Integrand size = 104, antiderivative size = 26 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\frac {4 \left (-11+x^2+\frac {x-\log (x)}{5+x+\log (4)}\right )}{3 x} \] Output:
4/3*(x^2-11+(x-ln(x))/(2*ln(2)+5+x))/x
Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\frac {4}{3} \left (x-\frac {55}{x (5+\log (4))}-\frac {22 \log (2)}{x (5+\log (4))}-\frac {\log (x)}{x (5+\log (4))}+\frac {5+\log (4 x)}{(5+\log (4)) (5+x+\log (4))}\right ) \] Input:
Integrate[(1080 + 436*x + 140*x^2 + 40*x^3 + 4*x^4 + (436 + 88*x + 40*x^2 + 8*x^3)*Log[4] + (44 + 4*x^2)*Log[4]^2 + (20 + 8*x + 4*Log[4])*Log[x])/(7 5*x^2 + 30*x^3 + 3*x^4 + (30*x^2 + 6*x^3)*Log[4] + 3*x^2*Log[4]^2),x]
Output:
(4*(x - 55/(x*(5 + Log[4])) - (22*Log[2])/(x*(5 + Log[4])) - Log[x]/(x*(5 + Log[4])) + (5 + Log[4*x])/((5 + Log[4])*(5 + x + Log[4]))))/3
Leaf count is larger than twice the leaf count of optimal. \(404\) vs. \(2(26)=52\).
Time = 1.30 (sec) , antiderivative size = 404, normalized size of antiderivative = 15.54, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 2026, 2007, 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.
| \(\displaystyle \int \frac {4 x^4+40 x^3+140 x^2+\left (4 x^2+44\right ) \log ^2(4)+\left (8 x^3+40 x^2+88 x+436\right ) \log (4)+436 x+(8 x+20+4 \log (4)) \log (x)+1080}{3 x^4+30 x^3+75 x^2+3 x^2 \log ^2(4)+\left (6 x^3+30 x^2\right ) \log (4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 x^4+40 x^3+140 x^2+\left (4 x^2+44\right ) \log ^2(4)+\left (8 x^3+40 x^2+88 x+436\right ) \log (4)+436 x+(8 x+20+4 \log (4)) \log (x)+1080}{3 x^4+30 x^3+x^2 \left (75+3 \log ^2(4)\right )+\left (6 x^3+30 x^2\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {4 x^4+40 x^3+140 x^2+\left (4 x^2+44\right ) \log ^2(4)+\left (8 x^3+40 x^2+88 x+436\right ) \log (4)+436 x+(8 x+20+4 \log (4)) \log (x)+1080}{x^2 \left (3 x^2+6 x (5+\log (4))+3 (5+\log (4))^2\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {4 x^4+40 x^3+140 x^2+\left (4 x^2+44\right ) \log ^2(4)+\left (8 x^3+40 x^2+88 x+436\right ) \log (4)+436 x+(8 x+20+4 \log (4)) \log (x)+1080}{x^2 \left (\sqrt {3} x+\sqrt {3} (5+\log (4))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (x^2+11\right ) \log ^2(4)}{3 x^2 (x+5+\log (4))^2}+\frac {4 x^2}{3 (x+5+\log (4))^2}+\frac {4 (2 x+5+\log (4)) \log (x)}{3 x^2 (x+5+\log (4))^2}+\frac {360}{x^2 (x+5+\log (4))^2}+\frac {4 \left (2 x^3+10 x^2+22 x+109\right ) \log (4)}{3 x^2 (x+5+\log (4))^2}+\frac {40 x}{3 (x+5+\log (4))^2}+\frac {140}{3 (x+5+\log (4))^2}+\frac {436}{3 x (x+5+\log (4))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 x}{3}-\frac {88 \log ^2(4) \log (x)}{3 (5+\log (4))^3}+\frac {88 \log ^2(4) \log (x+5+\log (4))}{3 (5+\log (4))^3}-\frac {4 \log ^2(4) \left (36+\log ^2(4)+10 \log (4)\right )}{3 (5+\log (4))^2 (x+5+\log (4))}-\frac {44 \log ^2(4)}{3 x (5+\log (4))^2}+\frac {8 \log (4) \left (179+\log ^3(4)+15 \log ^2(4)+64 \log (4)\right ) \log (x+5+\log (4))}{3 (5+\log (4))^3}+\frac {4 \log (4) \left (1+2 \log ^3(4)+20 \log ^2(4)+72 \log (4)\right )}{3 (5+\log (4))^2 (x+5+\log (4))}-\frac {4 x \log (x)}{3 (5+\log (4))^2 (x+5+\log (4))}+\frac {436 \log (x)}{3 (5+\log (4))^2}-\frac {8 (54-11 \log (4)) \log (4) \log (x)}{3 (5+\log (4))^3}-\frac {720 \log (x)}{(5+\log (4))^3}-\frac {8}{3} (5+\log (4)) \log (x+5+\log (4))-\frac {144 \log (x+5+\log (4))}{(5+\log (4))^2}+\frac {720 \log (x+5+\log (4))}{(5+\log (4))^3}+\frac {40}{3} \log (x+5+\log (4))-\frac {4 (5+\log (4))^2}{3 (x+5+\log (4))}+\frac {40 (5+\log (4))}{3 (x+5+\log (4))}+\frac {436}{3 (5+\log (4)) (x+5+\log (4))}-\frac {360}{(5+\log (4))^2 (x+5+\log (4))}-\frac {140}{3 (x+5+\log (4))}-\frac {4 \log (x)}{3 x (5+\log (4))}-\frac {4}{3 x (5+\log (4))}-\frac {436 \log (4)}{3 x (5+\log (4))^2}-\frac {360}{x (5+\log (4))^2}\) |
Input:
Int[(1080 + 436*x + 140*x^2 + 40*x^3 + 4*x^4 + (436 + 88*x + 40*x^2 + 8*x^ 3)*Log[4] + (44 + 4*x^2)*Log[4]^2 + (20 + 8*x + 4*Log[4])*Log[x])/(75*x^2 + 30*x^3 + 3*x^4 + (30*x^2 + 6*x^3)*Log[4] + 3*x^2*Log[4]^2),x]
Output:
(4*x)/3 - 360/(x*(5 + Log[4])^2) - (436*Log[4])/(3*x*(5 + Log[4])^2) - (44 *Log[4]^2)/(3*x*(5 + Log[4])^2) - 4/(3*x*(5 + Log[4])) - 140/(3*(5 + x + L og[4])) - 360/((5 + Log[4])^2*(5 + x + Log[4])) + 436/(3*(5 + Log[4])*(5 + x + Log[4])) + (40*(5 + Log[4]))/(3*(5 + x + Log[4])) - (4*(5 + Log[4])^2 )/(3*(5 + x + Log[4])) - (4*Log[4]^2*(36 + 10*Log[4] + Log[4]^2))/(3*(5 + Log[4])^2*(5 + x + Log[4])) + (4*Log[4]*(1 + 72*Log[4] + 20*Log[4]^2 + 2*L og[4]^3))/(3*(5 + Log[4])^2*(5 + x + Log[4])) - (720*Log[x])/(5 + Log[4])^ 3 - (8*(54 - 11*Log[4])*Log[4]*Log[x])/(3*(5 + Log[4])^3) - (88*Log[4]^2*L og[x])/(3*(5 + Log[4])^3) + (436*Log[x])/(3*(5 + Log[4])^2) - (4*Log[x])/( 3*x*(5 + Log[4])) - (4*x*Log[x])/(3*(5 + Log[4])^2*(5 + x + Log[4])) + (40 *Log[5 + x + Log[4]])/3 + (720*Log[5 + x + Log[4]])/(5 + Log[4])^3 + (88*L og[4]^2*Log[5 + x + Log[4]])/(3*(5 + Log[4])^3) - (144*Log[5 + x + Log[4]] )/(5 + Log[4])^2 - (8*(5 + Log[4])*Log[5 + x + Log[4]])/3 + (8*Log[4]*(179 + 64*Log[4] + 15*Log[4]^2 + Log[4]^3)*Log[5 + x + Log[4]])/(3*(5 + Log[4] )^3)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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[(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])
Time = 1.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65
| method | result | size |
| norman | \(\frac {\left (-\frac {16 \ln \left (2\right )^{2}}{3}-\frac {80 \ln \left (2\right )}{3}-\frac {140}{3}\right ) x +\frac {4 x^{3}}{3}-\frac {220}{3}-\frac {4 \ln \left (x \right )}{3}-\frac {88 \ln \left (2\right )}{3}}{x \left (2 \ln \left (2\right )+5+x \right )}\) | \(43\) |
| parallelrisch | \(\frac {-16 x \ln \left (2\right )^{2}-220+4 x^{3}-80 x \ln \left (2\right )-88 \ln \left (2\right )-140 x -4 \ln \left (x \right )}{3 x \left (2 \ln \left (2\right )+5+x \right )}\) | \(45\) |
