Integrand size = 164, antiderivative size = 32 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=\left (-x-x^2+x \log (5)\right ) \left (x-\left (-3+\frac {2}{\log \left (e^5+x\right )}\right )^2\right ) \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.79 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.38 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=12 \operatorname {ExpIntegralEi}\left (\log \left (e^5+x\right )\right ) \left (-1+2 e^5+\log (5)\right )-\frac {x (1+x-\log (5)) \left (-4+12 \log \left (e^5+x\right )+(-9+x) \log ^2\left (e^5+x\right )\right )}{\log ^2\left (e^5+x\right )}-12 \left (-1+2 e^5+\log (5)\right ) \operatorname {LogIntegral}\left (e^5+x\right ) \]
Integrate[(-8*x - 8*x^2 + 8*x*Log[5] + (16*x + 20*x^2 + E^5*(4 + 8*x) + (- 4*E^5 - 16*x)*Log[5])*Log[E^5 + x] + (E^5*(-12 - 24*x) - 12*x - 24*x^2 + ( 12*E^5 + 12*x)*Log[5])*Log[E^5 + x]^2 + (9*x + 16*x^2 - 3*x^3 + E^5*(9 + 1 6*x - 3*x^2) + (-9*x + 2*x^2 + E^5*(-9 + 2*x))*Log[5])*Log[E^5 + x]^3)/((E ^5 + x)*Log[E^5 + x]^3),x]
12*ExpIntegralEi[Log[E^5 + x]]*(-1 + 2*E^5 + Log[5]) - (x*(1 + x - Log[5]) *(-4 + 12*Log[E^5 + x] + (-9 + x)*Log[E^5 + x]^2))/Log[E^5 + x]^2 - 12*(-1 + 2*E^5 + Log[5])*LogIntegral[E^5 + x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 8.34, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6, 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.
| \(\displaystyle \int \frac {-8 x^2+\left (-24 x^2-12 x+e^5 (-24 x-12)+\left (12 x+12 e^5\right ) \log (5)\right ) \log ^2\left (x+e^5\right )+\left (20 x^2+16 x+e^5 (8 x+4)+\left (-16 x-4 e^5\right ) \log (5)\right ) \log \left (x+e^5\right )+\left (-3 x^3+16 x^2+e^5 \left (-3 x^2+16 x+9\right )+\left (2 x^2-9 x+e^5 (2 x-9)\right ) \log (5)+9 x\right ) \log ^3\left (x+e^5\right )-8 x+8 x \log (5)}{\left (x+e^5\right ) \log ^3\left (x+e^5\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-8 x^2+\left (-24 x^2-12 x+e^5 (-24 x-12)+\left (12 x+12 e^5\right ) \log (5)\right ) \log ^2\left (x+e^5\right )+\left (20 x^2+16 x+e^5 (8 x+4)+\left (-16 x-4 e^5\right ) \log (5)\right ) \log \left (x+e^5\right )+\left (-3 x^3+16 x^2+e^5 \left (-3 x^2+16 x+9\right )+\left (2 x^2-9 x+e^5 (2 x-9)\right ) \log (5)+9 x\right ) \log ^3\left (x+e^5\right )+x (8 \log (5)-8)}{\left (x+e^5\right ) \log ^3\left (x+e^5\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-3 x^2+\frac {4 \left (5 x^2+2 x \left (2+e^5-2 \log (5)\right )+e^5 (1-\log (5))\right )}{\left (x+e^5\right ) \log ^2\left (x+e^5\right )}-\frac {8 x (x+1-\log (5))}{\left (x+e^5\right ) \log ^3\left (x+e^5\right )}+16 x \left (1+\frac {\log (5)}{8}\right )-\frac {12 (2 x+1-\log (5))}{\log \left (x+e^5\right )}+9 (1-\log (5))\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -16 e^5 \operatorname {LogIntegral}\left (x+e^5\right )-4 \left (1-e^5-\log (5)\right ) \operatorname {LogIntegral}\left (x+e^5\right )-12 \left (1-2 e^5-\log (5)\right ) \operatorname {LogIntegral}\left (x+e^5\right )+4 \left (4-3 e^5-4 \log (5)\right ) \operatorname {LogIntegral}\left (x+e^5\right )-x^3+x^2 (8+\log (5))+\frac {4 \left (x+e^5\right ) x}{\log ^2\left (x+e^5\right )}+\frac {4 \left (x+e^5\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (x+e^5\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (x+e^5\right )}-\frac {12 \left (x+e^5\right ) x}{\log \left (x+e^5\right )}+9 x (1-\log (5))+\frac {4 e^5 \left (x+e^5\right )}{\log \left (x+e^5\right )}+\frac {4 \left (x+e^5\right ) \left (1-e^5-\log (5)\right )}{\log \left (x+e^5\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (x+e^5\right )}-\frac {4 \left (x+e^5\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (x+e^5\right )}\) |
