Integrand size = 148, antiderivative size = 24 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\frac {1}{2} x \log \left (1+x \log (3)-\frac {1}{x+\frac {\log (625)}{5}}\right ) \] Output:
1/2*ln(x*ln(3)+1-1/(4/5*ln(5)+x))*x
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\frac {1}{2} x \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right ) \] Input:
Integrate[(25*x + 25*x^3*Log[3] + 10*x^2*Log[3]*Log[625] + x*Log[3]*Log[62 5]^2 + (-25*x + 25*x^2 + 25*x^3*Log[3] + (-5 + 10*x + 10*x^2*Log[3])*Log[6 25] + (1 + x*Log[3])*Log[625]^2)*Log[(-5 + 5*x + 5*x^2*Log[3] + (1 + x*Log [3])*Log[625])/(5*x + Log[625])])/(-50*x + 50*x^2 + 50*x^3*Log[3] + (-10 + 20*x + 20*x^2*Log[3])*Log[625] + (2 + 2*x*Log[3])*Log[625]^2),x]
Output:
(x*Log[(-5 + 5*x + 5*x^2*Log[3] + (1 + x*Log[3])*Log[625])/(5*x + Log[625] )])/2
Leaf count is larger than twice the leaf count of optimal. \(506\) vs. \(2(24)=48\).
Time = 3.44 (sec) , antiderivative size = 506, normalized size of antiderivative = 21.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6, 2463, 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.
| \(\displaystyle \int \frac {25 x^3 \log (3)+10 x^2 \log (3) \log (625)+\left (25 x^3 \log (3)+25 x^2+\log (625) \left (10 x^2 \log (3)+10 x-5\right )-25 x+\log ^2(625) (x \log (3)+1)\right ) \log \left (\frac {5 x^2 \log (3)+5 x+\log (625) (x \log (3)+1)-5}{5 x+\log (625)}\right )+25 x+x \log (3) \log ^2(625)}{50 x^3 \log (3)+50 x^2+\log (625) \left (20 x^2 \log (3)+20 x-10\right )-50 x+\log ^2(625) (2 x \log (3)+2)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {25 x^3 \log (3)+10 x^2 \log (3) \log (625)+\left (25 x^3 \log (3)+25 x^2+\log (625) \left (10 x^2 \log (3)+10 x-5\right )-25 x+\log ^2(625) (x \log (3)+1)\right ) \log \left (\frac {5 x^2 \log (3)+5 x+\log (625) (x \log (3)+1)-5}{5 x+\log (625)}\right )+x \left (25+\log (3) \log ^2(625)\right )}{50 x^3 \log (3)+50 x^2+\log (625) \left (20 x^2 \log (3)+20 x-10\right )-50 x+\log ^2(625) (2 x \log (3)+2)}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {(x (-\log (3))-1) \left (25 x^3 \log (3)+10 x^2 \log (3) \log (625)+\left (25 x^3 \log (3)+25 x^2+\log (625) \left (10 x^2 \log (3)+10 x-5\right )-25 x+\log ^2(625) (x \log (3)+1)\right ) \log \left (\frac {5 x^2 \log (3)+5 x+\log (625) (x \log (3)+1)-5}{5 x+\log (625)}\right )+x \left (25+\log (3) \log ^2(625)\right )\right )}{10 \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}-\frac {25 x^3 \log (3)+10 x^2 \log (3) \log (625)+\left (25 x^3 \log (3)+25 x^2+\log (625) \left (10 x^2 \log (3)+10 x-5\right )-25 x+\log ^2(625) (x \log (3)+1)\right ) \log \left (\frac {5 x^2 \log (3)+5 x+\log (625) (x \log (3)+1)-5}{5 x+\log (625)}\right )+x \left (25+\log (3) \log ^2(625)\right )}{10 (5 x+\log (625))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (625) \sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))} \text {arctanh}\left (\frac {10 x \log (3)+5+\log (3) \log (625)}{\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))}}\right )}{50 \log (3)}+\frac {(5-4 \log (5)) \sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))} \text {arctanh}\left (\frac {10 x \log (3)+5+\log (3) \log (625)}{\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))}}\right )}{50 \log (3)}-\frac {\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))} \text {arctanh}\left (\frac {10 x \log (3)+5+\log (3) \log (625)}{\sqrt {25+\log ^2(3) \log ^2(625)+10 \log (3) (10-\log (625))}}\right )}{10 \log (3)}+\frac {1}{10} x \log (625) \log \left (-\frac {-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)}{5 x+\log (625)}\right )+\frac {1}{10} x (5-\log (625)) \log \left (-\frac {-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)}{5 x+\log (625)}\right )+\frac {\log (625) (5+\log (3) \log (625)) \log \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}{100 \log (3)}+\frac {(5-4 \log (5)) (5+\log (3) \log (625)) \log \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}{100 \log (3)}-\frac {(5+\log (3) \log (625)) \log \left (-5 x^2 \log (3)-x (5+\log (3) \log (625))+5-\log (625)\right )}{20 \log (3)}+\frac {x}{2}-\frac {1}{50} \log ^2(625) \log (5 x+\log (625))-\frac {1}{10} x \log (625)-\frac {1}{10} x (5-\log (625))-\frac {1}{50} (5-4 \log (5)) \log (625) \log (5 x+\log (625))+\frac {1}{10} \log (625) \log (5 x+\log (625))\) |
Input:
Int[(25*x + 25*x^3*Log[3] + 10*x^2*Log[3]*Log[625] + x*Log[3]*Log[625]^2 + (-25*x + 25*x^2 + 25*x^3*Log[3] + (-5 + 10*x + 10*x^2*Log[3])*Log[625] + (1 + x*Log[3])*Log[625]^2)*Log[(-5 + 5*x + 5*x^2*Log[3] + (1 + x*Log[3])*L og[625])/(5*x + Log[625])])/(-50*x + 50*x^2 + 50*x^3*Log[3] + (-10 + 20*x + 20*x^2*Log[3])*Log[625] + (2 + 2*x*Log[3])*Log[625]^2),x]
Output:
x/2 - (x*(5 - Log[625]))/10 - (x*Log[625])/10 - (ArcTanh[(5 + 10*x*Log[3] + Log[3]*Log[625])/Sqrt[25 + 10*Log[3]*(10 - Log[625]) + Log[3]^2*Log[625] ^2]]*Sqrt[25 + 10*Log[3]*(10 - Log[625]) + Log[3]^2*Log[625]^2])/(10*Log[3 ]) + (ArcTanh[(5 + 10*x*Log[3] + Log[3]*Log[625])/Sqrt[25 + 10*Log[3]*(10 - Log[625]) + Log[3]^2*Log[625]^2]]*(5 - 4*Log[5])*Sqrt[25 + 10*Log[3]*(10 - Log[625]) + Log[3]^2*Log[625]^2])/(50*Log[3]) + (ArcTanh[(5 + 10*x*Log[ 3] + Log[3]*Log[625])/Sqrt[25 + 10*Log[3]*(10 - Log[625]) + Log[3]^2*Log[6 25]^2]]*Log[625]*Sqrt[25 + 10*Log[3]*(10 - Log[625]) + Log[3]^2*Log[625]^2 ])/(50*Log[3]) + (Log[625]*Log[5*x + Log[625]])/10 - ((5 - 4*Log[5])*Log[6 25]*Log[5*x + Log[625]])/50 - (Log[625]^2*Log[5*x + Log[625]])/50 - ((5 + Log[3]*Log[625])*Log[5 - 5*x^2*Log[3] - Log[625] - x*(5 + Log[3]*Log[625]) ])/(20*Log[3]) + ((5 - 4*Log[5])*(5 + Log[3]*Log[625])*Log[5 - 5*x^2*Log[3 ] - Log[625] - x*(5 + Log[3]*Log[625])])/(100*Log[3]) + (Log[625]*(5 + Log [3]*Log[625])*Log[5 - 5*x^2*Log[3] - Log[625] - x*(5 + Log[3]*Log[625])])/ (100*Log[3]) + (x*(5 - Log[625])*Log[-((5 - 5*x^2*Log[3] - Log[625] - x*(5 + Log[3]*Log[625]))/(5*x + Log[625]))])/10 + (x*Log[625]*Log[-((5 - 5*x^2 *Log[3] - Log[625] - x*(5 + Log[3]*Log[625]))/(5*x + Log[625]))])/10
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[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 5.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58
| method | result | size |
| norman | \(\frac {x \ln \left (\frac {4 \left (x \ln \left (3\right )+1\right ) \ln \left (5\right )+5 x^{2} \ln \left (3\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right )}{2}\) | \(38\) |
