3.34.68 \(\int \frac {-32 x^7+(x^2)^x (32 x^7-16 x^8-8 x^8 \log (x^2))+32 x^7 \log (\log (4))}{-1+(x^2)^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+(x^2)^{2 x} (-3+3 \log (\log (4)))+(x^2)^x (3-6 \log (\log (4))+3 \log ^2(\log (4)))} \, dx\)

Optimal. Leaf size=19 \[ 1+\frac {4 x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \]

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Rubi [F]  time = 1.83, 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 {-32 x^7+\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+32 x^7 \log (\log (4))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-32*x^7 + (x^2)^x*(32*x^7 - 16*x^8 - 8*x^8*Log[x^2]) + 32*x^7*Log[Log[4]])/(-1 + (x^2)^(3*x) + 3*Log[Log[
4]] - 3*Log[Log[4]]^2 + Log[Log[4]]^3 + (x^2)^(2*x)*(-3 + 3*Log[Log[4]]) + (x^2)^x*(3 - 6*Log[Log[4]] + 3*Log[
Log[4]]^2)),x]

[Out]

-16*(1 - Log[Log[4]])*Defer[Int][x^8/(-1 + (x^2)^x + Log[Log[4]])^3, x] - 8*Log[x^2]*(1 - Log[Log[4]])*Defer[I
nt][x^8/(-1 + (x^2)^x + Log[Log[4]])^3, x] + 32*Defer[Int][x^7/(-1 + (x^2)^x + Log[Log[4]])^2, x] - 16*Defer[I
nt][x^8/(-1 + (x^2)^x + Log[Log[4]])^2, x] - 8*Log[x^2]*Defer[Int][x^8/(-1 + (x^2)^x + Log[Log[4]])^2, x] + 16
*(1 - Log[Log[4]])*Defer[Int][Defer[Int][x^8/(-1 + (x^2)^x + Log[Log[4]])^3, x]/x, x] + 16*Defer[Int][Defer[In
t][x^8/(-1 + (x^2)^x + Log[Log[4]])^2, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (x^2\right )^x \left (32 x^7-16 x^8-8 x^8 \log \left (x^2\right )\right )+x^7 (-32+32 \log (\log (4)))}{-1+\left (x^2\right )^{3 x}+3 \log (\log (4))-3 \log ^2(\log (4))+\log ^3(\log (4))+\left (x^2\right )^{2 x} (-3+3 \log (\log (4)))+\left (x^2\right )^x \left (3-6 \log (\log (4))+3 \log ^2(\log (4))\right )} \, dx\\ &=\int \frac {8 x^7 \left (2 x \left (x^2\right )^x+x \left (x^2\right )^x \log \left (x^2\right )-4 \left (-1+\left (x^2\right )^x+\log (\log (4))\right )\right )}{\left (1-\left (x^2\right )^x-\log (\log (4))\right )^3} \, dx\\ &=8 \int \frac {x^7 \left (2 x \left (x^2\right )^x+x \left (x^2\right )^x \log \left (x^2\right )-4 \left (-1+\left (x^2\right )^x+\log (\log (4))\right )\right )}{\left (1-\left (x^2\right )^x-\log (\log (4))\right )^3} \, dx\\ &=8 \int \left (\frac {x^8 \left (2+\log \left (x^2\right )\right ) (-1+\log (\log (4)))}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3}-\frac {x^7 \left (-4+2 x+x \log \left (x^2\right )\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {x^7 \left (-4+2 x+x \log \left (x^2\right )\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx\right )-(8 (1-\log (\log (4)))) \int \frac {x^8 \left (2+\log \left (x^2\right )\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx\\ &=-\left (8 \int \left (-\frac {4 x^7}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2}+\frac {2 x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2}+\frac {x^8 \log \left (x^2\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2}\right ) \, dx\right )-(8 (1-\log (\log (4)))) \int \left (\frac {2 x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3}+\frac {x^8 \log \left (x^2\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3}\right ) \, dx\\ &=-\left (8 \int \frac {x^8 \log \left (x^2\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx\right )-16 \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx+32 \int \frac {x^7}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx-(8 (1-\log (\log (4)))) \int \frac {x^8 \log \left (x^2\right )}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx-(16 (1-\log (\log (4)))) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx\\ &=8 \int \frac {2 \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx}{x} \, dx-16 \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx+32 \int \frac {x^7}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx-\left (8 \log \left (x^2\right )\right ) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx+(8 (1-\log (\log (4)))) \int \frac {2 \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx}{x} \, dx-(16 (1-\log (\log (4)))) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx-\left (8 \log \left (x^2\right ) (1-\log (\log (4)))\right ) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx\\ &=-\left (16 \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx\right )+16 \int \frac {\int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx}{x} \, dx+32 \int \frac {x^7}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx-\left (8 \log \left (x^2\right )\right ) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \, dx-(16 (1-\log (\log (4)))) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx+(16 (1-\log (\log (4)))) \int \frac {\int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx}{x} \, dx-\left (8 \log \left (x^2\right ) (1-\log (\log (4)))\right ) \int \frac {x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 17, normalized size = 0.89 \begin {gather*} \frac {4 x^8}{\left (-1+\left (x^2\right )^x+\log (\log (4))\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-32*x^7 + (x^2)^x*(32*x^7 - 16*x^8 - 8*x^8*Log[x^2]) + 32*x^7*Log[Log[4]])/(-1 + (x^2)^(3*x) + 3*Lo
g[Log[4]] - 3*Log[Log[4]]^2 + Log[Log[4]]^3 + (x^2)^(2*x)*(-3 + 3*Log[Log[4]]) + (x^2)^x*(3 - 6*Log[Log[4]] +
3*Log[Log[4]]^2)),x]