Int[(-8*x - 8*x^2 + 8*x*Log[5] + (16*x + 20*x^2 + E^5*(4 + 8*x) + (-4*E^5 - 16*x)*Log[5])*Log[E^5 + x] + (E^5*(-12 - 24*x) - 12*x - 24*x^2 + (12*E^5 + 12*x)*Log[5])*Log[E^5 + x]^2 + (9*x + 16*x^2 - 3*x^3 + E^5*(9 + 16*x - 3*x^2) + (-9*x + 2*x^2 + E^5*(-9 + 2*x))*Log[5])*Log[E^5 + x]^3)/((E^5 + x )*Log[E^5 + x]^3),x]
-x^3 + 9*x*(1 - Log[5]) + x^2*(8 + Log[5]) + (4*x*(E^5 + x))/Log[E^5 + x]^ 2 - (4*E^5*(1 - E^5 - Log[5]))/Log[E^5 + x]^2 + (4*(E^5 + x)*(1 - E^5 - Lo g[5]))/Log[E^5 + x]^2 + (4*E^5*(E^5 + x))/Log[E^5 + x] - (12*x*(E^5 + x))/ Log[E^5 + x] - (4*(E^5 + x)*(4 - 3*E^5 - 4*Log[5]))/Log[E^5 + x] + (12*E^5 *(1 - E^5 - Log[5]))/Log[E^5 + x] + (4*(E^5 + x)*(1 - E^5 - Log[5]))/Log[E ^5 + x] - 16*E^5*LogIntegral[E^5 + x] + 4*(4 - 3*E^5 - 4*Log[5])*LogIntegr al[E^5 + x] - 12*(1 - 2*E^5 - Log[5])*LogIntegral[E^5 + x] - 4*(1 - E^5 - Log[5])*LogIntegral[E^5 + x]
3.16.89.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(31)=62\).
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09
| method | result | size |
| risch | \(x^{2} \ln \left (5\right )-x^{3}-9 x \ln \left (5\right )+8 x^{2}+9 x +\frac {4 x \left (3 \ln \left (5\right ) \ln \left ({\mathrm e}^{5}+x \right )-3 \ln \left ({\mathrm e}^{5}+x \right ) x -\ln \left (5\right )+x -3 \ln \left ({\mathrm e}^{5}+x \right )+1\right )}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\) | \(67\) |
| norman | \(\frac {\left (4-4 \ln \left (5\right )\right ) x +\left (-12+12 \ln \left (5\right )\right ) x \ln \left ({\mathrm e}^{5}+x \right )+\left (8+\ln \left (5\right )\right ) x^{2} \ln \left ({\mathrm e}^{5}+x \right )^{2}+\left (-9 \ln \left (5\right )+9\right ) x \ln \left ({\mathrm e}^{5}+x \right )^{2}+4 x^{2}-x^{3} \ln \left ({\mathrm e}^{5}+x \right )^{2}-12 \ln \left ({\mathrm e}^{5}+x \right ) x^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\) | \(88\) |
| parallelrisch | \(-\frac {{\mathrm e}^{10} \ln \left (5\right ) \ln \left ({\mathrm e}^{5}+x \right )^{2}-\ln \left (5\right ) x^{2} \ln \left ({\mathrm e}^{5}+x \right )^{2}+x^{3} \ln \left ({\mathrm e}^{5}+x \right )^{2}+8 \ln \left ({\mathrm e}^{5}+x \right )^{2} {\mathrm e}^{10}-18 \ln \left ({\mathrm e}^{5}+x \right )^{2} {\mathrm e}^{5} \ln \left (5\right )+9 \ln \left ({\mathrm e}^{5}+x \right )^{2} \ln \left (5\right ) x -8 \ln \left ({\mathrm e}^{5}+x \right )^{2} x^{2}+18 \ln \left ({\mathrm e}^{5}+x \right )^{2} {\mathrm e}^{5}-12 \ln \left ({\mathrm e}^{5}+x \right ) \ln \left (5\right ) x +12 \ln \left ({\mathrm e}^{5}+x \right ) x^{2}-9 \ln \left ({\mathrm e}^{5}+x \right )^{2} x +4 x \ln \left (5\right )-4 x^{2}+12 \ln \left ({\mathrm e}^{5}+x \right ) x -4 x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\) | \(162\) |
| parts | \(-\frac {8 \,{\mathrm e}^{5} x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {4 \ln \left (5\right ) x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+9 x +\frac {12 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {4 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+\frac {4 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {4 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+\frac {4 \,{\mathrm e}^{5}+4 x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \left ({\mathrm e}^{5}+x \right )}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {12 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )}+x^{2} \ln \left (5\right )-x^{3}+8 x^{2}+\frac {24 \,{\mathrm e}^{5} x}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {12 \ln \left (5\right ) x}{\ln \left ({\mathrm e}^{5}+x \right )}-9 x \ln \left (5\right )\) | \(178\) |