| risch | \(\frac {x \ln \left (\frac {4 \left (x \ln \left (3\right )+1\right ) \ln \left (5\right )+5 x^{2} \ln \left (3\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right )}{2}\) | \(38\) |
| default | \(\frac {\ln \left (\frac {4 \ln \left (5\right ) \ln \left (3\right ) x +5 x^{2} \ln \left (3\right )+4 \ln \left (5\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right ) x}{2}\) | \(39\) |
| parts | \(\frac {\ln \left (\frac {4 \ln \left (5\right ) \ln \left (3\right ) x +5 x^{2} \ln \left (3\right )+4 \ln \left (5\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right ) x}{2}\) | \(39\) |
| parallelrisch | \(\frac {160000 \ln \left (3\right )^{2} \ln \left (5\right )^{4} x \ln \left (\frac {4 \ln \left (5\right ) \ln \left (3\right ) x +5 x^{2} \ln \left (3\right )+4 \ln \left (5\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right )-400000 \ln \left (3\right )^{2} \ln \left (5\right )^{3} x \ln \left (\frac {4 \ln \left (5\right ) \ln \left (3\right ) x +5 x^{2} \ln \left (3\right )+4 \ln \left (5\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right )+250000 \ln \left (3\right )^{2} \ln \left (5\right )^{2} x \ln \left (\frac {4 \ln \left (5\right ) \ln \left (3\right ) x +5 x^{2} \ln \left (3\right )+4 \ln \left (5\right )+5 x -5}{4 \ln \left (5\right )+5 x}\right )}{20000 \ln \left (3\right )^{2} \left (4 \ln \left (5\right )-5\right )^{2} \ln \left (5\right )^{2}}\) | \(158\) |
Input:
int(((16*(x*ln(3)+1)*ln(5)^2+4*(10*x^2*ln(3)+10*x-5)*ln(5)+25*x^3*ln(3)+25 *x^2-25*x)*ln((4*(x*ln(3)+1)*ln(5)+5*x^2*ln(3)+5*x-5)/(4*ln(5)+5*x))+16*x* ln(3)*ln(5)^2+40*x^2*ln(3)*ln(5)+25*x^3*ln(3)+25*x)/(16*(2*x*ln(3)+2)*ln(5 )^2+4*(20*x^2*ln(3)+20*x-10)*ln(5)+50*x^3*ln(3)+50*x^2-50*x),x,method=_RET URNVERBOSE)
Output:
1/2*x*ln((4*(x*ln(3)+1)*ln(5)+5*x^2*ln(3)+5*x-5)/(4*ln(5)+5*x))
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\frac {1}{2} \, x \log \left (\frac {5 \, x^{2} \log \left (3\right ) + 4 \, {\left (x \log \left (3\right ) + 1\right )} \log \left (5\right ) + 5 \, x - 5}{5 \, x + 4 \, \log \left (5\right )}\right ) \] Input:
integrate(((16*(x*log(3)+1)*log(5)^2+4*(10*x^2*log(3)+10*x-5)*log(5)+25*x^ 3*log(3)+25*x^2-25*x)*log((4*(x*log(3)+1)*log(5)+5*x^2*log(3)+5*x-5)/(4*lo g(5)+5*x))+16*x*log(3)*log(5)^2+40*x^2*log(3)*log(5)+25*x^3*log(3)+25*x)/( 16*(2*x*log(3)+2)*log(5)^2+4*(20*x^2*log(3)+20*x-10)*log(5)+50*x^3*log(3)+ 50*x^2-50*x),x, algorithm="fricas")
Output:
1/2*x*log((5*x^2*log(3) + 4*(x*log(3) + 1)*log(5) + 5*x - 5)/(5*x + 4*log( 5)))
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\frac {x \log {\left (\frac {5 x^{2} \log {\left (3 \right )} + 5 x + \left (4 x \log {\left (3 \right )} + 4\right ) \log {\left (5 \right )} - 5}{5 x + 4 \log {\left (5 \right )}} \right )}}{2} \] Input:
integrate(((16*(x*ln(3)+1)*ln(5)**2+4*(10*x**2*ln(3)+10*x-5)*ln(5)+25*x**3 *ln(3)+25*x**2-25*x)*ln((4*(x*ln(3)+1)*ln(5)+5*x**2*ln(3)+5*x-5)/(4*ln(5)+ 5*x))+16*x*ln(3)*ln(5)**2+40*x**2*ln(3)*ln(5)+25*x**3*ln(3)+25*x)/(16*(2*x *ln(3)+2)*ln(5)**2+4*(20*x**2*ln(3)+20*x-10)*ln(5)+50*x**3*ln(3)+50*x**2-5 0*x),x)
Output:
x*log((5*x**2*log(3) + 5*x + (4*x*log(3) + 4)*log(5) - 5)/(5*x + 4*log(5)) )/2
Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (22) = 44\).
