Integrand size = 141, antiderivative size = 20 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=\left (x-16 x^4-\frac {4}{x+\log (4) \log (5)}\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(221\) vs. \(2(20)=40\).
Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 11.05 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=\frac {16-32 x^7+256 x^{10}-64 x^6 \log (4) \log (5)+512 x^9 \log (4) \log (5)+8 \log ^2(4) \log ^2(5)+256 x^8 \log ^2(4) \log ^2(5)-\log ^4(4) \log ^4(5)+512 \log ^5(4) \log ^5(5)-32 \log ^7(4) \log ^7(5)-256 \log ^{10}(4) \log ^{10}(5)+x^4 (1+128 \log (4) \log (5))-32 x^5 \left (-4+\log ^2(4) \log ^2(5)\right )-32 x^2 \log ^3(4) \log ^3(5) \left (-16+\log ^2(4) \log ^2(5)+8 \log ^5(4) \log ^5(5)\right )-2 x \log (4) \log (5) \left (-4+\log ^2(4) \log ^2(5)-512 \log ^3(4) \log ^3(5)+32 \log ^5(4) \log ^5(5)+256 \log ^8(4) \log ^8(5)\right )+x^3 \log (5) \log (16)}{(x+\log (4) \log (5))^2} \]
Integrate[(-32 + 2*x^4 + 384*x^5 - 160*x^7 + 2048*x^10 + (-8*x + 6*x^3 + 8 96*x^4 - 480*x^6 + 6144*x^9)*Log[4]*Log[5] + (-8 + 6*x^2 + 512*x^3 - 480*x ^5 + 6144*x^8)*Log[4]^2*Log[5]^2 + (2*x - 160*x^4 + 2048*x^7)*Log[4]^3*Log [5]^3)/(x^3 + 3*x^2*Log[4]*Log[5] + 3*x*Log[4]^2*Log[5]^2 + Log[4]^3*Log[5 ]^3),x]
(16 - 32*x^7 + 256*x^10 - 64*x^6*Log[4]*Log[5] + 512*x^9*Log[4]*Log[5] + 8 *Log[4]^2*Log[5]^2 + 256*x^8*Log[4]^2*Log[5]^2 - Log[4]^4*Log[5]^4 + 512*L og[4]^5*Log[5]^5 - 32*Log[4]^7*Log[5]^7 - 256*Log[4]^10*Log[5]^10 + x^4*(1 + 128*Log[4]*Log[5]) - 32*x^5*(-4 + Log[4]^2*Log[5]^2) - 32*x^2*Log[4]^3* Log[5]^3*(-16 + Log[4]^2*Log[5]^2 + 8*Log[4]^5*Log[5]^5) - 2*x*Log[4]*Log[ 5]*(-4 + Log[4]^2*Log[5]^2 - 512*Log[4]^3*Log[5]^3 + 32*Log[4]^5*Log[5]^5 + 256*Log[4]^8*Log[5]^8) + x^3*Log[5]*Log[16])/(x + Log[4]*Log[5])^2
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(20)=40\).
Time = 0.43 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2007, 2389, 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.