| derivativedivides | \(-2 \,{\mathrm e}^{5} \ln \left (5\right ) \left ({\mathrm e}^{5}+x \right )-3 \,{\mathrm e}^{10} \left ({\mathrm e}^{5}+x \right )-\frac {12 \,{\mathrm e}^{5} \ln \left (5\right )}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \,{\mathrm e}^{5} \ln \left (5\right )}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+9 x +\frac {12 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {4 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+\frac {4 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+3 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )^{2}-\frac {12 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {12 \left ({\mathrm e}^{5}+x \right )}{\ln \left ({\mathrm e}^{5}+x \right )}+16 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )+8 \ln \left (5\right ) \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )-32 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )\right )-16 \ln \left (5\right ) \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )\right )-12 \ln \left (5\right ) \operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )-24 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )-16 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )-9 \left ({\mathrm e}^{5}+x \right ) \ln \left (5\right )+\ln \left (5\right ) \left ({\mathrm e}^{5}+x \right )^{2}+8 \left ({\mathrm e}^{5}+x \right )^{2}+9 \,{\mathrm e}^{5}-\left ({\mathrm e}^{5}+x \right )^{3}+\frac {4 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \,{\mathrm e}^{5}+4 x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\) | \(378\) |
| default | \(-2 \,{\mathrm e}^{5} \ln \left (5\right ) \left ({\mathrm e}^{5}+x \right )-3 \,{\mathrm e}^{10} \left ({\mathrm e}^{5}+x \right )-\frac {12 \,{\mathrm e}^{5} \ln \left (5\right )}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \,{\mathrm e}^{5} \ln \left (5\right )}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+9 x +\frac {12 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {4 \,{\mathrm e}^{5}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+\frac {4 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}+3 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )^{2}-\frac {12 \left ({\mathrm e}^{5}+x \right )^{2}}{\ln \left ({\mathrm e}^{5}+x \right )}-\frac {12 \left ({\mathrm e}^{5}+x \right )}{\ln \left ({\mathrm e}^{5}+x \right )}+16 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )+8 \ln \left (5\right ) \left (-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {{\mathrm e}^{5}+x}{2 \ln \left ({\mathrm e}^{5}+x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )}{2}\right )-32 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )\right )-16 \ln \left (5\right ) \left (-\frac {{\mathrm e}^{5}+x}{\ln \left ({\mathrm e}^{5}+x \right )}-\operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )\right )-12 \ln \left (5\right ) \operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )-24 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\ln \left ({\mathrm e}^{5}+x \right )\right )-16 \,{\mathrm e}^{5} \left ({\mathrm e}^{5}+x \right )-9 \left ({\mathrm e}^{5}+x \right ) \ln \left (5\right )+\ln \left (5\right ) \left ({\mathrm e}^{5}+x \right )^{2}+8 \left ({\mathrm e}^{5}+x \right )^{2}+9 \,{\mathrm e}^{5}-\left ({\mathrm e}^{5}+x \right )^{3}+\frac {4 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}-\frac {12 \,{\mathrm e}^{10}}{\ln \left ({\mathrm e}^{5}+x \right )}+\frac {4 \,{\mathrm e}^{5}+4 x}{\ln \left ({\mathrm e}^{5}+x \right )^{2}}\) | \(378\) |
