3.42.9 \(\int \frac {30 x^2-10 x^4+(30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)) \log (x)+(18 x^3-12 x^5+2 x^7-120 x^4 \log (3)) \log ^2(x)+(9 x-6 x^3+x^5-60 x^2 \log (3)) \log ^3(x)+((-30-50 x^2+40 x^4) \log (x)+20 x^2 \log ^2(x)) \log (\log (x))}{(9 x^5-6 x^7+x^9) \log (x)+(18 x^3-12 x^5+2 x^7) \log ^2(x)+(9 x-6 x^3+x^5) \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ x-\frac {10 \left (-3 \log (3)+\frac {\log (\log (x))}{x^2+\log (x)}\right )}{-3+x^2} \]

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Rubi [F]  time = 16.09, 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 {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{\left (9 x^5-6 x^7+x^9\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7\right ) \log ^2(x)+\left (9 x-6 x^3+x^5\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(30*x^2 - 10*x^4 + (30 - 10*x^2 + 9*x^5 - 6*x^7 + x^9 - 60*x^6*Log[3])*Log[x] + (18*x^3 - 12*x^5 + 2*x^7 -
120*x^4*Log[3])*Log[x]^2 + (9*x - 6*x^3 + x^5 - 60*x^2*Log[3])*Log[x]^3 + ((-30 - 50*x^2 + 40*x^4)*Log[x] + 2
0*x^2*Log[x]^2)*Log[Log[x]])/((9*x^5 - 6*x^7 + x^9)*Log[x] + (18*x^3 - 12*x^5 + 2*x^7)*Log[x]^2 + (9*x - 6*x^3
+ x^5)*Log[x]^3),x]

[Out]