| risch | \(-\frac {4 \ln \left (x \right )}{3 \left (2 \ln \left (2\right )+5+x \right ) x}+\frac {\frac {8 x^{2} \ln \left (2\right )}{3}+\frac {4 x^{3}}{3}+\frac {20 x^{2}}{3}-\frac {88 \ln \left (2\right )}{3}-\frac {40 x}{3}-\frac {220}{3}}{x \left (2 \ln \left (2\right )+5+x \right )}\) | \(56\) |
| default | \(\frac {4 x}{3}+\frac {4}{3 \left (2 \ln \left (2\right )+5+x \right )}+\frac {4 \ln \left (x \right )}{3 \left (2 \ln \left (2\right )+5\right )^{2}}-\frac {4 \left (22 \ln \left (2\right )+54\right )}{3 \left (2 \ln \left (2\right )+5\right ) x}-\frac {4 \ln \left (x \right ) x}{3 \left (2 \ln \left (2\right )+5\right )^{2} \left (2 \ln \left (2\right )+5+x \right )}-\frac {4 \ln \left (x \right )}{3 \left (2 \ln \left (2\right )+5\right ) x}-\frac {4}{3 \left (2 \ln \left (2\right )+5\right ) x}\) | \(97\) |
| parts | \(\frac {4 x}{3}+\frac {4}{3 \left (2 \ln \left (2\right )+5+x \right )}+\frac {4 \ln \left (x \right )}{3 \left (2 \ln \left (2\right )+5\right )^{2}}-\frac {4 \left (22 \ln \left (2\right )+54\right )}{3 \left (2 \ln \left (2\right )+5\right ) x}-\frac {4 \ln \left (x \right ) x}{3 \left (2 \ln \left (2\right )+5\right )^{2} \left (2 \ln \left (2\right )+5+x \right )}-\frac {4 \ln \left (x \right )}{3 \left (2 \ln \left (2\right )+5\right ) x}-\frac {4}{3 \left (2 \ln \left (2\right )+5\right ) x}\) | \(97\) |
Input:
int(((8*ln(2)+8*x+20)*ln(x)+4*(4*x^2+44)*ln(2)^2+2*(8*x^3+40*x^2+88*x+436) *ln(2)+4*x^4+40*x^3+140*x^2+436*x+1080)/(12*x^2*ln(2)^2+2*(6*x^3+30*x^2)*l n(2)+3*x^4+30*x^3+75*x^2),x,method=_RETURNVERBOSE)
Output:
((-16/3*ln(2)^2-80/3*ln(2)-140/3)*x+4/3*x^3-220/3-4/3*ln(x)-88/3*ln(2))/x/ (2*ln(2)+5+x)
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\frac {4 \, {\left (x^{3} + 5 \, x^{2} + 2 \, {\left (x^{2} - 11\right )} \log \left (2\right ) - 10 \, x - \log \left (x\right ) - 55\right )}}{3 \, {\left (x^{2} + 2 \, x \log \left (2\right ) + 5 \, x\right )}} \] Input:
integrate(((8*log(2)+8*x+20)*log(x)+4*(4*x^2+44)*log(2)^2+2*(8*x^3+40*x^2+ 88*x+436)*log(2)+4*x^4+40*x^3+140*x^2+436*x+1080)/(12*x^2*log(2)^2+2*(6*x^ 3+30*x^2)*log(2)+3*x^4+30*x^3+75*x^2),x, algorithm="fricas")
Output:
4/3*(x^3 + 5*x^2 + 2*(x^2 - 11)*log(2) - 10*x - log(x) - 55)/(x^2 + 2*x*lo g(2) + 5*x)
Time = 0.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\frac {4 x}{3} - \frac {4 \log {\left (x \right )}}{3 x^{2} + 6 x \log {\left (2 \right )} + 15 x} + \frac {- 40 x - 220 - 88 \log {\left (2 \right )}}{3 x^{2} + x \left (6 \log {\left (2 \right )} + 15\right )} \] Input:
integrate(((8*ln(2)+8*x+20)*ln(x)+4*(4*x**2+44)*ln(2)**2+2*(8*x**3+40*x**2 +88*x+436)*ln(2)+4*x**4+40*x**3+140*x**2+436*x+1080)/(12*x**2*ln(2)**2+2*( 6*x**3+30*x**2)*ln(2)+3*x**4+30*x**3+75*x**2),x)
Output:
4*x/3 - 4*log(x)/(3*x**2 + 6*x*log(2) + 15*x) + (-40*x - 220 - 88*log(2))/ (3*x**2 + x*(6*log(2) + 15))
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (26) = 52\).