Time = 0.24 (sec) , antiderivative size = 961, normalized size of antiderivative = 40.04 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\text {Too large to display} \] Input:
integrate(((16*(x*log(3)+1)*log(5)^2+4*(10*x^2*log(3)+10*x-5)*log(5)+25*x^ 3*log(3)+25*x^2-25*x)*log((4*(x*log(3)+1)*log(5)+5*x^2*log(3)+5*x-5)/(4*lo g(5)+5*x))+16*x*log(3)*log(5)^2+40*x^2*log(3)*log(5)+25*x^3*log(3)+25*x)/( 16*(2*x*log(3)+2)*log(5)^2+4*(20*x^2*log(3)+20*x-10)*log(5)+50*x^3*log(3)+ 50*x^2-50*x),x, algorithm="maxima")
Output:
-8/125*(2*log(5)*log(5*x^2*log(3) + (4*log(5)*log(3) + 5)*x + 4*log(5) - 5 ) - 4*log(5)*log(5*x + 4*log(5)) - (8*log(5)^2*log(3) - 10*log(5) + 25)*lo g((10*x*log(3) + 4*log(5)*log(3) - sqrt(16*log(5)^2*log(3)^2 - 40*log(5)*l og(3) + 100*log(3) + 25) + 5)/(10*x*log(3) + 4*log(5)*log(3) + sqrt(16*log (5)^2*log(3)^2 - 40*log(5)*log(3) + 100*log(3) + 25) + 5))/sqrt(16*log(5)^ 2*log(3)^2 - 40*log(5)*log(3) + 100*log(3) + 25))*log(5)^2*log(3) - 2/125* (32*log(5)^2*log(5*x + 4*log(5)) - (16*log(5)^2*log(3) + 25)*log(5*x^2*log (3) + (4*log(5)*log(3) + 5)*x + 4*log(5) - 5)/log(3) + (64*log(5)^3*log(3) ^2 - 80*log(5)^2*log(3) + 300*log(5)*log(3) + 125)*log((10*x*log(3) + 4*lo g(5)*log(3) - sqrt(16*log(5)^2*log(3)^2 - 40*log(5)*log(3) + 100*log(3) + 25) + 5)/(10*x*log(3) + 4*log(5)*log(3) + sqrt(16*log(5)^2*log(3)^2 - 40*l og(5)*log(3) + 100*log(3) + 25) + 5))/(sqrt(16*log(5)^2*log(3)^2 - 40*log( 5)*log(3) + 100*log(3) + 25)*log(3)))*log(5)*log(3) + 1/500*(128*log(5)^3* log(5*x + 4*log(5)) + 250*x/log(3) - (64*log(5)^3*log(3)^2 + 200*log(5)*lo g(3) + 125)*log(5*x^2*log(3) + (4*log(5)*log(3) + 5)*x + 4*log(5) - 5)/log (3)^2 + (256*log(5)^4*log(3)^3 - 320*log(5)^3*log(3)^2 + 1600*log(5)^2*log (3)^2 + 500*log(5)*log(3) + 1250*log(3) + 625)*log((10*x*log(3) + 4*log(5) *log(3) - sqrt(16*log(5)^2*log(3)^2 - 40*log(5)*log(3) + 100*log(3) + 25) + 5)/(10*x*log(3) + 4*log(5)*log(3) + sqrt(16*log(5)^2*log(3)^2 - 40*log(5 )*log(3) + 100*log(3) + 25) + 5))/(sqrt(16*log(5)^2*log(3)^2 - 40*log(5...