x - (30*Log[3])/(3 - x^2) + 30*Defer[Int][1/(x^3*(-3 + x^2)^2*Log[x]), x] - 10*Defer[Int][1/(x*(-3 + x^2)^2*Lo
g[x]), x] - (5*Defer[Int][1/((Sqrt[3] - x)*(x^2 + Log[x])^2), x])/3 - 180*Log[3]*Defer[Int][1/((Sqrt[3] - x)*(
x^2 + Log[x])^2), x] + (5*(1 + 108*Log[3])*Defer[Int][1/((Sqrt[3] - x)*(x^2 + Log[x])^2), x])/3 + (5*Defer[Int
][1/((Sqrt[3] + x)*(x^2 + Log[x])^2), x])/3 + 180*Log[3]*Defer[Int][1/((Sqrt[3] + x)*(x^2 + Log[x])^2), x] - (
5*(1 + 108*Log[3])*Defer[Int][1/((Sqrt[3] + x)*(x^2 + Log[x])^2), x])/3 - (5*Defer[Int][1/((Sqrt[3] - x)*(x^2
+ Log[x])), x])/9 - (10*Defer[Int][1/(x^3*(x^2 + Log[x])), x])/3 - (10*Defer[Int][1/(x*(x^2 + Log[x])), x])/9
+ (5*Defer[Int][1/((Sqrt[3] + x)*(x^2 + Log[x])), x])/9 - (65*Defer[Int][Log[Log[x]]/((Sqrt[3] - x)*(x^2 + Log
[x])^2), x])/3 - (10*Defer[Int][Log[Log[x]]/(x*(x^2 + Log[x])^2), x])/3 + (65*Defer[Int][Log[Log[x]]/((Sqrt[3]
+ x)*(x^2 + Log[x])^2), x])/3 + 60*Defer[Int][(x*Log[Log[x]])/((-3 + x^2)^2*(x^2 + Log[x])^2), x] + 20*Defer[
Int][(x*Log[x]*Log[Log[x]])/((-3 + x^2)^2*(x^2 + Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {30 x^2-10 x^4+\left (30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)\right ) \log (x)+\left (18 x^3-12 x^5+2 x^7-120 x^4 \log (3)\right ) \log ^2(x)+\left (9 x-6 x^3+x^5-60 x^2 \log (3)\right ) \log ^3(x)+\left (\left (-30-50 x^2+40 x^4\right ) \log (x)+20 x^2 \log ^2(x)\right ) \log (\log (x))}{x \left (3-x^2\right )^2 \log (x) \left (x^2+\log (x)\right )^2} \, dx\\ &=\int \left (\frac {30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}+\frac {30 x}{\left (-3+x^2\right )^2 \log (x) \left (x^2+\log (x)\right )^2}-\frac {10 x^3}{\left (-3+x^2\right )^2 \log (x) \left (x^2+\log (x)\right )^2}+\frac {2 x^2 \left (9-6 x^2+x^4-60 x \log (3)\right ) \log (x)}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}+\frac {\left (9-6 x^2+x^4-60 x \log (3)\right ) \log ^2(x)}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}+\frac {10 \left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}\right ) \, dx\\ &=2 \int \frac {x^2 \left (9-6 x^2+x^4-60 x \log (3)\right ) \log (x)}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx-10 \int \frac {x^3}{\left (-3+x^2\right )^2 \log (x) \left (x^2+\log (x)\right )^2} \, dx+10 \int \frac {\left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx+30 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (x^2+\log (x)\right )^2} \, dx+\int \frac {30-10 x^2+9 x^5-6 x^7+x^9-60 x^6 \log (3)}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx+\int \frac {\left (9-6 x^2+x^4-60 x \log (3)\right ) \log ^2(x)}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx\\ &=2 \int \left (-\frac {x^4 \left (9-6 x^2+x^4-60 x \log (3)\right )}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}+\frac {x^2 \left (9-6 x^2+x^4-60 x \log (3)\right )}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )}\right ) \, dx-10 \int \left (\frac {1}{x \left (-3+x^2\right )^2 \log (x)}-\frac {x}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}-\frac {1}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )}\right ) \, dx+10 \int \left (\frac {\left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{9 x \left (x^2+\log (x)\right )^2}+\frac {x \left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{3 \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}-\frac {x \left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{9 \left (-3+x^2\right ) \left (x^2+\log (x)\right )^2}\right ) \, dx+30 \int \left (\frac {1}{x^3 \left (-3+x^2\right )^2 \log (x)}-\frac {1}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}-\frac {1}{x^3 \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )}\right ) \, dx+\int \left (\frac {10}{3 x \left (x^2+\log (x)\right )^2}+\frac {x^4}{\left (x^2+\log (x)\right )^2}-\frac {60 x \log (3)}{\left (x^2+\log (x)\right )^2}-\frac {540 x \log (3)}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}+\frac {10 x (1+108 \log (3))}{3 \left (3-x^2\right ) \left (x^2+\log (x)\right )^2}\right ) \, dx+\int \left (\frac {9-6 x^2+x^4-60 x \log (3)}{\left (-3+x^2\right )^2}+\frac {x^4 \left (9-6 x^2+x^4-60 x \log (3)\right )}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2}-\frac {2 x^2 \left (9-6 x^2+x^4-60 x \log (3)\right )}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )}\right ) \, dx\\ &=\frac {10}{9} \int \frac {\left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{x \left (x^2+\log (x)\right )^2} \, dx-\frac {10}{9} \int \frac {x \left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{\left (-3+x^2\right ) \left (x^2+\log (x)\right )^2} \, dx-2 \int \frac {x^4 \left (9-6 x^2+x^4-60 x \log (3)\right )}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx+\frac {10}{3} \int \frac {1}{x \left (x^2+\log (x)\right )^2} \, dx+\frac {10}{3} \int \frac {x \left (-3-5 x^2+4 x^4+2 x^2 \log (x)\right ) \log (\log (x))}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx-10 \int \frac {1}{x \left (-3+x^2\right )^2 \log (x)} \, dx+10 \int \frac {x}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx+10 \int \frac {1}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )} \, dx+30 \int \frac {1}{x^3 \left (-3+x^2\right )^2 \log (x)} \, dx-30 \int \frac {1}{x \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx-30 \int \frac {1}{x^3 \left (-3+x^2\right )^2 \left (x^2+\log (x)\right )} \, dx-(60 \log (3)) \int \frac {x}{\left (x^2+\log (x)\right )^2} \, dx-(540 \log (3)) \int \frac {x}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx+\frac {1}{3} (10 (1+108 \log (3))) \int \frac {x}{\left (3-x^2\right ) \left (x^2+\log (x)\right )^2} \, dx+\int \frac {9-6 x^2+x^4-60 x \log (3)}{\left (-3+x^2\right )^2} \, dx+\int \frac {x^4}{\left (x^2+\log (x)\right )^2} \, dx+\int \frac {x^4 \left (9-6 x^2+x^4-60 x \log (3)\right )}{\left (-3+x^2\right )^2 \left (x^2+\log (x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 33, normalized size = 1.18 \begin {gather*} x+\frac {30 \log (3)}{-3+x^2}-\frac {10 \log (\log (x))}{\left (-3+x^2\right ) \left (x^2+\log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30*x^2 - 10*x^4 + (30 - 10*x^2 + 9*x^5 - 6*x^7 + x^9 - 60*x^6*Log[3])*Log[x] + (18*x^3 - 12*x^5 + 2
*x^7 - 120*x^4*Log[3])*Log[x]^2 + (9*x - 6*x^3 + x^5 - 60*x^2*Log[3])*Log[x]^3 + ((-30 - 50*x^2 + 40*x^4)*Log[
x] + 20*x^2*Log[x]^2)*Log[Log[x]])/((9*x^5 - 6*x^7 + x^9)*Log[x] + (18*x^3 - 12*x^5 + 2*x^7)*Log[x]^2 + (9*x -
6*x^3 + x^5)*Log[x]^3),x]

