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

Optimal. Leaf size=22 \[ \frac {5 \left (x+\frac {x}{\log (4)}\right ) \log ^2\left (x+\log ^2(x)\right )}{x^2} \]

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Rubi [F]  time = 0.90, 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 {(10 x+10 x \log (4)+(20+20 \log (4)) \log (x)) \log \left (x+\log ^2(x)\right )+\left (-5 x-5 x \log (4)+(-5-5 \log (4)) \log ^2(x)\right ) \log ^2\left (x+\log ^2(x)\right )}{x^3 \log (4)+x^2 \log (4) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(10*(1 + Log[4])*Defer[Int][Log[x + Log[x]^2]/(x*(x + Log[x]^2)), x])/Log[4] + (20*(1 + Log[4])*Defer[Int][(Lo
g[x]*Log[x + Log[x]^2])/(x^2*(x + Log[x]^2)), x])/Log[4] - (5*(1 + Log[4])*Defer[Int][Log[x + Log[x]^2]^2/x^2,
x])/Log[4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(10 x+10 x \log (4)+(20+20 \log (4)) \log (x)) \log \left (x+\log ^2(x)\right )+\left (-5 x-5 x \log (4)+(-5-5 \log (4)) \log ^2(x)\right ) \log ^2\left (x+\log ^2(x)\right )}{x^2 \left (x \log (4)+\log (4) \log ^2(x)\right )} \, dx\\ &=\int \frac {5 (1+\log (4)) \log \left (x+\log ^2(x)\right ) \left (2 x+4 \log (x)-x \log \left (x+\log ^2(x)\right )-\log ^2(x) \log \left (x+\log ^2(x)\right )\right )}{x^2 \log (4) \left (x+\log ^2(x)\right )} \, dx\\ &=\frac {(5 (1+\log (4))) \int \frac {\log \left (x+\log ^2(x)\right ) \left (2 x+4 \log (x)-x \log \left (x+\log ^2(x)\right )-\log ^2(x) \log \left (x+\log ^2(x)\right )\right )}{x^2 \left (x+\log ^2(x)\right )} \, dx}{\log (4)}\\ &=\frac {(5 (1+\log (4))) \int \left (\frac {2 (x+2 \log (x)) \log \left (x+\log ^2(x)\right )}{x^2 \left (x+\log ^2(x)\right )}-\frac {\log ^2\left (x+\log ^2(x)\right )}{x^2}\right ) \, dx}{\log (4)}\\ &=-\frac {(5 (1+\log (4))) \int \frac {\log ^2\left (x+\log ^2(x)\right )}{x^2} \, dx}{\log (4)}+\frac {(10 (1+\log (4))) \int \frac {(x+2 \log (x)) \log \left (x+\log ^2(x)\right )}{x^2 \left (x+\log ^2(x)\right )} \, dx}{\log (4)}\\ &=-\frac {(5 (1+\log (4))) \int \frac {\log ^2\left (x+\log ^2(x)\right )}{x^2} \, dx}{\log (4)}+\frac {(10 (1+\log (4))) \int \left (\frac {\log \left (x+\log ^2(x)\right )}{x \left (x+\log ^2(x)\right )}+\frac {2 \log (x) \log \left (x+\log ^2(x)\right )}{x^2 \left (x+\log ^2(x)\right )}\right ) \, dx}{\log (4)}\\ &=-\frac {(5 (1+\log (4))) \int \frac {\log ^2\left (x+\log ^2(x)\right )}{x^2} \, dx}{\log (4)}+\frac {(10 (1+\log (4))) \int \frac {\log \left (x+\log ^2(x)\right )}{x \left (x+\log ^2(x)\right )} \, dx}{\log (4)}+\frac {(20 (1+\log (4))) \int \frac {\log (x) \log \left (x+\log ^2(x)\right )}{x^2 \left (x+\log ^2(x)\right )} \, dx}{\log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(10 x+10 x \log (4)+(20+20 \log (4)) \log (x)) \log \left (x+\log ^2(x)\right )+\left (-5 x-5 x \log (4)+(-5-5 \log (4)) \log ^2(x)\right ) \log ^2\left (x+\log ^2(x)\right )}{x^3 \log (4)+x^2 \log (4) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 24, normalized size = 1.09 \begin {gather*} \frac {5 \, {\left (2 \, \log \relax (2) + 1\right )} \log \left (\log \relax (x)^{2} + x\right )^{2}}{2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-10*log(2)-5)*log(x)^2-10*x*log(2)-5*x)*log(log(x)^2+x)^2+((40*log(2)+20)*log(x)+20*x*log(2)+10*x
)*log(log(x)^2+x))/(2*x^2*log(2)*log(x)^2+2*x^3*log(2)),x, algorithm="fricas")

