Integrand size = 129, antiderivative size = 26 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=2+x+\left (3+x^2\right )^4-\frac {\log (5)}{-x+\log (5)}+\frac {1}{\log (x)} \]
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.92 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=\frac {x-\log (5)+\left (108 x^3+54 x^5+12 x^7+x^9+x^2 (1-108 \log (5))-54 x^4 \log (5)-12 x^6 \log (5)-x^8 \log (5)-x \log (5) \left (2+108 \log (5)+54 \log ^3(5)+12 \log ^5(5)+\log ^7(5)\right )+\log (5) \left (1+\log (5)+108 \log ^2(5)+54 \log ^4(5)+12 \log ^6(5)+\log ^8(5)\right )\right ) \log (x)}{(x-\log (5)) \log (x)} \]
Integrate[(-x^2 + 2*x*Log[5] - Log[5]^2 + (x^3 + 216*x^4 + 216*x^6 + 72*x^ 8 + 8*x^10 + (-x - 2*x^2 - 432*x^3 - 432*x^5 - 144*x^7 - 16*x^9)*Log[5] + (x + 216*x^2 + 216*x^4 + 72*x^6 + 8*x^8)*Log[5]^2)*Log[x]^2)/((x^3 - 2*x^2 *Log[5] + x*Log[5]^2)*Log[x]^2),x]
(x - Log[5] + (108*x^3 + 54*x^5 + 12*x^7 + x^9 + x^2*(1 - 108*Log[5]) - 54 *x^4*Log[5] - 12*x^6*Log[5] - x^8*Log[5] - x*Log[5]*(2 + 108*Log[5] + 54*L og[5]^3 + 12*Log[5]^5 + Log[5]^7) + Log[5]*(1 + Log[5] + 108*Log[5]^2 + 54 *Log[5]^4 + 12*Log[5]^6 + Log[5]^8))*Log[x])/((x - Log[5])*Log[x])
Time = 1.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2026, 7277, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-x^2+\left (8 x^{10}+72 x^8+216 x^6+216 x^4+x^3+\left (8 x^8+72 x^6+216 x^4+216 x^2+x\right ) \log ^2(5)+\left (-16 x^9-144 x^7-432 x^5-432 x^3-2 x^2-x\right ) \log (5)\right ) \log ^2(x)+2 x \log (5)-\log ^2(5)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-x^2+\left (8 x^{10}+72 x^8+216 x^6+216 x^4+x^3+\left (8 x^8+72 x^6+216 x^4+216 x^2+x\right ) \log ^2(5)+\left (-16 x^9-144 x^7-432 x^5-432 x^3-2 x^2-x\right ) \log (5)\right ) \log ^2(x)+2 x \log (5)-\log ^2(5)}{x \left (x^2-2 x \log (5)+\log ^2(5)\right ) \log ^2(x)}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int -\frac {x^2-2 \log (5) x-\left (8 x^{10}+72 x^8+216 x^6+216 x^4+x^3+\left (8 x^8+72 x^6+216 x^4+216 x^2+x\right ) \log ^2(5)-\left (16 x^9+144 x^7+432 x^5+432 x^3+2 x^2+x\right ) \log (5)\right ) \log ^2(x)+\log ^2(5)}{4 x (x-\log (5))^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {x^2-2 \log (5) x-\left (8 x^{10}+72 x^8+216 x^6+216 x^4+x^3+\left (8 x^8+72 x^6+216 x^4+216 x^2+x\right ) \log ^2(5)-\left (16 x^9+144 x^7+432 x^5+432 x^3+2 x^2+x\right ) \log (5)\right ) \log ^2(x)+\log ^2(5)}{x (x-\log (5))^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\int \left (\frac {-8 x^9+16 \log (5) x^8-72 \left (1+\frac {\log ^2(5)}{9}\right ) x^7+144 \log (5) x^6-216 \left (1+\frac {\log ^2(5)}{3}\right ) x^5+432 \log (5) x^4-216 \left (1+\log ^2(5)\right ) x^3-(1-432 \log (5)) x^2+2 (1-108 \log (5)) \log (5) x+(1-\log (5)) \log (5)}{(x-\log (5))^2}+\frac {1}{x \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^8+12 x^6+54 x^4+108 x^2+x+\frac {1}{\log (x)}+\frac {\log (5)}{x-\log (5)}\) |
Int[(-x^2 + 2*x*Log[5] - Log[5]^2 + (x^3 + 216*x^4 + 216*x^6 + 72*x^8 + 8* x^10 + (-x - 2*x^2 - 432*x^3 - 432*x^5 - 144*x^7 - 16*x^9)*Log[5] + (x + 2 16*x^2 + 216*x^4 + 72*x^6 + 8*x^8)*Log[5]^2)*Log[x]^2)/((x^3 - 2*x^2*Log[5 ] + x*Log[5]^2)*Log[x]^2),x]
3.6.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38
method | result | size |
default | \(x^{8}+12 x^{6}+54 x^{4}+108 x^{2}+x +\frac {\ln \left (5\right )}{-\ln \left (5\right )+x}+\frac {1}{\ln \left (x \right )}\) | \(36\) |
parts | \(x^{8}+12 x^{6}+54 x^{4}+108 x^{2}+x +\frac {\ln \left (5\right )}{-\ln \left (5\right )+x}+\frac {1}{\ln \left (x \right )}\) | \(36\) |