[Out]

x + (30*Log[3])/(-3 + x^2) - (10*Log[Log[x]])/((-3 + x^2)*(x^2 + Log[x]))

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fricas [A]  time = 0.73, size = 55, normalized size = 1.96 \begin {gather*} \frac {x^{5} - 3 \, x^{3} + 30 \, x^{2} \log \relax (3) + {\left (x^{3} - 3 \, x + 30 \, \log \relax (3)\right )} \log \relax (x) - 10 \, \log \left (\log \relax (x)\right )}{x^{4} - 3 \, x^{2} + {\left (x^{2} - 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^2*log(3)+x^5-6*x^3+9*x)*log(x)^3+(-1
20*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log(x)^2+(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/(
(x^5-6*x^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*log(x)),x, algorithm="fricas")

[Out]

(x^5 - 3*x^3 + 30*x^2*log(3) + (x^3 - 3*x + 30*log(3))*log(x) - 10*log(log(x)))/(x^4 - 3*x^2 + (x^2 - 3)*log(x
))

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giac [A]  time = 3.13, size = 39, normalized size = 1.39 \begin {gather*} x + \frac {30 \, \log \relax (3)}{x^{2} - 3} - \frac {10 \, \log \left (\log \relax (x)\right )}{x^{4} + x^{2} \log \relax (x) - 3 \, x^{2} - 3 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^2*log(3)+x^5-6*x^3+9*x)*log(x)^3+(-1
20*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log(x)^2+(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/(
(x^5-6*x^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*log(x)),x, algorithm="giac")

[Out]

x + 30*log(3)/(x^2 - 3) - 10*log(log(x))/(x^4 + x^2*log(x) - 3*x^2 - 3*log(x))

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maple [A]  time = 0.06, size = 41, normalized size = 1.46

method	result	size
risch	\(-\frac {10 \ln \left (\ln \relax (x )\right )}{\left (x^{2}-3\right ) \left (\ln \relax (x )+x^{2}\right )}+\frac {x^{3}+30 \ln \relax (3)-3 x}{x^{2}-3}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^2*ln(x)^2+(40*x^4-50*x^2-30)*ln(x))*ln(ln(x))+(-60*x^2*ln(3)+x^5-6*x^3+9*x)*ln(x)^3+(-120*x^4*ln(3)
+2*x^7-12*x^5+18*x^3)*ln(x)^2+(-60*x^6*ln(3)+x^9-6*x^7+9*x^5-10*x^2+30)*ln(x)-10*x^4+30*x^2)/((x^5-6*x^3+9*x)*
ln(x)^3+(2*x^7-12*x^5+18*x^3)*ln(x)^2+(x^9-6*x^7+9*x^5)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-10/(x^2-3)/(ln(x)+x^2)*ln(ln(x))+(x^3+30*ln(3)-3*x)/(x^2-3)

