3.20.45 \(\int \frac {4+(1+x) \log (\frac {3}{x^2}) \log ^2(\log ^2(\frac {3}{x^2}))}{x \log (\frac {3}{x^2}) \log (\log ^2(\frac {3}{x^2}))+((-x+x^2) \log (\frac {3}{x^2})+x \log (\frac {3}{x^2}) \log (x)) \log ^2(\log ^2(\frac {3}{x^2}))} \, dx\)

Optimal. Leaf size=17 \[ \log \left (-1+x+\log (x)+\frac {1}{\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )}\right ) \]

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Rubi [F]  time = 3.65, 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 {4+(1+x) \log \left (\frac {3}{x^2}\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )}{x \log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+\left (\left (-x+x^2\right ) \log \left (\frac {3}{x^2}\right )+x \log \left (\frac {3}{x^2}\right ) \log (x)\right ) \log ^2\left (\log ^2\left (\frac {3}{x^2}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 0.64, size = 80, normalized size = 4.71 \begin {gather*} -\log \left (\log \left (\log ^2\left (\frac {3}{x^2}\right )\right )\right )+\log \left (2-2 \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+2 x \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )-\log \left (\frac {3}{x^2}\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )+2 \left (\frac {1}{2} \log \left (\frac {3}{x^2}\right )+\log (x)\right ) \log \left (\log ^2\left (\frac {3}{x^2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.82, size = 75, normalized size = 4.41 \begin {gather*} \log \left (-2 \, x - \log \relax (3) + \log \left (\frac {3}{x^{2}}\right ) + 2\right ) + \log \left (\frac {{\left (2 \, x + \log \relax (3) - \log \left (\frac {3}{x^{2}}\right ) - 2\right )} \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right ) + 2}{2 \, x + \log \relax (3) - \log \left (\frac {3}{x^{2}}\right ) - 2}\right ) - \log \left (\log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x + 1\right )} \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right )^{2} \log \left (\frac {3}{x^{2}}\right ) + 4}{{\left (x \log \relax (x) \log \left (\frac {3}{x^{2}}\right ) + {\left (x^{2} - x\right )} \log \left (\frac {3}{x^{2}}\right )\right )} \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right )^{2} + x \log \left (\log \left (\frac {3}{x^{2}}\right )^{2}\right ) \log \left (\frac {3}{x^{2}}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((x + 1)*log(log(3/x^2)^2)^2*log(3/x^2) + 4)/((x*log(x)*log(3/x^2) + (x^2 - x)*log(3/x^2))*log(log(3
/x^2)^2)^2 + x*log(log(3/x^2)^2)*log(3/x^2)), x)

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maple [C]  time = 8.05, size = 1769, normalized size = 104.06

method	result	size
risch	\(\ln \left (-1+\ln \relax (x )+x \right )+\ln \left (\ln \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )-\frac {i \left (\pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{3}-\pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}+2 \pi -2 \pi x +2 i-4 i x \ln \relax (2)+4 i \ln \relax (2)+\pi x \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )-2 \pi x \,\mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}+\ln \relax (x ) \pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )-2 \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}-4 i \ln \relax (x ) \ln \relax (2)-2 \ln \relax (x ) \pi -\pi x \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{3}-2 \pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}-\ln \relax (x ) \pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{3}+2 \ln \relax (x ) \pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}+2 \pi x \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}\right )}{4 \left (-1+\ln \relax (x )+x \right )}\right )-\ln \left (\ln \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )-\frac {i \left (2 \pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right )^{2} \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )\right ) \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (-\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}+2 i \ln \relax (3)-4 i \ln \relax (x )\right )^{2}\right )^{3}-4 i \ln \relax (2)-2 \pi \right )}{4}\right )\)	\(1769\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(-1+ln(x)+x)+ln(ln(-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+2*I*ln(3)-4*I*l
n(x))-1/4*I*(Pi*csgn(I*(-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+2*I*ln(3)-4*
I*ln(x))^2)^3-Pi*csgn(I*(-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+2*I*ln(3)-4
*I*ln(x)))^2*csgn(I*(-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+2*I*ln(3)-4*I*l
n(x))^2)+2*Pi*csgn(I*(-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+2*I*ln(3)-4*I*
ln(x)))*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))
^2)^2+2*Pi-2*Pi*x+2*I-4*I*x*ln(2)+4*I*ln(2)+Pi*x*csgn(I*(-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+2*I*ln(3)-4*I*ln(x)))^2*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)-2*Pi*x*csgn(I*(-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+2*I*ln(3)-4*I*ln(x)))*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^2+ln(x)*Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x)))^2*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)-2*ln(x)*Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x)))*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^2-4*I*ln(x)*ln(2)-2*ln(x)*Pi-Pi*x*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^3-2*Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^2-ln(x)*Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^3+2*ln(x)*Pi*csgn(I*(-Pi*csg
n(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+2*I*ln(3)-4*I*ln(x))^2)^2+2*Pi*x*csgn(I*(-P
i*csgn(I*x)^2*csgn(I*x^2)+2*Pi*csgn(I*x)*csgn(I*x^2)^2-Pi*csgn(I*x^2)^3+2*I*ln(3)-4*I*ln(x))^2)^2)/(-1+ln(x)+x
))-ln(ln(-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+2*I*ln(3)-4*I*ln(x))-1/4*I*
(2*Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2
)^2+Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x)))
^2*csgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)-2
*Pi*csgn(I*(-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+2*I*ln(3)-4*I*ln(x)))*cs
gn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^2-Pi*c
sgn(I*(-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+2*I*ln(3)-4*I*ln(x))^2)^3-4*I
*ln(2)-2*Pi))

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maxima [B]  time = 0.60, size = 49, normalized size = 2.88 \begin {gather*} \log \left (x + \log \relax (x) - 1\right ) + \log \left (\frac {2 \, {\left (x + \log \relax (x) - 1\right )} \log \left (-\log \relax (3) + 2 \, \log \relax (x)\right ) + 1}{2 \, {\left (x + \log \relax (x) - 1\right )}}\right ) - \log \left (\log \left (-\log \relax (3) + 2 \, \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x) - 1) + log(1/2*(2*(x + log(x) - 1)*log(-log(3) + 2*log(x)) + 1)/(x + log(x) - 1)) - log(log(-lo
g(3) + 2*log(x)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} -\int \frac {\ln \left (\frac {3}{x^2}\right )\,\left (x+1\right )\,{\ln \left ({\ln \left (\frac {3}{x^2}\right )}^2\right )}^2+4}{{\ln \left ({\ln \left (\frac {3}{x^2}\right )}^2\right )}^2\,\left (\ln \left (\frac {3}{x^2}\right )\,\left (x-x^2\right )-x\,\ln \left (\frac {3}{x^2}\right )\,\ln \relax (x)\right )-x\,\ln \left ({\ln \left (\frac {3}{x^2}\right )}^2\right )\,\ln \left (\frac {3}{x^2}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(3/x^2)^2)^2*log(3/x^2)*(x + 1) + 4)/(log(log(3/x^2)^2)^2*(log(3/x^2)*(x - x^2) - x*log(3/x^2)*lo
g(x)) - x*log(log(3/x^2)^2)*log(3/x^2)),x)

[Out]

-int((log(log(3/x^2)^2)^2*log(3/x^2)*(x + 1) + 4)/(log(log(3/x^2)^2)^2*(log(3/x^2)*(x - x^2) - x*log(3/x^2)*lo
g(x)) - x*log(log(3/x^2)^2)*log(3/x^2)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: PolynomialError

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