3.42.21 \(\int \frac {-2 x \log (x)+2 \log (x) \log ^2(x^2)+((-x-x \log (x)) \log (x^2)+(2+2 \log (x)) \log ^2(x^2)) \log (\frac {-x+2 \log (x^2)}{5 x \log (x^2)})+(-5 x \log (x^2)+10 \log ^2(x^2)) \log ^2(\frac {-x+2 \log (x^2)}{5 x \log (x^2)})}{(-5 x \log (x^2)+10 \log ^2(x^2)) \log ^2(\frac {-x+2 \log (x^2)}{5 x \log (x^2)})} \, dx\)

Optimal. Leaf size=33 \[ x+\frac {1}{5} \left (3+\frac {x \log (x)}{\log \left (\frac {2-\frac {x}{\log \left (x^2\right )}}{5 x}\right )}\right ) \]

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Rubi [F]  time = 2.18, 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 \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x*Log[x] + 2*Log[x]*Log[x^2]^2 + ((-x - x*Log[x])*Log[x^2] + (2 + 2*Log[x])*Log[x^2]^2)*Log[(-x + 2*Lo
g[x^2])/(5*x*Log[x^2])] + (-5*x*Log[x^2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2)/((-5*x*Log[
x^2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \log (x)-2 \log (x) \log ^2\left (x^2\right )-\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )-\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{5 \left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx\\ &=\frac {1}{5} \int \frac {2 x \log (x)-2 \log (x) \log ^2\left (x^2\right )-\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )-\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx\\ &=\frac {1}{5} \int \left (5+\frac {2 \log (x) \left (x-\log ^2\left (x^2\right )\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}+\frac {1+\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \, dx\\ &=x+\frac {1}{5} \int \frac {1+\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx+\frac {2}{5} \int \frac {\log (x) \left (x-\log ^2\left (x^2\right )\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx\\ &=x+\frac {1}{5} \int \left (\frac {1}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}+\frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \, dx+\frac {2}{5} \int \left (\frac {x \log (x)}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}-\frac {\log (x) \log \left (x^2\right )}{\left (x-2 \log \left (x^2\right )\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \, dx\\ &=x+\frac {1}{5} \int \frac {1}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx+\frac {1}{5} \int \frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx+\frac {2}{5} \int \frac {x \log (x)}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx-\frac {2}{5} \int \frac {\log (x) \log \left (x^2\right )}{\left (x-2 \log \left (x^2\right )\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 32, normalized size = 0.97 \begin {gather*} -\frac {1}{5} x \left (-5-\frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x*Log[x] + 2*Log[x]*Log[x^2]^2 + ((-x - x*Log[x])*Log[x^2] + (2 + 2*Log[x])*Log[x^2]^2)*Log[(-x
+ 2*Log[x^2])/(5*x*Log[x^2])] + (-5*x*Log[x^2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2)/((-5*
x*Log[x^2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2),x]

[Out]

-1/5*(x*(-5 - Log[x]/Log[2/(5*x) - 1/(5*Log[x^2])]))

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fricas [A]  time = 0.68, size = 44, normalized size = 1.33 \begin {gather*} \frac {x \log \relax (x) + 5 \, x \log \left (-\frac {x - 4 \, \log \relax (x)}{10 \, x \log \relax (x)}\right )}{5 \, \log \left (-\frac {x - 4 \, \log \relax (x)}{10 \, x \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*log(x^2)^2-5*x*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2))^2+((2*log(x)+2)*log(x^2)^2+(-x*log(
x)-x)*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2))+2*log(x)*log(x^2)^2-2*x*log(x))/(10*log(x^2)^2-5*x*log(x^2)
)/log(1/5*(2*log(x^2)-x)/x/log(x^2))^2,x, algorithm="fricas")

[Out]

1/5*(x*log(x) + 5*x*log(-1/10*(x - 4*log(x))/(x*log(x))))/log(-1/10*(x - 4*log(x))/(x*log(x)))