| \(\displaystyle \int \frac {2048 x^{10}-160 x^7+384 x^5+2 x^4+\left (2048 x^7-160 x^4+2 x\right ) \log ^3(4) \log ^3(5)+\left (6144 x^9-480 x^6+896 x^4+6 x^3-8 x\right ) \log (4) \log (5)+\left (6144 x^8-480 x^5+512 x^3+6 x^2-8\right ) \log ^2(4) \log ^2(5)-32}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {2048 x^{10}-160 x^7+384 x^5+2 x^4+\left (2048 x^7-160 x^4+2 x\right ) \log ^3(4) \log ^3(5)+\left (6144 x^9-480 x^6+896 x^4+6 x^3-8 x\right ) \log (4) \log (5)+\left (6144 x^8-480 x^5+512 x^3+6 x^2-8\right ) \log ^2(4) \log ^2(5)-32}{(x+\log (4) \log (5))^3}dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (2048 x^7-160 x^4+384 x^2-\frac {8 \left (16 \log ^4(4) \log ^4(5)+\log (4) \log (5)\right )}{(x+\log (4) \log (5))^2}-2 x (128 \log (4) \log (5)-1)-\frac {32}{(x+\log (4) \log (5))^3}+128 \log ^2(4) \log ^2(5)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 256 x^8-32 x^5+128 x^3+x^2 (1-128 \log (4) \log (5))+\frac {8 \log (4) \log (5) \left (1+16 \log ^3(4) \log ^3(5)\right )}{x+\log (4) \log (5)}+128 x \log ^2(4) \log ^2(5)+\frac {16}{(x+\log (4) \log (5))^2}\) |
Int[(-32 + 2*x^4 + 384*x^5 - 160*x^7 + 2048*x^10 + (-8*x + 6*x^3 + 896*x^4 - 480*x^6 + 6144*x^9)*Log[4]*Log[5] + (-8 + 6*x^2 + 512*x^3 - 480*x^5 + 6 144*x^8)*Log[4]^2*Log[5]^2 + (2*x - 160*x^4 + 2048*x^7)*Log[4]^3*Log[5]^3) /(x^3 + 3*x^2*Log[4]*Log[5] + 3*x*Log[4]^2*Log[5]^2 + Log[4]^3*Log[5]^3),x ]
128*x^3 - 32*x^5 + 256*x^8 + 128*x*Log[4]^2*Log[5]^2 + x^2*(1 - 128*Log[4] *Log[5]) + 16/(x + Log[4]*Log[5])^2 + (8*Log[4]*Log[5]*(1 + 16*Log[4]^3*Lo g[5]^3))/(x + Log[4]*Log[5])
3.20.10.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(21)=42\).
Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.00
| method | result | size |
| default | \(256 x^{8}+512 x \ln \left (2\right )^{2} \ln \left (5\right )^{2}-32 x^{5}-256 x^{2} \ln \left (2\right ) \ln \left (5\right )+128 x^{3}+x^{2}+\frac {16}{\left (2 \ln \left (2\right ) \ln \left (5\right )+x \right )^{2}}+\frac {16 \ln \left (2\right ) \ln \left (5\right ) \left (128 \ln \left (2\right )^{3} \ln \left (5\right )^{3}+1\right )}{2 \ln \left (2\right ) \ln \left (5\right )+x}\) | \(80\) |
| risch | \(256 x^{8}+512 x \ln \left (2\right )^{2} \ln \left (5\right )^{2}-32 x^{5}-256 x^{2} \ln \left (2\right ) \ln \left (5\right )+128 x^{3}+x^{2}+\frac {\left (512 \ln \left (5\right )^{4} \ln \left (2\right )^{4}+4 \ln \left (2\right ) \ln \left (5\right )\right ) x +1024 \ln \left (2\right )^{5} \ln \left (5\right )^{5}+8 \ln \left (2\right )^{2} \ln \left (5\right )^{2}+4}{\ln \left (2\right )^{2} \ln \left (5\right )^{2}+x \ln \left (2\right ) \ln \left (5\right )+\frac {x^{2}}{4}}\) | \(105\) |
