3.94.31 \(\int \frac {-32 \log (3)+32 \log (3) \log (\frac {x}{3})+(-8+16 x) \log (3) \log ^2(\frac {x}{3})-8 \log (3) \log ^2(\frac {x}{3}) \log (x)}{(-4 x \log (\frac {x}{3})-x^2 \log ^2(\frac {x}{3})+x \log ^2(\frac {x}{3}) \log (x)) \log ^2(\frac {16 x^2+8 x^3 \log (\frac {x}{3})+x^4 \log ^2(\frac {x}{3})+(-8 x^2 \log (\frac {x}{3})-2 x^3 \log ^2(\frac {x}{3})) \log (x)+x^2 \log ^2(\frac {x}{3}) \log ^2(x)}{\log ^2(\frac {x}{3})})} \, dx\)

Optimal. Leaf size=29 \[ \frac {4 \log (3)}{\log \left (x^2 \left (x+\frac {4}{\log \left (\frac {x}{3}\right )}-\log (x)\right )^2\right )} \]

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Rubi [F]  time = 4.32, 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 {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-32*Log[3] + 32*Log[3]*Log[x/3] + (-8 + 16*x)*Log[3]*Log[x/3]^2 - 8*Log[3]*Log[x/3]^2*Log[x])/((-4*x*Log[
x/3] - x^2*Log[x/3]^2 + x*Log[x/3]^2*Log[x])*Log[(16*x^2 + 8*x^3*Log[x/3] + x^4*Log[x/3]^2 + (-8*x^2*Log[x/3]
- 2*x^3*Log[x/3]^2)*Log[x] + x^2*Log[x/3]^2*Log[x]^2)/Log[x/3]^2]^2),x]

[Out]

32*Log[3]*(1 + Log[3])*Defer[Int][1/(x*Log[x/3]*(4 + x*Log[x/3] - Log[x/3]*Log[x])*Log[(x^2*(4 + Log[x/3]*(x -
Log[x]))^2)/Log[x/3]^2]^2), x] - 16*Log[3]*Defer[Int][Log[x/3]/((4 + x*Log[x/3] - Log[x/3]*Log[x])*Log[(x^2*(
4 + Log[x/3]*(x - Log[x]))^2)/Log[x/3]^2]^2), x] + 8*Log[3]*Defer[Int][Log[x/3]/(x*(4 + x*Log[x/3] - Log[x/3]*
Log[x])*Log[(x^2*(4 + Log[x/3]*(x - Log[x]))^2)/Log[x/3]^2]^2), x] - 32*Log[3]*Defer[Int][Log[x]/(x*Log[x/3]*(
4 + x*Log[x/3] - Log[x/3]*Log[x])*Log[(x^2*(4 + Log[x/3]*(x - Log[x]))^2)/Log[x/3]^2]^2), x] + 8*Log[3]*Defer[
Int][(Log[x/3]*Log[x])/(x*(4 + x*Log[x/3] - Log[x/3]*Log[x])*Log[(x^2*(4 + Log[x/3]*(x - Log[x]))^2)/Log[x/3]^
2]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \log (3) \left (4 (1+\log (3))-4 \log (x)+\log ^2\left (\frac {x}{3}\right ) (1-2 x+\log (x))\right )}{x \log \left (\frac {x}{3}\right ) \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx\\ &=(8 \log (3)) \int \frac {4 (1+\log (3))-4 \log (x)+\log ^2\left (\frac {x}{3}\right ) (1-2 x+\log (x))}{x \log \left (\frac {x}{3}\right ) \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx\\ &=(8 \log (3)) \int \left (\frac {4 (1+\log (3))}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}-\frac {2 \log \left (\frac {x}{3}\right )}{\left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}+\frac {\log \left (\frac {x}{3}\right )}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}-\frac {4 \log (x)}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}+\frac {\log \left (\frac {x}{3}\right ) \log (x)}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}\right ) \, dx\\ &=(8 \log (3)) \int \frac {\log \left (\frac {x}{3}\right )}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx+(8 \log (3)) \int \frac {\log \left (\frac {x}{3}\right ) \log (x)}{x \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx-(16 \log (3)) \int \frac {\log \left (\frac {x}{3}\right )}{\left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx-(32 \log (3)) \int \frac {\log (x)}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx+(32 \log (3) (1+\log (3))) \int \frac {1}{x \log \left (\frac {x}{3}\right ) \left (4+x \log \left (\frac {x}{3}\right )-\log \left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.16, size = 94, normalized size = 3.24 \begin {gather*} \frac {4 \log (3) \left (-4-\log (81)+\log ^2\left (\frac {x}{3}\right ) (-1+2 x-\log (x))+4 \log (x)\right )}{\left (-4+4 \log \left (\frac {x}{3}\right )+\log ^2\left (\frac {x}{3}\right ) (-1+2 x-\log (x))\right ) \log \left (\frac {x^2 \left (4+\log \left (\frac {x}{3}\right ) (x-\log (x))\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32*Log[3] + 32*Log[3]*Log[x/3] + (-8 + 16*x)*Log[3]*Log[x/3]^2 - 8*Log[3]*Log[x/3]^2*Log[x])/((-4*
x*Log[x/3] - x^2*Log[x/3]^2 + x*Log[x/3]^2*Log[x])*Log[(16*x^2 + 8*x^3*Log[x/3] + x^4*Log[x/3]^2 + (-8*x^2*Log
[x/3] - 2*x^3*Log[x/3]^2)*Log[x] + x^2*Log[x/3]^2*Log[x]^2)/Log[x/3]^2]^2),x]