int(((((2*x-9)*exp(5)+2*x^2-9*x)*ln(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3+16*x^2 +9*x)*ln(exp(5)+x)^3+((12*exp(5)+12*x)*ln(5)+(-24*x-12)*exp(5)-24*x^2-12*x )*ln(exp(5)+x)^2+((-4*exp(5)-16*x)*ln(5)+(8*x+4)*exp(5)+20*x^2+16*x)*ln(ex p(5)+x)+8*x*ln(5)-8*x^2-8*x)/(exp(5)+x)/ln(exp(5)+x)^3,x,method=_RETURNVER BOSE)
x^2*ln(5)-x^3-9*x*ln(5)+8*x^2+9*x+4*x*(3*ln(5)*ln(exp(5)+x)-3*ln(exp(5)+x) *x-ln(5)+x-3*ln(exp(5)+x)+1)/ln(exp(5)+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {{\left (x^{3} - 8 \, x^{2} - {\left (x^{2} - 9 \, x\right )} \log \left (5\right ) - 9 \, x\right )} \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x \log \left (5\right ) + 12 \, {\left (x^{2} - x \log \left (5\right ) + x\right )} \log \left (x + e^{5}\right ) - 4 \, x}{\log \left (x + e^{5}\right )^{2}} \]
integrate(((((2*x-9)*exp(5)+2*x^2-9*x)*log(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3 +16*x^2+9*x)*log(exp(5)+x)^3+((12*exp(5)+12*x)*log(5)+(-24*x-12)*exp(5)-24 *x^2-12*x)*log(exp(5)+x)^2+((-4*exp(5)-16*x)*log(5)+(8*x+4)*exp(5)+20*x^2+ 16*x)*log(exp(5)+x)+8*x*log(5)-8*x^2-8*x)/(exp(5)+x)/log(exp(5)+x)^3,x, al gorithm=\
-((x^3 - 8*x^2 - (x^2 - 9*x)*log(5) - 9*x)*log(x + e^5)^2 - 4*x^2 + 4*x*lo g(5) + 12*(x^2 - x*log(5) + x)*log(x + e^5) - 4*x)/log(x + e^5)^2
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=- x^{3} + x^{2} \left (\log {\left (5 \right )} + 8\right ) + x \left (9 - 9 \log {\left (5 \right )}\right ) + \frac {4 x^{2} - 4 x \log {\left (5 \right )} + 4 x + \left (- 12 x^{2} - 12 x + 12 x \log {\left (5 \right )}\right ) \log {\left (x + e^{5} \right )}}{\log {\left (x + e^{5} \right )}^{2}} \]
integrate(((((2*x-9)*exp(5)+2*x**2-9*x)*ln(5)+(-3*x**2+16*x+9)*exp(5)-3*x* *3+16*x**2+9*x)*ln(exp(5)+x)**3+((12*exp(5)+12*x)*ln(5)+(-24*x-12)*exp(5)- 24*x**2-12*x)*ln(exp(5)+x)**2+((-4*exp(5)-16*x)*ln(5)+(8*x+4)*exp(5)+20*x* *2+16*x)*ln(exp(5)+x)+8*x*ln(5)-8*x**2-8*x)/(exp(5)+x)/ln(exp(5)+x)**3,x)
-x**3 + x**2*(log(5) + 8) + x*(9 - 9*log(5)) + (4*x**2 - 4*x*log(5) + 4*x + (-12*x**2 - 12*x + 12*x*log(5))*log(x + exp(5)))/log(x + exp(5))**2
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {{\left (x^{3} - x^{2} {\left (\log \left (5\right ) + 8\right )} + 9 \, x {\left (\log \left (5\right ) - 1\right )}\right )} \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x {\left (\log \left (5\right ) - 1\right )} + 12 \, {\left (x^{2} - x {\left (\log \left (5\right ) - 1\right )}\right )} \log \left (x + e^{5}\right )}{\log \left (x + e^{5}\right )^{2}} \]
integrate(((((2*x-9)*exp(5)+2*x^2-9*x)*log(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3 +16*x^2+9*x)*log(exp(5)+x)^3+((12*exp(5)+12*x)*log(5)+(-24*x-12)*exp(5)-24 *x^2-12*x)*log(exp(5)+x)^2+((-4*exp(5)-16*x)*log(5)+(8*x+4)*exp(5)+20*x^2+ 16*x)*log(exp(5)+x)+8*x*log(5)-8*x^2-8*x)/(exp(5)+x)/log(exp(5)+x)^3,x, al gorithm=\
-((x^3 - x^2*(log(5) + 8) + 9*x*(log(5) - 1))*log(x + e^5)^2 - 4*x^2 + 4*x *(log(5) - 1) + 12*(x^2 - x*(log(5) - 1))*log(x + e^5))/log(x + e^5)^2
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (28) = 56\).