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maxima [A]  time = 0.51, size = 55, normalized size = 1.96 \begin {gather*} \frac {x^{5} - 3 \, x^{3} + 30 \, x^{2} \log \relax (3) + {\left (x^{3} - 3 \, x + 30 \, \log \relax (3)\right )} \log \relax (x) - 10 \, \log \left (\log \relax (x)\right )}{x^{4} - 3 \, x^{2} + {\left (x^{2} - 3\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2*log(x)^2+(40*x^4-50*x^2-30)*log(x))*log(log(x))+(-60*x^2*log(3)+x^5-6*x^3+9*x)*log(x)^3+(-1
20*x^4*log(3)+2*x^7-12*x^5+18*x^3)*log(x)^2+(-60*x^6*log(3)+x^9-6*x^7+9*x^5-10*x^2+30)*log(x)-10*x^4+30*x^2)/(
(x^5-6*x^3+9*x)*log(x)^3+(2*x^7-12*x^5+18*x^3)*log(x)^2+(x^9-6*x^7+9*x^5)*log(x)),x, algorithm="maxima")

[Out]

(x^5 - 3*x^3 + 30*x^2*log(3) + (x^3 - 3*x + 30*log(3))*log(x) - 10*log(log(x)))/(x^4 - 3*x^2 + (x^2 - 3)*log(x
))

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mupad [B]  time = 3.99, size = 37, normalized size = 1.32 \begin {gather*} x+\frac {30\,x^2\,\ln \relax (3)-10\,\ln \left (\ln \relax (x)\right )+30\,\ln \relax (3)\,\ln \relax (x)}{\left (\ln \relax (x)+x^2\right )\,\left (x^2-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(60*x^6*log(3) + 10*x^2 - 9*x^5 + 6*x^7 - x^9 - 30) + log(x)^2*(120*x^4*log(3) - 18*x^3 + 12*x^5
- 2*x^7) - log(x)^3*(9*x - 60*x^2*log(3) - 6*x^3 + x^5) + log(log(x))*(log(x)*(50*x^2 - 40*x^4 + 30) - 20*x^2*
log(x)^2) - 30*x^2 + 10*x^4)/(log(x)^2*(18*x^3 - 12*x^5 + 2*x^7) + log(x)^3*(9*x - 6*x^3 + x^5) + log(x)*(9*x^
5 - 6*x^7 + x^9)),x)

[Out]

x + (30*x^2*log(3) - 10*log(log(x)) + 30*log(3)*log(x))/((log(x) + x^2)*(x^2 - 3))

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sympy [A]  time = 0.53, size = 37, normalized size = 1.32 \begin {gather*} x - \frac {10 \log {\left (\log {\relax (x )} \right )}}{x^{4} + x^{2} \log {\relax (x )} - 3 x^{2} - 3 \log {\relax (x )}} + \frac {30 \log {\relax (3 )}}{x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**2*ln(x)**2+(40*x**4-50*x**2-30)*ln(x))*ln(ln(x))+(-60*x**2*ln(3)+x**5-6*x**3+9*x)*ln(x)**3+(
-120*x**4*ln(3)+2*x**7-12*x**5+18*x**3)*ln(x)**2+(-60*x**6*ln(3)+x**9-6*x**7+9*x**5-10*x**2+30)*ln(x)-10*x**4+
30*x**2)/((x**5-6*x**3+9*x)*ln(x)**3+(2*x**7-12*x**5+18*x**3)*ln(x)**2+(x**9-6*x**7+9*x**5)*ln(x)),x)

[Out]

x - 10*log(log(x))/(x**4 + x**2*log(x) - 3*x**2 - 3*log(x)) + 30*log(3)/(x**2 - 3)

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