| norman | \(\frac {\left (256 \ln \left (2\right ) \ln \left (5\right )+1\right ) x^{4}+\left (-128 \ln \left (2\right )^{2} \ln \left (5\right )^{2}+128\right ) x^{5}+\left (-16 \ln \left (2\right )^{3} \ln \left (5\right )^{3}+16 \ln \left (2\right ) \ln \left (5\right )\right ) x -32 x^{7}+256 x^{10}+4 \ln \left (5\right ) \ln \left (2\right ) x^{3}-128 \ln \left (5\right ) \ln \left (2\right ) x^{6}+1024 \ln \left (5\right ) \ln \left (2\right ) x^{9}+1024 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{8}+16-16 \ln \left (5\right )^{4} \ln \left (2\right )^{4}+32 \ln \left (2\right )^{2} \ln \left (5\right )^{2}}{\left (2 \ln \left (2\right ) \ln \left (5\right )+x \right )^{2}}\) | \(131\) |
| gosper | \(-\frac {-1024 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{8}-1024 \ln \left (5\right ) \ln \left (2\right ) x^{9}-256 x^{10}+128 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{5}+16 \ln \left (5\right )^{4} \ln \left (2\right )^{4}+128 \ln \left (5\right ) \ln \left (2\right ) x^{6}+16 \ln \left (5\right )^{3} \ln \left (2\right )^{3} x +32 x^{7}-256 x^{4} \ln \left (5\right ) \ln \left (2\right )-4 \ln \left (5\right ) \ln \left (2\right ) x^{3}-128 x^{5}-32 \ln \left (2\right )^{2} \ln \left (5\right )^{2}-x^{4}-16 x \ln \left (2\right ) \ln \left (5\right )-16}{4 \ln \left (2\right )^{2} \ln \left (5\right )^{2}+4 x \ln \left (2\right ) \ln \left (5\right )+x^{2}}\) | \(148\) |
| parallelrisch | \(-\frac {-1024 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{8}-1024 \ln \left (5\right ) \ln \left (2\right ) x^{9}-256 x^{10}+128 \ln \left (5\right )^{2} \ln \left (2\right )^{2} x^{5}+16 \ln \left (5\right )^{4} \ln \left (2\right )^{4}+128 \ln \left (5\right ) \ln \left (2\right ) x^{6}+16 \ln \left (5\right )^{3} \ln \left (2\right )^{3} x +32 x^{7}-256 x^{4} \ln \left (5\right ) \ln \left (2\right )-4 \ln \left (5\right ) \ln \left (2\right ) x^{3}-128 x^{5}-32 \ln \left (2\right )^{2} \ln \left (5\right )^{2}-x^{4}-16 x \ln \left (2\right ) \ln \left (5\right )-16}{4 \ln \left (2\right )^{2} \ln \left (5\right )^{2}+4 x \ln \left (2\right ) \ln \left (5\right )+x^{2}}\) | \(148\) |
int((8*(2048*x^7-160*x^4+2*x)*ln(2)^3*ln(5)^3+4*(6144*x^8-480*x^5+512*x^3+ 6*x^2-8)*ln(2)^2*ln(5)^2+2*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)*ln(2)*ln(5 )+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*ln(2)^3*ln(5)^3+12*x*ln(2)^2*ln(5 )^2+6*x^2*ln(2)*ln(5)+x^3),x,method=_RETURNVERBOSE)
256*x^8+512*x*ln(2)^2*ln(5)^2-32*x^5-256*x^2*ln(2)*ln(5)+128*x^3+x^2+16/(2 *ln(2)*ln(5)+x)^2+16*ln(2)*ln(5)*(128*ln(2)^3*ln(5)^3+1)/(2*ln(2)*ln(5)+x)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 131, normalized size of antiderivative = 6.55 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=\frac {256 \, x^{10} + 4096 \, \log \left (5\right )^{5} \log \left (2\right )^{5} + 4096 \, x \log \left (5\right )^{4} \log \left (2\right )^{4} + 1024 \, x^{2} \log \left (5\right )^{3} \log \left (2\right )^{3} - 32 \, x^{7} + 128 \, x^{5} + 4 \, {\left (256 \, x^{8} - 32 \, x^{5} + x^{2} + 8\right )} \log \left (5\right )^{2} \log \left (2\right )^{2} + x^{4} + 4 \, {\left (256 \, x^{9} - 32 \, x^{6} + 64 \, x^{4} + x^{3} + 4 \, x\right )} \log \left (5\right ) \log \left (2\right ) + 16}{4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 4 \, x \log \left (5\right ) \log \left (2\right ) + x^{2}} \]