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\frac {1}{2} \, x \log \left (5 \, x^{2} \log \left (3\right ) + 4 \, x \log \left (5\right ) \log \left (3\right ) + 5 \, x + 4 \, \log \left (5\right ) - 5\right ) - \frac {1}{2} \, x \log \left (5 \, x + 4 \, \log \left (5\right )\right ) \] Input:
integrate(((16*(x*log(3)+1)*log(5)^2+4*(10*x^2*log(3)+10*x-5)*log(5)+25*x^ 3*log(3)+25*x^2-25*x)*log((4*(x*log(3)+1)*log(5)+5*x^2*log(3)+5*x-5)/(4*lo g(5)+5*x))+16*x*log(3)*log(5)^2+40*x^2*log(3)*log(5)+25*x^3*log(3)+25*x)/( 16*(2*x*log(3)+2)*log(5)^2+4*(20*x^2*log(3)+20*x-10)*log(5)+50*x^3*log(3)+ 50*x^2-50*x),x, algorithm="giac")
Output:
1/2*x*log(5*x^2*log(3) + 4*x*log(5)*log(3) + 5*x + 4*log(5) - 5) - 1/2*x*l og(5*x + 4*log(5))
Time = 24.49 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx=\frac {x\,\ln \left (\frac {5\,x+4\,\ln \left (5\right )\,\left (x\,\ln \left (3\right )+1\right )+5\,x^2\,\ln \left (3\right )-5}{5\,x+4\,\ln \left (5\right )}\right )}{2} \] Input:
int((25*x + log((5*x + 4*log(5)*(x*log(3) + 1) + 5*x^2*log(3) - 5)/(5*x + 4*log(5)))*(25*x^3*log(3) - 25*x + 4*log(5)*(10*x + 10*x^2*log(3) - 5) + 1 6*log(5)^2*(x*log(3) + 1) + 25*x^2) + 25*x^3*log(3) + 16*x*log(3)*log(5)^2 + 40*x^2*log(3)*log(5))/(50*x^3*log(3) - 50*x + 4*log(5)*(20*x + 20*x^2*l og(3) - 10) + 16*log(5)^2*(2*x*log(3) + 2) + 50*x^2),x)
Output:
(x*log((5*x + 4*log(5)*(x*log(3) + 1) + 5*x^2*log(3) - 5)/(5*x + 4*log(5)) ))/2
Time = 0.19 (sec) , antiderivative size = 395, normalized size of antiderivative = 16.46 \[ \int \frac {25 x+25 x^3 \log (3)+10 x^2 \log (3) \log (625)+x \log (3) \log ^2(625)+\left (-25 x+25 x^2+25 x^3 \log (3)+\left (-5+10 x+10 x^2 \log (3)\right ) \log (625)+(1+x \log (3)) \log ^2(625)\right ) \log \left (\frac {-5+5 x+5 x^2 \log (3)+(1+x \log (3)) \log (625)}{5 x+\log (625)}\right )}{-50 x+50 x^2+50 x^3 \log (3)+\left (-10+20 x+20 x^2 \log (3)\right ) \log (625)+(2+2 x \log (3)) \log ^2(625)} \, dx =\text {Too large to display} \] Input:
int(((16*(x*log(3)+1)*log(5)^2+4*(10*x^2*log(3)+10*x-5)*log(5)+25*x^3*log( 3)+25*x^2-25*x)*log((4*(x*log(3)+1)*log(5)+5*x^2*log(3)+5*x-5)/(4*log(5)+5 *x))+16*x*log(3)*log(5)^2+40*x^2*log(3)*log(5)+25*x^3*log(3)+25*x)/(16*(2* x*log(3)+2)*log(5)^2+4*(20*x^2*log(3)+20*x-10)*log(5)+50*x^3*log(3)+50*x^2 -50*x),x)
Output:
(5*log(16*log(5)**2*log(3)*x + 16*log(5)**2 + 40*log(5)*log(3)*x**2 + 40*l og(5)*x - 20*log(5) + 25*log(3)*x**3 + 25*x**2 - 25*x)*log(3) - 4*log(4*lo g(5)*log(3)*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5)*log(5)*log(3) + 10*log (4*log(5)*log(3)*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5)*log(3) + 5*log(4* log(5)*log(3)*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5) + 4*log(4*log(5) + 5 *x)*log(5)*log(3) - 20*log(4*log(5) + 5*x)*log(3) - 5*log(4*log(5) + 5*x) + 4*log((4*log(5)*log(3)*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5)/(4*log(5) + 5*x))*log(5)*log(3)**2*x + 4*log((4*log(5)*log(3)*x + 4*log(5) + 5*log( 3)*x**2 + 5*x - 5)/(4*log(5) + 5*x))*log(5)*log(3) - 5*log((4*log(5)*log(3 )*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5)/(4*log(5) + 5*x))*log(3)*x - 15* log((4*log(5)*log(3)*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5)/(4*log(5) + 5 *x))*log(3) - 5*log((4*log(5)*log(3)*x + 4*log(5) + 5*log(3)*x**2 + 5*x - 5)/(4*log(5) + 5*x)))/(2*log(3)*(4*log(5)*log(3) - 5))