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giac [B]  time = 1.18, size = 218, normalized size = 6.61 \begin {gather*} x - \frac {4 \, x^{2} \log \left (x^{2}\right ) \log \relax (x)^{2} - 8 \, x \log \left (x^{2}\right )^{2} \log \relax (x)^{2} - x^{3} \log \left (x^{2}\right ) + 2 \, x^{2} \log \left (x^{2}\right )^{2}}{10 \, {\left (x \log \left (x^{2}\right )^{2} \log \relax (x) - 4 \, \log \left (x^{2}\right )^{2} \log \relax (x)^{2} - x \log \left (x^{2}\right )^{2} \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) + 4 \, \log \left (x^{2}\right )^{2} \log \relax (x) \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) + x \log \left (x^{2}\right )^{2} \log \left (5 \, \log \left (x^{2}\right )\right ) - 4 \, \log \left (x^{2}\right )^{2} \log \relax (x) \log \left (5 \, \log \left (x^{2}\right )\right ) - x^{2} \log \relax (x) + 4 \, x \log \relax (x)^{2} + x^{2} \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) - 4 \, x \log \relax (x) \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) - x^{2} \log \left (5 \, \log \left (x^{2}\right )\right ) + 4 \, x \log \relax (x) \log \left (5 \, \log \left (x^{2}\right )\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*log(x^2)^2-5*x*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2))^2+((2*log(x)+2)*log(x^2)^2+(-x*log(
x)-x)*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2))+2*log(x)*log(x^2)^2-2*x*log(x))/(10*log(x^2)^2-5*x*log(x^2)
)/log(1/5*(2*log(x^2)-x)/x/log(x^2))^2,x, algorithm="giac")

[Out]

x - 1/10*(4*x^2*log(x^2)*log(x)^2 - 8*x*log(x^2)^2*log(x)^2 - x^3*log(x^2) + 2*x^2*log(x^2)^2)/(x*log(x^2)^2*l
og(x) - 4*log(x^2)^2*log(x)^2 - x*log(x^2)^2*log(-x + 2*log(x^2)) + 4*log(x^2)^2*log(x)*log(-x + 2*log(x^2)) +
x*log(x^2)^2*log(5*log(x^2)) - 4*log(x^2)^2*log(x)*log(5*log(x^2)) - x^2*log(x) + 4*x*log(x)^2 + x^2*log(-x +
2*log(x^2)) - 4*x*log(x)*log(-x + 2*log(x^2)) - x^2*log(5*log(x^2)) + 4*x*log(x)*log(5*log(x^2)))

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maple [C]  time = 0.65, size = 1531, normalized size = 46.39

method	result	size
risch	\(x -\frac {2 x \ln \relax (x )}{5 \left (-2 \ln \relax (2)+2 \ln \relax (5)+2 \ln \relax (x )+2 \ln \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )-2 \ln \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )^{3}-i \pi \,\mathrm {csgn}\left (i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{x \left (\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}\right )}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right ) \mathrm {csgn}\left (\frac {i \left (-i x +\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}+4 i \ln \relax (x )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right )}{\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )+\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}}\right )^{2}\right )}\)	\(1531\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*ln(x^2)^2-5*x*ln(x^2))*ln(1/5*(2*ln(x^2)-x)/x/ln(x^2))^2+((2*ln(x)+2)*ln(x^2)^2+(-x*ln(x)-x)*ln(x^2))
*ln(1/5*(2*ln(x^2)-x)/x/ln(x^2))+2*ln(x)*ln(x^2)^2-2*x*ln(x))/(10*ln(x^2)^2-5*x*ln(x^2))/ln(1/5*(2*ln(x^2)-x)/
x/ln(x^2))^2,x,method=_RETURNVERBOSE)

[Out]

x-2/5*x*ln(x)/(-2*ln(2)+2*ln(5)+2*ln(x)+2*ln(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I
*x)*csgn(I*x^2)^2)-2*ln(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^
3)+I*Pi*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))*csgn(I*(-
I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))*csgn(I/(Pi*csgn(I*x^2
)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*c
sgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))+I*Pi*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*c
sgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(
x)+Pi*csgn(I*x^2)^3))^3-I*Pi*csgn(I*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi
*csgn(I*x^2)^3))*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(
-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))^2-I*Pi*csgn(I/x)*csg
n(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^
2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))^2-I*Pi*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln
(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*c
sgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi
*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^
2)^3))^2+I*Pi*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*
x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))^3+I*Pi*csgn(I/x)*csgn(I
/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^2*csg
n(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))*csgn(I/x/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csg
n(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)
^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))-I*Pi*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I
*x)*csgn(I*x^2)^2))*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2
)*(-I*x+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+4*I*ln(x)+Pi*csgn(I*x^2)^3))^2)

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maxima [B]  time = 0.51, size = 57, normalized size = 1.73 \begin {gather*} \frac {5 \, x {\left (\log \relax (5) + \log \relax (2)\right )} + 4 \, x \log \relax (x) - 5 \, x \log \left (-x + 4 \, \log \relax (x)\right ) + 5 \, x \log \left (\log \relax (x)\right )}{5 \, {\left (\log \relax (5) + \log \relax (2) + \log \relax (x) - \log \left (-x + 4 \, \log \relax (x)\right ) + \log \left (\log \relax (x)\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*log(x^2)^2-5*x*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2))^2+((2*log(x)+2)*log(x^2)^2+(-x*log(
x)-x)*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2))+2*log(x)*log(x^2)^2-2*x*log(x))/(10*log(x^2)^2-5*x*log(x^2)
)/log(1/5*(2*log(x^2)-x)/x/log(x^2))^2,x, algorithm="maxima")