[Out]

(4*Log[3]*(-4 - Log[81] + Log[x/3]^2*(-1 + 2*x - Log[x]) + 4*Log[x]))/((-4 + 4*Log[x/3] + Log[x/3]^2*(-1 + 2*x
- Log[x]))*Log[(x^2*(4 + Log[x/3]*(x - Log[x]))^2)/Log[x/3]^2])

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fricas [B]  time = 0.55, size = 97, normalized size = 3.34 \begin {gather*} \frac {4 \, \log \relax (3)}{\log \left (\frac {x^{2} \log \left (\frac {1}{3} \, x\right )^{4} - 2 \, {\left (x^{3} - x^{2} \log \relax (3)\right )} \log \left (\frac {1}{3} \, x\right )^{3} + {\left (x^{4} - 2 \, x^{3} \log \relax (3) + x^{2} \log \relax (3)^{2} - 8 \, x^{2}\right )} \log \left (\frac {1}{3} \, x\right )^{2} + 16 \, x^{2} + 8 \, {\left (x^{3} - x^{2} \log \relax (3)\right )} \log \left (\frac {1}{3} \, x\right )}{\log \left (\frac {1}{3} \, x\right )^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*log(3)*log(1/3*x)-32*log(3))/(x*log(1
/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*log(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(
1/3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="fricas")

[Out]

4*log(3)/log((x^2*log(1/3*x)^4 - 2*(x^3 - x^2*log(3))*log(1/3*x)^3 + (x^4 - 2*x^3*log(3) + x^2*log(3)^2 - 8*x^
2)*log(1/3*x)^2 + 16*x^2 + 8*(x^3 - x^2*log(3))*log(1/3*x))/log(1/3*x)^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*log(3)*log(1/3*x)-32*log(3))/(x*log(1
/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*log(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(
1/3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 1.31, size = 1028, normalized size = 35.45

method	result	size
risch	\(\frac {8 i \ln \relax (3)}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{2} \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2} \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}\right )^{3}+\pi \mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )\right )^{2} \mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )\right ) \mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{2} \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i x^{2} \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )^{2}}{\left (2 i \ln \relax (3)-2 i \ln \relax (x )\right )^{2}}\right )^{3}+4 i \ln \left (-2 i x \ln \relax (3)+2 i \ln \relax (3) \ln \relax (x )+2 i x \ln \relax (x )-2 i \ln \relax (x )^{2}+8 i\right )+4 i \ln \relax (x )-4 i \ln \left (-2 i \ln \relax (3)+2 i \ln \relax (x )\right )}\)	\(1028\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*ln(3)*ln(1/3*x)^2*ln(x)+(16*x-8)*ln(3)*ln(1/3*x)^2+32*ln(3)*ln(1/3*x)-32*ln(3))/(x*ln(1/3*x)^2*ln(x)-x
^2*ln(1/3*x)^2-4*x*ln(1/3*x))/ln((x^2*ln(1/3*x)^2*ln(x)^2+(-2*x^3*ln(1/3*x)^2-8*x^2*ln(1/3*x))*ln(x)+x^4*ln(1/
3*x)^2+8*x^3*ln(1/3*x)+16*x^2)/ln(1/3*x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

8*I*ln(3)/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+Pi*csgn(I*x^2)*csgn(I/(2*I
*ln(3)-2*I*ln(x))^2*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)*csgn(I*x^2*(-2*I*x*ln(3)+2*I
*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2/(2*I*ln(3)-2*I*ln(x))^2)-Pi*csgn(I*x^2)*csgn(I*x^2*(-2*I*x*ln(3)+2
*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2/(2*I*ln(3)-2*I*ln(x))^2)^2+Pi*csgn(I/(2*I*ln(3)-2*I*ln(x))^2)*cs
gn(I*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)*csgn(I/(2*I*ln(3)-2*I*ln(x))^2*(-2*I*x*ln(3
)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)-Pi*csgn(I/(2*I*ln(3)-2*I*ln(x))^2)*csgn(I/(2*I*ln(3)-2*I*ln(
x))^2*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)^2-Pi*csgn(I*(2*I*ln(3)-2*I*ln(x)))^2*csgn(
I*(2*I*ln(3)-2*I*ln(x))^2)-2*Pi*csgn(I*(2*I*ln(3)-2*I*ln(x)))*csgn(I*(2*I*ln(3)-2*I*ln(x))^2)^2-Pi*csgn(I*(2*I
*ln(3)-2*I*ln(x))^2)^3+Pi*csgn(I*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I))^2*csgn(I*(-2*I*x*
ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)-2*Pi*csgn(I*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*
I*ln(x)^2+8*I))*csgn(I*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)^2+Pi*csgn(I*(-2*I*x*ln(3)
+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)^3-Pi*csgn(I*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(
x)^2+8*I)^2)*csgn(I/(2*I*ln(3)-2*I*ln(x))^2*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)^2+Pi
*csgn(I/(2*I*ln(3)-2*I*ln(x))^2*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)^3-Pi*csgn(I/(2*I
*ln(3)-2*I*ln(x))^2*(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2)*csgn(I*x^2*(-2*I*x*ln(3)+2*I
*ln(3)*ln(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2/(2*I*ln(3)-2*I*ln(x))^2)^2+Pi*csgn(I*x^2*(-2*I*x*ln(3)+2*I*ln(3)*l
n(x)+2*I*x*ln(x)-2*I*ln(x)^2+8*I)^2/(2*I*ln(3)-2*I*ln(x))^2)^3+4*I*ln(-2*I*x*ln(3)+2*I*ln(3)*ln(x)+2*I*x*ln(x)
-2*I*ln(x)^2+8*I)+4*I*ln(x)-4*I*ln(-2*I*ln(3)+2*I*ln(x)))

