3.13.72 \(\int \frac {-8+10 \log (2 x)+(4-3 x+(20+10 x) \log (2 x)) \log (x^2)+(1+5 \log (2 x)) \log (x^2) \log (\log (x^2))}{10 \log (x^2)} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{10} x (-4+5 \log (2 x)) \left (4+x+\log \left (\log \left (x^2\right )\right )\right ) \]

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Rubi [F]  time = 0.39, 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 {-8+10 \log (2 x)+(4-3 x+(20+10 x) \log (2 x)) \log \left (x^2\right )+(1+5 \log (2 x)) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{10 \log \left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 + 10*Log[2*x] + (4 - 3*x + (20 + 10*x)*Log[2*x])*Log[x^2] + (1 + 5*Log[2*x])*Log[x^2]*Log[Log[x^2]])/(
10*Log[x^2]),x]

[Out]

(-11*x)/10 - (2*x^2)/5 - (x*ExpIntegralEi[Log[x^2]/2])/(10*Sqrt[x^2]) - (x*ExpIntegralEi[Log[x^2]/2]*(4 - 5*Lo
g[2*x]))/(10*Sqrt[x^2]) + 2*x*Log[2*x] + (x^2*Log[2*x])/2 - (x*ExpIntegralEi[Log[x^2]/2]*Log[x^2])/(4*Sqrt[x^2
]) + (x*Log[Log[x^2]])/10 + Defer[Int][Log[2*x]*Log[Log[x^2]], x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int \frac {-8+10 \log (2 x)+(4-3 x+(20+10 x) \log (2 x)) \log \left (x^2\right )+(1+5 \log (2 x)) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{\log \left (x^2\right )} \, dx\\ &=\frac {1}{10} \int \left (\frac {-8+10 \log (2 x)+4 \log \left (x^2\right )-3 x \log \left (x^2\right )+20 \log (2 x) \log \left (x^2\right )+10 x \log (2 x) \log \left (x^2\right )}{\log \left (x^2\right )}+(1+5 \log (2 x)) \log \left (\log \left (x^2\right )\right )\right ) \, dx\\ &=\frac {1}{10} \int \frac {-8+10 \log (2 x)+4 \log \left (x^2\right )-3 x \log \left (x^2\right )+20 \log (2 x) \log \left (x^2\right )+10 x \log (2 x) \log \left (x^2\right )}{\log \left (x^2\right )} \, dx+\frac {1}{10} \int (1+5 \log (2 x)) \log \left (\log \left (x^2\right )\right ) \, dx\\ &=\frac {1}{10} \int \frac {-8+(4-3 x) \log \left (x^2\right )+10 \log (2 x) \left (1+(2+x) \log \left (x^2\right )\right )}{\log \left (x^2\right )} \, dx+\frac {1}{10} \int \left (\log \left (\log \left (x^2\right )\right )+5 \log (2 x) \log \left (\log \left (x^2\right )\right )\right ) \, dx\\ &=\frac {1}{10} \int \left (4-3 x+20 \log (2 x)+10 x \log (2 x)+\frac {2 (-4+5 \log (2 x))}{\log \left (x^2\right )}\right ) \, dx+\frac {1}{10} \int \log \left (\log \left (x^2\right )\right ) \, dx+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx\\ &=\frac {2 x}{5}-\frac {3 x^2}{20}+\frac {1}{10} x \log \left (\log \left (x^2\right )\right )-\frac {1}{5} \int \frac {1}{\log \left (x^2\right )} \, dx+\frac {1}{5} \int \frac {-4+5 \log (2 x)}{\log \left (x^2\right )} \, dx+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx+2 \int \log (2 x) \, dx+\int x \log (2 x) \, dx\\ &=-\frac {8 x}{5}-\frac {2 x^2}{5}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right ) (4-5 \log (2 x))}{10 \sqrt {x^2}}+2 x \log (2 x)+\frac {1}{2} x^2 \log (2 x)+\frac {1}{10} x \log \left (\log \left (x^2\right )\right )+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx-\frac {x \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{10 \sqrt {x^2}}-\int \frac {\text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{2 \sqrt {x^2}} \, dx\\ &=-\frac {8 x}{5}-\frac {2 x^2}{5}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{10 \sqrt {x^2}}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right ) (4-5 \log (2 x))}{10 \sqrt {x^2}}+2 x \log (2 x)+\frac {1}{2} x^2 \log (2 x)+\frac {1}{10} x \log \left (\log \left (x^2\right )\right )-\frac {1}{2} \int \frac {\text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{\sqrt {x^2}} \, dx+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx\\ &=-\frac {8 x}{5}-\frac {2 x^2}{5}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{10 \sqrt {x^2}}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right ) (4-5 \log (2 x))}{10 \sqrt {x^2}}+2 x \log (2 x)+\frac {1}{2} x^2 \log (2 x)+\frac {1}{10} x \log \left (\log \left (x^2\right )\right )+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx-\frac {x \int \frac {\text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{x} \, dx}{2 \sqrt {x^2}}\\ &=-\frac {8 x}{5}-\frac {2 x^2}{5}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{10 \sqrt {x^2}}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right ) (4-5 \log (2 x))}{10 \sqrt {x^2}}+2 x \log (2 x)+\frac {1}{2} x^2 \log (2 x)+\frac {1}{10} x \log \left (\log \left (x^2\right )\right )+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx-\frac {x \operatorname {Subst}\left (\int \text {Ei}\left (\frac {x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{4 \sqrt {x^2}}\\ &=-\frac {11 x}{10}-\frac {2 x^2}{5}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{10 \sqrt {x^2}}-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right ) (4-5 \log (2 x))}{10 \sqrt {x^2}}+2 x \log (2 x)+\frac {1}{2} x^2 \log (2 x)-\frac {x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right ) \log \left (x^2\right )}{4 \sqrt {x^2}}+\frac {1}{10} x \log \left (\log \left (x^2\right )\right )+\frac {1}{2} \int \log (2 x) \log \left (\log \left (x^2\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{10} x (-4+5 \log (2 x)) \left (4+x+\log \left (\log \left (x^2\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 10*Log[2*x] + (4 - 3*x + (20 + 10*x)*Log[2*x])*Log[x^2] + (1 + 5*Log[2*x])*Log[x^2]*Log[Log[x^
2]])/(10*Log[x^2]),x]

