3.81.6 \(\int \frac {2 x-x^2-4 x^4+2 x^4 \log (5)+(-1-4 x^2+2 x^2 \log (5)) \log (\log (2))+(4 x^3-2 x^3 \log (5)+(4 x-2 x \log (5)) \log (\log (2))) \log (x^2+\log (\log (2)))}{-x^3-x \log (\log (2))+(x^2+\log (\log (2))) \log (x^2+\log (\log (2)))} \, dx\)

Optimal. Leaf size=25 \[ 2+x^2 (2-\log (5))+\log \left (x-\log \left (x^2+\log (\log (2))\right )\right ) \]

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Rubi [A]  time = 0.50, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6, 6688, 6725, 6684} \begin {gather*} x^2 (2-\log (5))+\log \left (x-\log \left (x^2+\log (\log (2))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x-x^2+x^4 (-4+2 \log (5))+\left (-1-4 x^2+2 x^2 \log (5)\right ) \log (\log (2))+\left (4 x^3-2 x^3 \log (5)+(4 x-2 x \log (5)) \log (\log (2))\right ) \log \left (x^2+\log (\log (2))\right )}{-x^3-x \log (\log (2))+\left (x^2+\log (\log (2))\right ) \log \left (x^2+\log (\log (2))\right )} \, dx\\ &=\int \frac {-2 x-2 x^4 (-2+\log (5))+\log (\log (2))+x^2 (1-2 (-2+\log (5)) \log (\log (2)))+2 x (-2+\log (5)) \left (x^2+\log (\log (2))\right ) \log \left (x^2+\log (\log (2))\right )}{\left (x^2+\log (\log (2))\right ) \left (x-\log \left (x^2+\log (\log (2))\right )\right )} \, dx\\ &=\int \left (-2 x (-2+\log (5))+\frac {-2 x+x^2+\log (\log (2))}{\left (x^2+\log (\log (2))\right ) \left (x-\log \left (x^2+\log (\log (2))\right )\right )}\right ) \, dx\\ &=x^2 (2-\log (5))+\int \frac {-2 x+x^2+\log (\log (2))}{\left (x^2+\log (\log (2))\right ) \left (x-\log \left (x^2+\log (\log (2))\right )\right )} \, dx\\ &=x^2 (2-\log (5))+\log \left (x-\log \left (x^2+\log (\log (2))\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.46, size = 24, normalized size = 0.96 \begin {gather*} x^2 (2-\log (5))+\log \left (x-\log \left (x^2+\log (\log (2))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.18, size = 26, normalized size = 1.04 \begin {gather*} -x^{2} \log \relax (5) + 2 \, x^{2} + \log \left (-x + \log \left (x^{2} + \log \left (\log \relax (2)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x*log(5)+4*x)*log(log(2))-2*x^3*log(5)+4*x^3)*log(log(log(2))+x^2)+(2*x^2*log(5)-4*x^2-1)*log(
log(2))+2*x^4*log(5)-4*x^4-x^2+2*x)/((log(log(2))+x^2)*log(log(log(2))+x^2)-x*log(log(2))-x^3),x, algorithm="f
ricas")

[Out]

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

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giac [A]  time = 0.16, size = 23, normalized size = 0.92 \begin {gather*} -x^{2} {\left (\log \relax (5) - 2\right )} + \log \left (x - \log \left (x^{2} + \log \left (\log \relax (2)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x*log(5)+4*x)*log(log(2))-2*x^3*log(5)+4*x^3)*log(log(log(2))+x^2)+(2*x^2*log(5)-4*x^2-1)*log(
log(2))+2*x^4*log(5)-4*x^4-x^2+2*x)/((log(log(2))+x^2)*log(log(log(2))+x^2)-x*log(log(2))-x^3),x, algorithm="g
iac")

[Out]

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

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maple [A]  time = 0.14, size = 25, normalized size = 1.00

method	result	size
norman	\(\left (2-\ln \relax (5)\right ) x^{2}+\ln \left (-\ln \left (\ln \left (\ln \relax (2)\right )+x^{2}\right )+x \right )\)	\(25\)
default	\(2 x^{2}+\ln \left (-\ln \left (\ln \left (\ln \relax (2)\right )+x^{2}\right )+x \right )-x^{2} \ln \relax (5)\)	\(27\)
risch	\(-x^{2} \ln \relax (5)+2 x^{2}+\ln \left (-x +\ln \left (\ln \left (\ln \relax (2)\right )+x^{2}\right )\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x*ln(5)+4*x)*ln(ln(2))-2*x^3*ln(5)+4*x^3)*ln(ln(ln(2))+x^2)+(2*x^2*ln(5)-4*x^2-1)*ln(ln(2))+2*x^4*ln
(5)-4*x^4-x^2+2*x)/((ln(ln(2))+x^2)*ln(ln(ln(2))+x^2)-x*ln(ln(2))-x^3),x,method=_RETURNVERBOSE)

[Out]

(2-ln(5))*x^2+ln(-ln(ln(ln(2))+x^2)+x)

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maxima [A]  time = 0.50, size = 23, normalized size = 0.92 \begin {gather*} -x^{2} {\left (\log \relax (5) - 2\right )} + \log \left (-x + \log \left (x^{2} + \log \left (\log \relax (2)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x*log(5)+4*x)*log(log(2))-2*x^3*log(5)+4*x^3)*log(log(log(2))+x^2)+(2*x^2*log(5)-4*x^2-1)*log(
log(2))+2*x^4*log(5)-4*x^4-x^2+2*x)/((log(log(2))+x^2)*log(log(log(2))+x^2)-x*log(log(2))-x^3),x, algorithm="m
axima")

[Out]

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

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mupad [B]  time = 0.50, size = 23, normalized size = 0.92 \begin {gather*} \ln \left (\ln \left (x^2+\ln \left (\ln \relax (2)\right )\right )-x\right )-x^2\,\left (\ln \relax (5)-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.24, size = 20, normalized size = 0.80 \begin {gather*} x^{2} \left (2 - \log {\relax (5 )}\right ) + \log {\left (- x + \log {\left (x^{2} + \log {\left (\log {\relax (2 )} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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