[Out]

5/2*(2*log(2) + 1)*log(log(x)^2 + x)^2/(x*log(2))

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giac [A]  time = 0.21, size = 24, normalized size = 1.09 \begin {gather*} \frac {5 \, {\left (2 \, \log \relax (2) + 1\right )} \log \left (\log \relax (x)^{2} + x\right )^{2}}{2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-10*log(2)-5)*log(x)^2-10*x*log(2)-5*x)*log(log(x)^2+x)^2+((40*log(2)+20)*log(x)+20*x*log(2)+10*x
)*log(log(x)^2+x))/(2*x^2*log(2)*log(x)^2+2*x^3*log(2)),x, algorithm="giac")

[Out]

5/2*(2*log(2) + 1)*log(log(x)^2 + x)^2/(x*log(2))

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maple [A]  time = 0.05, size = 25, normalized size = 1.14

method	result	size
risch	\(\frac {5 \left (1+2 \ln \relax (2)\right ) \ln \left (\ln \relax (x )^{2}+x \right )^{2}}{2 x \ln \relax (2)}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-10*ln(2)-5)*ln(x)^2-10*x*ln(2)-5*x)*ln(ln(x)^2+x)^2+((40*ln(2)+20)*ln(x)+20*x*ln(2)+10*x)*ln(ln(x)^2+x
))/(2*x^2*ln(2)*ln(x)^2+2*x^3*ln(2)),x,method=_RETURNVERBOSE)

[Out]

5/2*(1+2*ln(2))/x/ln(2)*ln(ln(x)^2+x)^2

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maxima [A]  time = 0.66, size = 24, normalized size = 1.09 \begin {gather*} \frac {5 \, {\left (2 \, \log \relax (2) + 1\right )} \log \left (\log \relax (x)^{2} + x\right )^{2}}{2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-10*log(2)-5)*log(x)^2-10*x*log(2)-5*x)*log(log(x)^2+x)^2+((40*log(2)+20)*log(x)+20*x*log(2)+10*x
)*log(log(x)^2+x))/(2*x^2*log(2)*log(x)^2+2*x^3*log(2)),x, algorithm="maxima")

[Out]

5/2*(2*log(2) + 1)*log(log(x)^2 + x)^2/(x*log(2))

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mupad [B]  time = 2.49, size = 24, normalized size = 1.09 \begin {gather*} \frac {{\ln \left ({\ln \relax (x)}^2+x\right )}^2\,\left (10\,\ln \relax (2)+5\right )}{2\,x\,\ln \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + log(x)^2)^2*(5*x + 10*x*log(2) + log(x)^2*(10*log(2) + 5)) - log(x + log(x)^2)*(10*x + 20*x*log(
2) + log(x)*(40*log(2) + 20)))/(2*x^3*log(2) + 2*x^2*log(2)*log(x)^2),x)

[Out]

(log(x + log(x)^2)^2*(10*log(2) + 5))/(2*x*log(2))

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sympy [A]  time = 0.42, size = 22, normalized size = 1.00 \begin {gather*} \frac {\left (5 + 10 \log {\relax (2 )}\right ) \log {\left (x + \log {\relax (x )}^{2} \right )}^{2}}{2 x \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-10*ln(2)-5)*ln(x)**2-10*x*ln(2)-5*x)*ln(ln(x)**2+x)**2+((40*ln(2)+20)*ln(x)+20*x*ln(2)+10*x)*ln(
ln(x)**2+x))/(2*x**2*ln(2)*ln(x)**2+2*x**3*ln(2)),x)

[Out]

(5 + 10*log(2))*log(x + log(x)**2)**2/(2*x*log(2))

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