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maxima [A]  time = 0.52, size = 38, normalized size = 1.31 \begin {gather*} \frac {2 \, \log \relax (3)}{\log \left (x \log \relax (3) - {\left (x + \log \relax (3)\right )} \log \relax (x) + \log \relax (x)^{2} - 4\right ) + \log \relax (x) - \log \left (-\log \relax (3) + \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*log(3)*log(1/3*x)-32*log(3))/(x*log(1
/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*log(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(
1/3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.28, size = 78, normalized size = 2.69 \begin {gather*} \frac {4\,\ln \relax (3)}{\ln \left (\frac {8\,x^3\,\ln \left (\frac {x}{3}\right )+16\,x^2+x^4\,{\ln \left (\frac {x}{3}\right )}^2-\ln \relax (x)\,\left (2\,x^3\,{\ln \left (\frac {x}{3}\right )}^2+8\,x^2\,\ln \left (\frac {x}{3}\right )\right )+x^2\,{\ln \left (\frac {x}{3}\right )}^2\,{\ln \relax (x)}^2}{{\ln \left (\frac {x}{3}\right )}^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*log(3) - 32*log(x/3)*log(3) + 8*log(x/3)^2*log(3)*log(x) - log(x/3)^2*log(3)*(16*x - 8))/(log((8*x^3*l
og(x/3) + 16*x^2 + x^4*log(x/3)^2 - log(x)*(8*x^2*log(x/3) + 2*x^3*log(x/3)^2) + x^2*log(x/3)^2*log(x)^2)/log(
x/3)^2)^2*(4*x*log(x/3) + x^2*log(x/3)^2 - x*log(x/3)^2*log(x))),x)

[Out]

(4*log(3))/log((8*x^3*log(x/3) + 16*x^2 + x^4*log(x/3)^2 - log(x)*(8*x^2*log(x/3) + 2*x^3*log(x/3)^2) + x^2*lo
g(x/3)^2*log(x)^2)/log(x/3)^2)

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sympy [B]  time = 1.29, size = 90, normalized size = 3.10 \begin {gather*} \frac {4 \log {\relax (3 )}}{\log {\left (\frac {x^{4} \left (\log {\relax (x )} - \log {\relax (3 )}\right )^{2} + 8 x^{3} \left (\log {\relax (x )} - \log {\relax (3 )}\right ) + x^{2} \left (\log {\relax (x )} - \log {\relax (3 )}\right )^{2} \log {\relax (x )}^{2} + 16 x^{2} + \left (- 2 x^{3} \left (\log {\relax (x )} - \log {\relax (3 )}\right )^{2} - 8 x^{2} \left (\log {\relax (x )} - \log {\relax (3 )}\right )\right ) \log {\relax (x )}}{\left (\log {\relax (x )} - \log {\relax (3 )}\right )^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*ln(3)*ln(1/3*x)**2*ln(x)+(16*x-8)*ln(3)*ln(1/3*x)**2+32*ln(3)*ln(1/3*x)-32*ln(3))/(x*ln(1/3*x)**
2*ln(x)-x**2*ln(1/3*x)**2-4*x*ln(1/3*x))/ln((x**2*ln(1/3*x)**2*ln(x)**2+(-2*x**3*ln(1/3*x)**2-8*x**2*ln(1/3*x)
)*ln(x)+x**4*ln(1/3*x)**2+8*x**3*ln(1/3*x)+16*x**2)/ln(1/3*x)**2)**2,x)

[Out]

4*log(3)/log((x**4*(log(x) - log(3))**2 + 8*x**3*(log(x) - log(3)) + x**2*(log(x) - log(3))**2*log(x)**2 + 16*
x**2 + (-2*x**3*(log(x) - log(3))**2 - 8*x**2*(log(x) - log(3)))*log(x))/(log(x) - log(3))**2)

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