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=-\frac {x^{3} \log \left (x + e^{5}\right )^{2} - x^{2} \log \left (5\right ) \log \left (x + e^{5}\right )^{2} - 8 \, x^{2} \log \left (x + e^{5}\right )^{2} + 9 \, x \log \left (5\right ) \log \left (x + e^{5}\right )^{2} + 12 \, x^{2} \log \left (x + e^{5}\right ) - 12 \, x \log \left (5\right ) \log \left (x + e^{5}\right ) - 9 \, x \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x \log \left (5\right ) + 12 \, x \log \left (x + e^{5}\right ) - 4 \, x}{\log \left (x + e^{5}\right )^{2}} \]
integrate(((((2*x-9)*exp(5)+2*x^2-9*x)*log(5)+(-3*x^2+16*x+9)*exp(5)-3*x^3 +16*x^2+9*x)*log(exp(5)+x)^3+((12*exp(5)+12*x)*log(5)+(-24*x-12)*exp(5)-24 *x^2-12*x)*log(exp(5)+x)^2+((-4*exp(5)-16*x)*log(5)+(8*x+4)*exp(5)+20*x^2+ 16*x)*log(exp(5)+x)+8*x*log(5)-8*x^2-8*x)/(exp(5)+x)/log(exp(5)+x)^3,x, al gorithm=\
-(x^3*log(x + e^5)^2 - x^2*log(5)*log(x + e^5)^2 - 8*x^2*log(x + e^5)^2 + 9*x*log(5)*log(x + e^5)^2 + 12*x^2*log(x + e^5) - 12*x*log(5)*log(x + e^5) - 9*x*log(x + e^5)^2 - 4*x^2 + 4*x*log(5) + 12*x*log(x + e^5) - 4*x)/log( x + e^5)^2
Time = 12.99 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.91 \[ \int \frac {-8 x-8 x^2+8 x \log (5)+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx=9\,x-\frac {12\,x^2}{\ln \left (x+{\mathrm {e}}^5\right )}+\frac {4\,x^2}{{\ln \left (x+{\mathrm {e}}^5\right )}^2}-9\,x\,\ln \left (5\right )+x^2\,\ln \left (5\right )+8\,x^2-x^3-\frac {12\,x}{\ln \left (x+{\mathrm {e}}^5\right )}+\frac {4\,x}{{\ln \left (x+{\mathrm {e}}^5\right )}^2}+\frac {12\,x\,\ln \left (5\right )}{\ln \left (x+{\mathrm {e}}^5\right )}-\frac {4\,x\,\ln \left (5\right )}{{\ln \left (x+{\mathrm {e}}^5\right )}^2} \]
int(-(8*x - log(x + exp(5))*(16*x - log(5)*(16*x + 4*exp(5)) + 20*x^2 + ex p(5)*(8*x + 4)) - 8*x*log(5) - log(x + exp(5))^3*(9*x + exp(5)*(16*x - 3*x ^2 + 9) + log(5)*(2*x^2 - 9*x + exp(5)*(2*x - 9)) + 16*x^2 - 3*x^3) + log( x + exp(5))^2*(12*x - log(5)*(12*x + 12*exp(5)) + 24*x^2 + exp(5)*(24*x + 12)) + 8*x^2)/(log(x + exp(5))^3*(x + exp(5))),x)