Time = 0.17 (sec) , antiderivative size = 683, normalized size of antiderivative = 26.27 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\text {Too large to display} \] Input:
integrate(((8*log(2)+8*x+20)*log(x)+4*(4*x^2+44)*log(2)^2+2*(8*x^3+40*x^2+ 88*x+436)*log(2)+4*x^4+40*x^3+140*x^2+436*x+1080)/(12*x^2*log(2)^2+2*(6*x^ 3+30*x^2)*log(2)+3*x^4+30*x^3+75*x^2),x, algorithm="maxima")
Output:
-176/3*((2*x + 2*log(2) + 5)/((4*log(2)^2 + 20*log(2) + 25)*x^2 + (8*log(2 )^3 + 60*log(2)^2 + 150*log(2) + 125)*x) - 2*log(x + 2*log(2) + 5)/(8*log( 2)^3 + 60*log(2)^2 + 150*log(2) + 125) + 2*log(x)/(8*log(2)^3 + 60*log(2)^ 2 + 150*log(2) + 125))*log(2)^2 - 872/3*((2*x + 2*log(2) + 5)/((4*log(2)^2 + 20*log(2) + 25)*x^2 + (8*log(2)^3 + 60*log(2)^2 + 150*log(2) + 125)*x) - 2*log(x + 2*log(2) + 5)/(8*log(2)^3 + 60*log(2)^2 + 150*log(2) + 125) + 2*log(x)/(8*log(2)^3 + 60*log(2)^2 + 150*log(2) + 125))*log(2) + 16/3*((2* log(2) + 5)/(x + 2*log(2) + 5) + log(x + 2*log(2) + 5))*log(2) - 176/3*(lo g(x + 2*log(2) + 5)/(4*log(2)^2 + 20*log(2) + 25) - log(x)/(4*log(2)^2 + 2 0*log(2) + 25) - 1/(x*(2*log(2) + 5) + 4*log(2)^2 + 20*log(2) + 25))*log(2 ) - 8/3*(2*log(2) + 5)*log(x + 2*log(2) + 5) + 4/3*x - 16/3*log(2)^2/(x + 2*log(2) + 5) - 4/3*(x*(2*log(2) + 5) + 4*log(2)^2 + (x^2 + x*(2*log(2) + 5) + 4*log(2)^2 + 20*log(2) + 25)*log(x) + 20*log(2) + 25)/((4*log(2)^2 + 20*log(2) + 25)*x^2 + (8*log(2)^3 + 60*log(2)^2 + 150*log(2) + 125)*x) - 3 60*(2*x + 2*log(2) + 5)/((4*log(2)^2 + 20*log(2) + 25)*x^2 + (8*log(2)^3 + 60*log(2)^2 + 150*log(2) + 125)*x) - 4/3*(4*log(2)^2 + 20*log(2) + 25)/(x + 2*log(2) + 5) + 40/3*(2*log(2) + 5)/(x + 2*log(2) + 5) - 80/3*log(2)/(x + 2*log(2) + 5) + 720*log(x + 2*log(2) + 5)/(8*log(2)^3 + 60*log(2)^2 + 1 50*log(2) + 125) - 144*log(x + 2*log(2) + 5)/(4*log(2)^2 + 20*log(2) + 25) - 720*log(x)/(8*log(2)^3 + 60*log(2)^2 + 150*log(2) + 125) + 436/3*log...