[Out]

(x*(-4 + 5*Log[2*x])*(4 + x + Log[Log[x^2]]))/10

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fricas [B]  time = 0.75, size = 47, normalized size = 2.24 \begin {gather*} -\frac {2}{5} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + 4 \, x\right )} \log \left (2 \, x\right ) + \frac {1}{10} \, {\left (5 \, x \log \left (2 \, x\right ) - 4 \, x\right )} \log \left (-2 \, \log \relax (2) + 2 \, \log \left (2 \, x\right )\right ) - \frac {8}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((5*log(2*x)+1)*log(x^2)*log(log(x^2))+((10*x+20)*log(2*x)+4-3*x)*log(x^2)+10*log(2*x)-8)/log(x
^2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.60, size = 56, normalized size = 2.67 \begin {gather*} \frac {1}{10} \, x^{2} {\left (5 \, \log \relax (2) - 4\right )} + \frac {1}{10} \, x {\left (20 \, \log \relax (2) - 11\right )} + \frac {1}{2} \, {\left (x^{2} + 4 \, x\right )} \log \relax (x) + \frac {1}{10} \, {\left (x {\left (5 \, \log \relax (2) - 4\right )} + 5 \, x \log \relax (x)\right )} \log \left (\log \left (x^{2}\right )\right ) - \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((5*log(2*x)+1)*log(x^2)*log(log(x^2))+((10*x+20)*log(2*x)+4-3*x)*log(x^2)+10*log(2*x)-8)/log(x
^2),x, algorithm="giac")