integrate((8*(2048*x^7-160*x^4+2*x)*log(2)^3*log(5)^3+4*(6144*x^8-480*x^5+ 512*x^3+6*x^2-8)*log(2)^2*log(5)^2+2*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)* log(2)*log(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*log(2)^3*log(5)^3+12* x*log(2)^2*log(5)^2+6*x^2*log(2)*log(5)+x^3),x, algorithm=\
(256*x^10 + 4096*log(5)^5*log(2)^5 + 4096*x*log(5)^4*log(2)^4 + 1024*x^2*l og(5)^3*log(2)^3 - 32*x^7 + 128*x^5 + 4*(256*x^8 - 32*x^5 + x^2 + 8)*log(5 )^2*log(2)^2 + x^4 + 4*(256*x^9 - 32*x^6 + 64*x^4 + x^3 + 4*x)*log(5)*log( 2) + 16)/(4*log(5)^2*log(2)^2 + 4*x*log(5)*log(2) + x^2)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 5.70 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=256 x^{8} - 32 x^{5} + 128 x^{3} + x^{2} \left (- 256 \log {\left (2 \right )} \log {\left (5 \right )} + 1\right ) + 512 x \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2} + \frac {x \left (16 \log {\left (2 \right )} \log {\left (5 \right )} + 2048 \log {\left (2 \right )}^{4} \log {\left (5 \right )}^{4}\right ) + 16 + 32 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2} + 4096 \log {\left (2 \right )}^{5} \log {\left (5 \right )}^{5}}{x^{2} + 4 x \log {\left (2 \right )} \log {\left (5 \right )} + 4 \log {\left (2 \right )}^{2} \log {\left (5 \right )}^{2}} \]
integrate((8*(2048*x**7-160*x**4+2*x)*ln(2)**3*ln(5)**3+4*(6144*x**8-480*x **5+512*x**3+6*x**2-8)*ln(2)**2*ln(5)**2+2*(6144*x**9-480*x**6+896*x**4+6* x**3-8*x)*ln(2)*ln(5)+2048*x**10-160*x**7+384*x**5+2*x**4-32)/(8*ln(2)**3* ln(5)**3+12*x*ln(2)**2*ln(5)**2+6*x**2*ln(2)*ln(5)+x**3),x)
256*x**8 - 32*x**5 + 128*x**3 + x**2*(-256*log(2)*log(5) + 1) + 512*x*log( 2)**2*log(5)**2 + (x*(16*log(2)*log(5) + 2048*log(2)**4*log(5)**4) + 16 + 32*log(2)**2*log(5)**2 + 4096*log(2)**5*log(5)**5)/(x**2 + 4*x*log(2)*log( 5) + 4*log(2)**2*log(5)**2)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (23) = 46\).
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.25 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=256 \, x^{8} - 32 \, x^{5} + 512 \, x \log \left (5\right )^{2} \log \left (2\right )^{2} - {\left (256 \, \log \left (5\right ) \log \left (2\right ) - 1\right )} x^{2} + 128 \, x^{3} + \frac {16 \, {\left (256 \, \log \left (5\right )^{5} \log \left (2\right )^{5} + 2 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + {\left (128 \, \log \left (5\right )^{4} \log \left (2\right )^{4} + \log \left (5\right ) \log \left (2\right )\right )} x + 1\right )}}{4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 4 \, x \log \left (5\right ) \log \left (2\right ) + x^{2}} \]
integrate((8*(2048*x^7-160*x^4+2*x)*log(2)^3*log(5)^3+4*(6144*x^8-480*x^5+ 512*x^3+6*x^2-8)*log(2)^2*log(5)^2+2*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)* log(2)*log(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*log(2)^3*log(5)^3+12* x*log(2)^2*log(5)^2+6*x^2*log(2)*log(5)+x^3),x, algorithm=\