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\frac {4}{3} \, {\left (\frac {1}{2 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 5 \, x + 20 \, \log \left (2\right ) + 25} - \frac {1}{2 \, x \log \left (2\right ) + 5 \, x}\right )} \log \left (x\right ) + \frac {4}{3} \, x + \frac {4}{3 \, {\left (x + 2 \, \log \left (2\right ) + 5\right )}} - \frac {44}{3 \, x} \] Input:
integrate(((8*log(2)+8*x+20)*log(x)+4*(4*x^2+44)*log(2)^2+2*(8*x^3+40*x^2+ 88*x+436)*log(2)+4*x^4+40*x^3+140*x^2+436*x+1080)/(12*x^2*log(2)^2+2*(6*x^ 3+30*x^2)*log(2)+3*x^4+30*x^3+75*x^2),x, algorithm="giac")
Output:
4/3*(1/(2*x*log(2) + 4*log(2)^2 + 5*x + 20*log(2) + 25) - 1/(2*x*log(2) + 5*x))*log(x) + 4/3*x + 4/3/(x + 2*log(2) + 5) - 44/3/x
Timed out. \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\int \frac {436\,x+\ln \left (x\right )\,\left (8\,x+8\,\ln \left (2\right )+20\right )+2\,\ln \left (2\right )\,\left (8\,x^3+40\,x^2+88\,x+436\right )+4\,{\ln \left (2\right )}^2\,\left (4\,x^2+44\right )+140\,x^2+40\,x^3+4\,x^4+1080}{12\,x^2\,{\ln \left (2\right )}^2+2\,\ln \left (2\right )\,\left (6\,x^3+30\,x^2\right )+75\,x^2+30\,x^3+3\,x^4} \,d x \] Input:
int((436*x + log(x)*(8*x + 8*log(2) + 20) + 2*log(2)*(88*x + 40*x^2 + 8*x^ 3 + 436) + 4*log(2)^2*(4*x^2 + 44) + 140*x^2 + 40*x^3 + 4*x^4 + 1080)/(12* x^2*log(2)^2 + 2*log(2)*(30*x^2 + 6*x^3) + 75*x^2 + 30*x^3 + 3*x^4),x)
Output:
int((436*x + log(x)*(8*x + 8*log(2) + 20) + 2*log(2)*(88*x + 40*x^2 + 8*x^ 3 + 436) + 4*log(2)^2*(4*x^2 + 44) + 140*x^2 + 40*x^3 + 4*x^4 + 1080)/(12* x^2*log(2)^2 + 2*log(2)*(30*x^2 + 6*x^3) + 75*x^2 + 30*x^3 + 3*x^4), x)
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {1080+436 x+140 x^2+40 x^3+4 x^4+\left (436+88 x+40 x^2+8 x^3\right ) \log (4)+\left (44+4 x^2\right ) \log ^2(4)+(20+8 x+4 \log (4)) \log (x)}{75 x^2+30 x^3+3 x^4+\left (30 x^2+6 x^3\right ) \log (4)+3 x^2 \log ^2(4)} \, dx=\frac {-\frac {8 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )}{3}-\frac {20 \,\mathrm {log}\left (x \right )}{3}+\frac {16 \mathrm {log}\left (2\right )^{2} x^{2}}{3}-\frac {176 \mathrm {log}\left (2\right )^{2}}{3}+\frac {8 \,\mathrm {log}\left (2\right ) x^{3}}{3}+\frac {80 \,\mathrm {log}\left (2\right ) x^{2}}{3}-\frac {880 \,\mathrm {log}\left (2\right )}{3}+\frac {20 x^{3}}{3}+\frac {140 x^{2}}{3}-\frac {1100}{3}}{x \left (4 \mathrm {log}\left (2\right )^{2}+2 \,\mathrm {log}\left (2\right ) x +20 \,\mathrm {log}\left (2\right )+5 x +25\right )} \] Input:
int(((8*log(2)+8*x+20)*log(x)+4*(4*x^2+44)*log(2)^2+2*(8*x^3+40*x^2+88*x+4 36)*log(2)+4*x^4+40*x^3+140*x^2+436*x+1080)/(12*x^2*log(2)^2+2*(6*x^3+30*x ^2)*log(2)+3*x^4+30*x^3+75*x^2),x)
Output:
(4*( - 2*log(x)*log(2) - 5*log(x) + 4*log(2)**2*x**2 - 44*log(2)**2 + 2*lo g(2)*x**3 + 20*log(2)*x**2 - 220*log(2) + 5*x**3 + 35*x**2 - 275))/(3*x*(4 *log(2)**2 + 2*log(2)*x + 20*log(2) + 5*x + 25))