3.49.90 \(\int \frac {2+x-2 x^2-x \log (16)+(-x+x^2+(-1+x) \log (16)-\log (x^2)) \log (x-x^2+(1-x) \log (16)+\log (x^2)) \log (\log (x-x^2+(1-x) \log (16)+\log (x^2)))+(-x+x^2+(-1+x) \log (16)-\log (x^2)) \log (x-x^2+(1-x) \log (16)+\log (x^2)) \log (\log (x-x^2+(1-x) \log (16)+\log (x^2))) \log (\log (\log (x-x^2+(1-x) \log (16)+\log (x^2))))}{(x^3-x^4+(x^2-x^3) \log (16)+x^2 \log (x^2)) \log (x-x^2+(1-x) \log (16)+\log (x^2)) \log (\log (x-x^2+(1-x) \log (16)+\log (x^2)))} \, dx\)

Optimal. Leaf size=24 \[ \frac {1+\log \left (\log \left (\log \left ((1-x) (x+\log (16))+\log \left (x^2\right )\right )\right )\right )}{x} \]

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Rubi [F]  time = 4.79, 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-2 x^2-x \log (16)+\left (-x+x^2+(-1+x) \log (16)-\log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )+\left (-x+x^2+(-1+x) \log (16)-\log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right ) \log \left (\log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )\right )}{\left (x^3-x^4+\left (x^2-x^3\right ) \log (16)+x^2 \log \left (x^2\right )\right ) \log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right ) \log \left (\log \left (x-x^2+(1-x) \log (16)+\log \left (x^2\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x^(-1) + 2*Defer[Int][1/((x^2 - x*(1 - Log[16]) - Log[16*x^2])*Log[-((-1 + x)*(x + Log[16])) + Log[x^2]]*Log[L
og[-((-1 + x)*(x + Log[16])) + Log[x^2]]]), x] - (1 - Log[16])*Defer[Int][1/(x*(x^2 - x*(1 - Log[16]) - Log[16
*x^2])*Log[-((-1 + x)*(x + Log[16])) + Log[x^2]]*Log[Log[-((-1 + x)*(x + Log[16])) + Log[x^2]]]), x] + 2*Defer
[Int][1/(x^2*(-x^2 + x*(1 - Log[16]) + Log[16*x^2])*Log[-((-1 + x)*(x + Log[16])) + Log[x^2]]*Log[Log[-((-1 +
x)*(x + Log[16])) + Log[x^2]]]), x] - Defer[Int][Log[Log[Log[-((-1 + x)*(x + Log[16])) + Log[x^2]]]]/x^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{x}+\frac {\log \left (\log \left (\log \left (-((-1+x) (x+\log (16)))+\log \left (x^2\right )\right )\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + x - 2*x^2 - x*Log[16] + (-x + x^2 + (-1 + x)*Log[16] - Log[x^2])*Log[x - x^2 + (1 - x)*Log[16]
+ Log[x^2]]*Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]] + (-x + x^2 + (-1 + x)*Log[16] - Log[x^2])*Log[x -
x^2 + (1 - x)*Log[16] + Log[x^2]]*Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]]*Log[Log[Log[x - x^2 + (1 - x)
*Log[16] + Log[x^2]]]])/((x^3 - x^4 + (x^2 - x^3)*Log[16] + x^2*Log[x^2])*Log[x - x^2 + (1 - x)*Log[16] + Log[
x^2]]*Log[Log[x - x^2 + (1 - x)*Log[16] + Log[x^2]]]),x]

[Out]

x^(-1) + Log[Log[Log[-((-1 + x)*(x + Log[16])) + Log[x^2]]]]/x

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fricas [A]  time = 0.79, size = 27, normalized size = 1.12 \begin {gather*} \frac {\log \left (\log \left (\log \left (-x^{2} - 4 \, {\left (x - 1\right )} \log \relax (2) + x + \log \left (x^{2}\right )\right )\right )\right ) + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 6.63, size = 31, normalized size = 1.29 \begin {gather*} \frac {\log \left (\log \left (\log \left (-x^{2} - 4 \, x \log \relax (2) + x + 4 \, \log \relax (2) + \log \left (x^{2}\right )\right )\right )\right )}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-\ln \left (x^{2}\right )+4 \left (x -1\right ) \ln \relax (2)+x^{2}-x \right ) \ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right ) \ln \left (\ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right )\right ) \ln \left (\ln \left (\ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right )\right )\right )+\left (-\ln \left (x^{2}\right )+4 \left (x -1\right ) \ln \relax (2)+x^{2}-x \right ) \ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right ) \ln \left (\ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right )\right )-4 x \ln \relax (2)-2 x^{2}+x +2}{\left (x^{2} \ln \left (x^{2}\right )+4 \left (-x^{3}+x^{2}\right ) \ln \relax (2)-x^{4}+x^{3}\right ) \ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right ) \ln \left (\ln \left (\ln \left (x^{2}\right )+4 \left (1-x \right ) \ln \relax (2)-x^{2}+x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 32, normalized size = 1.33 \begin {gather*} \frac {\log \left (\log \left (\log \left (-x^{2} - x {\left (4 \, \log \relax (2) - 1\right )} + 4 \, \log \relax (2) + 2 \, \log \relax (x)\right )\right )\right ) + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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

[Out]

Timed out

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