1/2*I*Pi*x*csgn(I*x^4/(x^4+6*x^3+9*x^2-1)^2*(x+ln(x))^2)^2*csgn(I*x^4)+1/2*I*Pi*x
*csgn(I*x)*csgn(I*x^4)^2+1/2*I*Pi*x*csgn(I*x^3)*csgn(I*x^4)^2+1/2*I*Pi*x*csgn(I*x^2)*csgn(I*x^3)^2+1/2*I*Pi*x*
csgn(I*x)*csgn(I*x^3)^2+1/2*I*Pi*x*csgn(I*(x^4+6*x^3+9*x^2-1))^2*csgn(I*(x^4+6*x^3+9*x^2-1)^2)-I*Pi*x*csgn(I*(
x^4+6*x^3+9*x^2-1))*csgn(I*(x^4+6*x^3+9*x^2-1)^2)^2+1/2*I*Pi*x*csgn(I*(x+ln(x))^2)*csgn(I*(x+ln(x))^2/(x^4+6*x
^3+9*x^2-1)^2)^2+1/2*I*Pi*x*csgn(I/(x^4+6*x^3+9*x^2-1)^2)*csgn(I*(x+ln(x))^2/(x^4+6*x^3+9*x^2-1)^2)^2-1/2*I*Pi
*x*csgn(I*(x+ln(x)))^2*csgn(I*(x+ln(x))^2)-1/2*I*Pi*x*csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)-1/2*I*Pi*x*csgn(I*(x+l
n(x))^2/(x^4+6*x^3+9*x^2-1)^2)*csgn(I*x^4/(x^4+6*x^3+9*x^2-1)^2*(x+ln(x))^2)*csgn(I*x^4)-1/2*I*Pi*x*csgn(I*(x+
ln(x))^2)*csgn(I/(x^4+6*x^3+9*x^2-1)^2)*csgn(I*(x+ln(x))^2/(x^4+6*x^3+9*x^2-1)^2)-1/2*I*Pi*x*csgn(I*x)*csgn(I*
x^2)*csgn(I*x^3)-1/2*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+I*Pi*x*csgn(I*x)*csgn(I*x^2)^2

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 41, normalized size = 1.64 \begin {gather*} -2 \, x \log \left (x^{2} + 3 \, x + 1\right ) - 2 \, x \log \left (x^{2} + 3 \, x - 1\right ) + 2 \, x \log \left (x + \log \relax (x)\right ) + 4 \, x \log \relax (x) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x
^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^
3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm="maxima")

[Out]

-2*x*log(x^2 + 3*x + 1) - 2*x*log(x^2 + 3*x - 1) + 2*x*log(x + log(x)) + 4*x*log(x) + 5*x

________________________________________________________________________________________

mupad [B]  time = 5.87, size = 62, normalized size = 2.48 \begin {gather*} x\,\left (\ln \left (\frac {x^6+2\,x^5\,\ln \relax (x)+x^4\,{\ln \relax (x)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+79\,x^4-12\,x^3-18\,x^2+1}\right )+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((2*x^5*log(x) + x^4*log(x)^2 + x^6)/(79*x^4 - 12*x^3 - 18*x^2 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + 1))
*(9*x^3 - x + 6*x^4 + x^5 + log(x)*(9*x^2 + 6*x^3 + x^4 - 1)) - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + log(
x)*(45*x^2 + 18*x^3 + x^4 - 9) - 2)/(9*x^3 - x + 6*x^4 + x^5 + log(x)*(9*x^2 + 6*x^3 + x^4 - 1)),x)

[Out]

x*(log((2*x^5*log(x) + x^4*log(x)^2 + x^6)/(79*x^4 - 12*x^3 - 18*x^2 + 108*x^5 + 54*x^6 + 12*x^7 + x^8 + 1)) +
5)

________________________________________________________________________________________

sympy [B]  time = 1.46, size = 102, normalized size = 4.08 \begin {gather*} 5 x + \left (x + \frac {1}{5}\right ) \log {\left (\frac {x^{6} + 2 x^{5} \log {\relax (x )} + x^{4} \log {\relax (x )}^{2}}{x^{8} + 12 x^{7} + 54 x^{6} + 108 x^{5} + 79 x^{4} - 12 x^{3} - 18 x^{2} + 1} \right )} - \frac {4 \log {\relax (x )}}{5} - \frac {2 \log {\left (x + \log {\relax (x )} \right )}}{5} + \frac {2 \log {\left (x^{4} + 6 x^{3} + 9 x^{2} - 1 \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4+6*x**3+9*x**2-1)*ln(x)+x**5+6*x**4+9*x**3-x)*ln((x**4*ln(x)**2+2*x**5*ln(x)+x**6)/(x**8+12*x
**7+54*x**6+108*x**5+79*x**4-12*x**3-18*x**2+1))+(x**4+18*x**3+45*x**2-9)*ln(x)+3*x**5+32*x**4+75*x**3+18*x**2
-11*x-2)/((x**4+6*x**3+9*x**2-1)*ln(x)+x**5+6*x**4+9*x**3-x),x)

[Out]

5*x + (x + 1/5)*log((x**6 + 2*x**5*log(x) + x**4*log(x)**2)/(x**8 + 12*x**7 + 54*x**6 + 108*x**5 + 79*x**4 - 1
2*x**3 - 18*x**2 + 1)) - 4*log(x)/5 - 2*log(x + log(x))/5 + 2*log(x**4 + 6*x**3 + 9*x**2 - 1)/5

________________________________________________________________________________________