[Out]

1/10*x^2*(5*log(2) - 4) + 1/10*x*(20*log(2) - 11) + 1/2*(x^2 + 4*x)*log(x) + 1/10*(x*(5*log(2) - 4) + 5*x*log(
x))*log(log(x^2)) - 1/2*x

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maple [C]  time = 0.11, size = 84, normalized size = 4.00

method	result	size
risch	\(\frac {\left (5 x \ln \relax (x )+5 x \ln \relax (2)-4 x \right ) \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )}{10}+\frac {x^{2} \ln \relax (x )}{2}+2 x \ln \relax (x )+\frac {x^{2} \ln \relax (2)}{2}+2 x \ln \relax (2)-\frac {2 x^{2}}{5}-\frac {8 x}{5}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10*((5*ln(2*x)+1)*ln(x^2)*ln(ln(x^2))+((10*x+20)*ln(2*x)+4-3*x)*ln(x^2)+10*ln(2*x)-8)/ln(x^2),x,method=_
RETURNVERBOSE)

[Out]

1/10*(5*x*ln(x)+5*x*ln(2)-4*x)*ln(2*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2)+1/2*x^2*ln(x)+2*x*l
n(x)+1/2*x^2*ln(2)+2*x*ln(2)-2/5*x^2-8/5*x

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maxima [B]  time = 0.60, size = 65, normalized size = 3.10 \begin {gather*} \frac {1}{2} \, x^{2} \log \left (2 \, x\right ) + \frac {1}{2} \, x \log \relax (2) \log \relax (x) + \frac {1}{10} \, {\left (5 \, \log \relax (2)^{2} - 4 \, \log \relax (2)\right )} x - \frac {2}{5} \, x^{2} + 2 \, x \log \left (2 \, x\right ) + \frac {1}{10} \, {\left (x {\left (5 \, \log \relax (2) - 4\right )} + 5 \, x \log \relax (x)\right )} \log \left (\log \relax (x)\right ) - \frac {8}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((5*log(2*x)+1)*log(x^2)*log(log(x^2))+((10*x+20)*log(2*x)+4-3*x)*log(x^2)+10*log(2*x)-8)/log(x
^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.16, size = 41, normalized size = 1.95 \begin {gather*} \ln \left (2\,x\right )\,\left (\frac {x^2}{2}+2\,x\right )-\frac {8\,x}{5}-\ln \left (\ln \left (x^2\right )\right )\,\left (\frac {2\,x}{5}-\frac {x\,\ln \left (2\,x\right )}{2}\right )-\frac {2\,x^2}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x) + (log(x^2)*(log(2*x)*(10*x + 20) - 3*x + 4))/10 + (log(x^2)*log(log(x^2))*(5*log(2*x) + 1))/10
- 4/5)/log(x^2),x)

[Out]

log(2*x)*(2*x + x^2/2) - (8*x)/5 - log(log(x^2))*((2*x)/5 - (x*log(2*x))/2) - (2*x^2)/5

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sympy [B]  time = 0.64, size = 49, normalized size = 2.33 \begin {gather*} - \frac {2 x^{2}}{5} - \frac {8 x}{5} + \left (\frac {x^{2}}{2} + 2 x\right ) \log {\left (2 x \right )} + \left (\frac {x \log {\left (2 x \right )}}{2} - \frac {2 x}{5}\right ) \log {\left (2 \log {\left (2 x \right )} - \log {\relax (4 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*((5*ln(2*x)+1)*ln(x**2)*ln(ln(x**2))+((10*x+20)*ln(2*x)+4-3*x)*ln(x**2)+10*ln(2*x)-8)/ln(x**2),
x)

[Out]

-2*x**2/5 - 8*x/5 + (x**2/2 + 2*x)*log(2*x) + (x*log(2*x)/2 - 2*x/5)*log(2*log(2*x) - log(4))

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