[Out]

(4*x^8)/(-1 + (x^2)^x + Log[Log[4]])^2

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fricas [B]  time = 0.49, size = 44, normalized size = 2.32 \begin {gather*} \frac {4 \, x^{8}}{2 \, {\left (x^{2}\right )}^{x} {\left (\log \left (2 \, \log \relax (2)\right ) - 1\right )} + \log \left (2 \, \log \relax (2)\right )^{2} + {\left (x^{2}\right )}^{2 \, x} - 2 \, \log \left (2 \, \log \relax (2)\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*log(2))-32*x^7)/(exp(x*log(x^2))^3+(3*
log(2*log(2))-3)*exp(x*log(x^2))^2+(3*log(2*log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log
(2*log(2))^2+3*log(2*log(2))-1),x, algorithm="fricas")

[Out]

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

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giac [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(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*log(2))-32*x^7)/(exp(x*log(x^2))^3+(3*
log(2*log(2))-3)*exp(x*log(x^2))^2+(3*log(2*log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log
(2*log(2))^2+3*log(2*log(2))-1),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 20, normalized size = 1.05

method	result	size
risch	\(\frac {4 x^{8}}{\left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )+\left (x^{2}\right )^{x}-1\right )^{2}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^8*ln(x^2)-16*x^8+32*x^7)*exp(x*ln(x^2))+32*x^7*ln(2*ln(2))-32*x^7)/(exp(x*ln(x^2))^3+(3*ln(2*ln(2))
-3)*exp(x*ln(x^2))^2+(3*ln(2*ln(2))^2-6*ln(2*ln(2))+3)*exp(x*ln(x^2))+ln(2*ln(2))^3-3*ln(2*ln(2))^2+3*ln(2*ln(
2))-1),x,method=_RETURNVERBOSE)

[Out]

4*x^8/(ln(2)+ln(ln(2))+(x^2)^x-1)^2

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maxima [B]  time = 0.64, size = 51, normalized size = 2.68 \begin {gather*} \frac {4 \, x^{8}}{2 \, x^{2 \, x} {\left (\log \relax (2) + \log \left (\log \relax (2)\right ) - 1\right )} + 2 \, {\left (\log \left (\log \relax (2)\right ) - 1\right )} \log \relax (2) + \log \relax (2)^{2} + \log \left (\log \relax (2)\right )^{2} + x^{4 \, x} - 2 \, \log \left (\log \relax (2)\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^8*log(x^2)-16*x^8+32*x^7)*exp(x*log(x^2))+32*x^7*log(2*log(2))-32*x^7)/(exp(x*log(x^2))^3+(3*
log(2*log(2))-3)*exp(x*log(x^2))^2+(3*log(2*log(2))^2-6*log(2*log(2))+3)*exp(x*log(x^2))+log(2*log(2))^3-3*log
(2*log(2))^2+3*log(2*log(2))-1),x, algorithm="maxima")

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x*log(x^2))*(8*x^8*log(x^2) - 32*x^7 + 16*x^8) - 32*x^7*log(2*log(2)) + 32*x^7)/(3*log(2*log(2)) + e
xp(3*x*log(x^2)) - 3*log(2*log(2))^2 + log(2*log(2))^3 + exp(x*log(x^2))*(3*log(2*log(2))^2 - 6*log(2*log(2))
+ 3) + exp(2*x*log(x^2))*(3*log(2*log(2)) - 3) - 1),x)

[Out]

\text{Hanged}

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sympy [B]  time = 0.42, size = 71, normalized size = 3.74 \begin {gather*} \frac {4 x^{8}}{e^{2 x \log {\left (x^{2} \right )}} + \left (-2 + 2 \log {\left (\log {\relax (2 )} \right )} + 2 \log {\relax (2 )}\right ) e^{x \log {\left (x^{2} \right )}} - 2 \log {\relax (2 )} + 2 \log {\relax (2 )} \log {\left (\log {\relax (2 )} \right )} + \log {\left (\log {\relax (2 )} \right )}^{2} + \log {\relax (2 )}^{2} - 2 \log {\left (\log {\relax (2 )} \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**8*ln(x**2)-16*x**8+32*x**7)*exp(x*ln(x**2))+32*x**7*ln(2*ln(2))-32*x**7)/(exp(x*ln(x**2))**3
+(3*ln(2*ln(2))-3)*exp(x*ln(x**2))**2+(3*ln(2*ln(2))**2-6*ln(2*ln(2))+3)*exp(x*ln(x**2))+ln(2*ln(2))**3-3*ln(2
*ln(2))**2+3*ln(2*ln(2))-1),x)

[Out]

4*x**8/(exp(2*x*log(x**2)) + (-2 + 2*log(log(2)) + 2*log(2))*exp(x*log(x**2)) - 2*log(2) + 2*log(2)*log(log(2)
) + log(log(2))**2 + log(2)**2 - 2*log(log(2)) + 1)

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