256*x^8 - 32*x^5 + 512*x*log(5)^2*log(2)^2 - (256*log(5)*log(2) - 1)*x^2 + 128*x^3 + 16*(256*log(5)^5*log(2)^5 + 2*log(5)^2*log(2)^2 + (128*log(5)^4 *log(2)^4 + log(5)*log(2))*x + 1)/(4*log(5)^2*log(2)^2 + 4*x*log(5)*log(2) + x^2)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.50 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=256 \, x^{8} - 32 \, x^{5} + 512 \, x \log \left (5\right )^{2} \log \left (2\right )^{2} - 256 \, x^{2} \log \left (5\right ) \log \left (2\right ) + 128 \, x^{3} + x^{2} + \frac {16 \, {\left (256 \, \log \left (5\right )^{5} \log \left (2\right )^{5} + 128 \, x \log \left (5\right )^{4} \log \left (2\right )^{4} + 2 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + x \log \left (5\right ) \log \left (2\right ) + 1\right )}}{{\left (2 \, \log \left (5\right ) \log \left (2\right ) + x\right )}^{2}} \]
integrate((8*(2048*x^7-160*x^4+2*x)*log(2)^3*log(5)^3+4*(6144*x^8-480*x^5+ 512*x^3+6*x^2-8)*log(2)^2*log(5)^2+2*(6144*x^9-480*x^6+896*x^4+6*x^3-8*x)* log(2)*log(5)+2048*x^10-160*x^7+384*x^5+2*x^4-32)/(8*log(2)^3*log(5)^3+12* x*log(2)^2*log(5)^2+6*x^2*log(2)*log(5)+x^3),x, algorithm=\
256*x^8 - 32*x^5 + 512*x*log(5)^2*log(2)^2 - 256*x^2*log(5)*log(2) + 128*x ^3 + x^2 + 16*(256*log(5)^5*log(2)^5 + 128*x*log(5)^4*log(2)^4 + 2*log(5)^ 2*log(2)^2 + x*log(5)*log(2) + 1)/(2*log(5)*log(2) + x)^2
Time = 9.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.75 \[ \int \frac {-32+2 x^4+384 x^5-160 x^7+2048 x^{10}+\left (-8 x+6 x^3+896 x^4-480 x^6+6144 x^9\right ) \log (4) \log (5)+\left (-8+6 x^2+512 x^3-480 x^5+6144 x^8\right ) \log ^2(4) \log ^2(5)+\left (2 x-160 x^4+2048 x^7\right ) \log ^3(4) \log ^3(5)}{x^3+3 x^2 \log (4) \log (5)+3 x \log ^2(4) \log ^2(5)+\log ^3(4) \log ^3(5)} \, dx=\frac {x\,\left (16\,\ln \left (2\right )\,\ln \left (5\right )+2048\,{\ln \left (2\right )}^4\,{\ln \left (5\right )}^4\right )+32\,{\ln \left (2\right )}^2\,{\ln \left (5\right )}^2+4096\,{\ln \left (2\right )}^5\,{\ln \left (5\right )}^5+16}{x^2+4\,\ln \left (2\right )\,\ln \left (5\right )\,x+4\,{\ln \left (2\right )}^2\,{\ln \left (5\right )}^2}+x\,\left (6\,\ln \left (2\right )\,\ln \left (5\right )\,\left (512\,\ln \left (2\right )\,\ln \left (5\right )-2\right )-4608\,{\ln \left (2\right )}^2\,{\ln \left (5\right )}^2+4\,\ln \left (2\right )\,\ln \left (5\right )\,\left (512\,\ln \left (2\right )\,\ln \left (5\right )+3\right )\right )-x^2\,\left (256\,\ln \left (2\right )\,\ln \left (5\right )-1\right )+128\,x^3-32\,x^5+256\,x^8 \]
int((2*x^4 + 384*x^5 - 160*x^7 + 2048*x^10 + 4*log(2)^2*log(5)^2*(6*x^2 + 512*x^3 - 480*x^5 + 6144*x^8 - 8) + 2*log(2)*log(5)*(6*x^3 - 8*x + 896*x^4 - 480*x^6 + 6144*x^9) + 8*log(2)^3*log(5)^3*(2*x - 160*x^4 + 2048*x^7) - 32)/(x^3 + 8*log(2)^3*log(5)^3 + 6*x^2*log(2)*log(5) + 12*x*log(2)^2*log(5 )^2),x)
(x*(16*log(2)*log(5) + 2048*log(2)^4*log(5)^4) + 32*log(2)^2*log(5)^2 + 40 96*log(2)^5*log(5)^5 + 16)/(x^2 + 4*log(2)^2*log(5)^2 + 4*x*log(2)*log(5)) + x*(6*log(2)*log(5)*(512*log(2)*log(5) - 2) - 4608*log(2)^2*log(5)^2 + 4 *log(2)*log(5)*(512*log(2)*log(5) + 3)) - x^2*(256*log(2)*log(5) - 1) + 12 8*x^3 - 32*x^5 + 256*x^8