risch | \(\frac {x^{8} \ln \left (5\right )-x^{9}+12 x^{6} \ln \left (5\right )-12 x^{7}+54 x^{4} \ln \left (5\right )-54 x^{5}+108 x^{2} \ln \left (5\right )-108 x^{3}+x \ln \left (5\right )-x^{2}-\ln \left (5\right )}{\ln \left (5\right )-x}+\frac {1}{\ln \left (x \right )}\) | \(76\) |
parallelrisch | \(\frac {\ln \left (5\right ) x^{8} \ln \left (x \right )-x^{9} \ln \left (x \right )+12 \ln \left (5\right ) x^{6} \ln \left (x \right )-12 x^{7} \ln \left (x \right )+54 \ln \left (5\right ) x^{4} \ln \left (x \right )-54 x^{5} \ln \left (x \right )+108 x^{2} \ln \left (5\right ) \ln \left (x \right )-108 x^{3} \ln \left (x \right )+\ln \left (x \right ) \ln \left (5\right )^{2}-x^{2} \ln \left (x \right )-\ln \left (5\right ) \ln \left (x \right )+\ln \left (5\right )-x}{\ln \left (x \right ) \left (\ln \left (5\right )-x \right )}\) | \(103\) |
int((((8*x^8+72*x^6+216*x^4+216*x^2+x)*ln(5)^2+(-16*x^9-144*x^7-432*x^5-43 2*x^3-2*x^2-x)*ln(5)+8*x^10+72*x^8+216*x^6+216*x^4+x^3)*ln(x)^2-ln(5)^2+2* x*ln(5)-x^2)/(x*ln(5)^2-2*x^2*ln(5)+x^3)/ln(x)^2,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=\frac {{\left (x^{9} + 12 \, x^{7} + 54 \, x^{5} + 108 \, x^{3} + x^{2} - {\left (x^{8} + 12 \, x^{6} + 54 \, x^{4} + 108 \, x^{2} + x - 1\right )} \log \left (5\right )\right )} \log \left (x\right ) + x - \log \left (5\right )}{{\left (x - \log \left (5\right )\right )} \log \left (x\right )} \]
integrate((((8*x^8+72*x^6+216*x^4+216*x^2+x)*log(5)^2+(-16*x^9-144*x^7-432 *x^5-432*x^3-2*x^2-x)*log(5)+8*x^10+72*x^8+216*x^6+216*x^4+x^3)*log(x)^2-l og(5)^2+2*x*log(5)-x^2)/(x*log(5)^2-2*x^2*log(5)+x^3)/log(x)^2,x, algorith m=\
((x^9 + 12*x^7 + 54*x^5 + 108*x^3 + x^2 - (x^8 + 12*x^6 + 54*x^4 + 108*x^2 + x - 1)*log(5))*log(x) + x - log(5))/((x - log(5))*log(x))
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=x^{8} + 12 x^{6} + 54 x^{4} + 108 x^{2} + x + \frac {1}{\log {\left (x \right )}} + \frac {\log {\left (5 \right )}}{x - \log {\left (5 \right )}} \]
integrate((((8*x**8+72*x**6+216*x**4+216*x**2+x)*ln(5)**2+(-16*x**9-144*x* *7-432*x**5-432*x**3-2*x**2-x)*ln(5)+8*x**10+72*x**8+216*x**6+216*x**4+x** 3)*ln(x)**2-ln(5)**2+2*x*ln(5)-x**2)/(x*ln(5)**2-2*x**2*ln(5)+x**3)/ln(x)* *2,x)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=\frac {{\left (x^{9} - x^{8} \log \left (5\right ) + 12 \, x^{7} - 12 \, x^{6} \log \left (5\right ) + 54 \, x^{5} - 54 \, x^{4} \log \left (5\right ) + 108 \, x^{3} - x^{2} {\left (108 \, \log \left (5\right ) - 1\right )} - x \log \left (5\right ) + \log \left (5\right )\right )} \log \left (x\right ) + x - \log \left (5\right )}{{\left (x - \log \left (5\right )\right )} \log \left (x\right )} \]
integrate((((8*x^8+72*x^6+216*x^4+216*x^2+x)*log(5)^2+(-16*x^9-144*x^7-432 *x^5-432*x^3-2*x^2-x)*log(5)+8*x^10+72*x^8+216*x^6+216*x^4+x^3)*log(x)^2-l og(5)^2+2*x*log(5)-x^2)/(x*log(5)^2-2*x^2*log(5)+x^3)/log(x)^2,x, algorith m=\
((x^9 - x^8*log(5) + 12*x^7 - 12*x^6*log(5) + 54*x^5 - 54*x^4*log(5) + 108 *x^3 - x^2*(108*log(5) - 1) - x*log(5) + log(5))*log(x) + x - log(5))/((x - log(5))*log(x))
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=x^{8} + 12 \, x^{6} + 54 \, x^{4} + 108 \, x^{2} + x + \frac {\log \left (5\right )}{x - \log \left (5\right )} + \frac {1}{\log \left (x\right )} \]
integrate((((8*x^8+72*x^6+216*x^4+216*x^2+x)*log(5)^2+(-16*x^9-144*x^7-432 *x^5-432*x^3-2*x^2-x)*log(5)+8*x^10+72*x^8+216*x^6+216*x^4+x^3)*log(x)^2-l og(5)^2+2*x*log(5)-x^2)/(x*log(5)^2-2*x^2*log(5)+x^3)/log(x)^2,x, algorith m=\
Time = 13.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{\left (x^3-2 x^2 \log (5)+x \log ^2(5)\right ) \log ^2(x)} \, dx=x+\frac {1}{\ln \left (x\right )}+\frac {\ln \left (5\right )}{x-\ln \left (5\right )}+108\,x^2+54\,x^4+12\,x^6+x^8 \]