[Out]

1/5*(5*x*(log(5) + log(2)) + 4*x*log(x) - 5*x*log(-x + 4*log(x)) + 5*x*log(log(x)))/(log(5) + log(2) + log(x)
- log(-x + 4*log(x)) + log(log(x)))

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mupad [B]  time = 4.02, size = 296, normalized size = 8.97 \begin {gather*} \frac {4\,x}{5}+\frac {\frac {x\,\ln \relax (x)}{5}-\frac {x\,\ln \left (-\frac {\frac {x}{5}-\frac {2\,\ln \left (x^2\right )}{5}}{x\,\ln \left (x^2\right )}\right )\,\ln \left (x^2\right )\,\left (\ln \relax (x)+1\right )\,\left (x-2\,\ln \left (x^2\right )\right )}{10\,\left (4\,\ln \relax (x)\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-x+4\,{\ln \relax (x)}^2+{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2\right )}}{\ln \left (-\frac {\frac {x}{5}-\frac {2\,\ln \left (x^2\right )}{5}}{x\,\ln \left (x^2\right )}\right )}-\frac {x\,\ln \relax (x)}{5}+\frac {x^2}{20}-\frac {\frac {2\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-32\,x^4\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )+16\,x^4\,{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2-x^5\,{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2+64\,x^4-20\,x^5+x^6}{20\,\left (16\,x^2-x^3\right )}+\frac {\ln \relax (x)\,\left (16\,x^4\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-x^5\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )\right )}{10\,\left (16\,x^2-x^3\right )}}{4\,\ln \relax (x)\,\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )-x+4\,{\ln \relax (x)}^2+{\left (\ln \left (x^2\right )-2\,\ln \relax (x)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(x/5 - (2*log(x^2))/5)/(x*log(x^2)))^2*(5*x*log(x^2) - 10*log(x^2)^2) + 2*x*log(x) + log(-(x/5 - (2*
log(x^2))/5)/(x*log(x^2)))*(log(x^2)*(x + x*log(x)) - log(x^2)^2*(2*log(x) + 2)) - 2*log(x^2)^2*log(x))/(log(-
(x/5 - (2*log(x^2))/5)/(x*log(x^2)))^2*(5*x*log(x^2) - 10*log(x^2)^2)),x)

[Out]

(4*x)/5 + ((x*log(x))/5 - (x*log(-(x/5 - (2*log(x^2))/5)/(x*log(x^2)))*log(x^2)*(log(x) + 1)*(x - 2*log(x^2)))
/(10*(4*log(x)*(log(x^2) - 2*log(x)) - x + 4*log(x)^2 + (log(x^2) - 2*log(x))^2)))/log(-(x/5 - (2*log(x^2))/5)
/(x*log(x^2))) - (x*log(x))/5 + x^2/20 - ((2*x^5*(log(x^2) - 2*log(x)) - 32*x^4*(log(x^2) - 2*log(x)) + 16*x^4
*(log(x^2) - 2*log(x))^2 - x^5*(log(x^2) - 2*log(x))^2 + 64*x^4 - 20*x^5 + x^6)/(20*(16*x^2 - x^3)) + (log(x)*
(16*x^4*(log(x^2) - 2*log(x)) - x^5*(log(x^2) - 2*log(x))))/(10*(16*x^2 - x^3)))/(4*log(x)*(log(x^2) - 2*log(x
)) - x + 4*log(x)^2 + (log(x^2) - 2*log(x))^2)

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sympy [A]  time = 0.35, size = 26, normalized size = 0.79 \begin {gather*} \frac {x \log {\relax (x )}}{5 \log {\left (\frac {- \frac {x}{5} + \frac {4 \log {\relax (x )}}{5}}{2 x \log {\relax (x )}} \right )}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*ln(x**2)**2-5*x*ln(x**2))*ln(1/5*(2*ln(x**2)-x)/x/ln(x**2))**2+((2*ln(x)+2)*ln(x**2)**2+(-x*ln(
x)-x)*ln(x**2))*ln(1/5*(2*ln(x**2)-x)/x/ln(x**2))+2*ln(x)*ln(x**2)**2-2*x*ln(x))/(10*ln(x**2)**2-5*x*ln(x**2))
/ln(1/5*(2*ln(x**2)-x)/x/ln(x**2))**2,x)

[Out]

x*log(x)/(5*log((-x/5 + 4*log(x)/5)/(2*x*log(x)))) + x

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