Integrand size = 134, antiderivative size = 28 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=\frac {e^x}{e^8 \left (4-\frac {x}{125}\right )^2-x}+6 \log (x) \]
Time = 3.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=\frac {15625 e^x}{e^8 (-500+x)^2-15625 x}+6 \log (x) \]
Integrate[(1464843750*x^2 + E^8*(-46875000000*x + 187500000*x^2 - 187500*x ^3) + E^16*(375000000000 - 3000000000*x + 9000000*x^2 - 12000*x^3 + 6*x^4) + E^x*(244140625*x - 244140625*x^2 + E^8*(3921875000*x - 15656250*x^2 + 1 5625*x^3)))/(244140625*x^3 + E^8*(-7812500000*x^2 + 31250000*x^3 - 31250*x ^4) + E^16*(62500000000*x - 500000000*x^2 + 1500000*x^3 - 2000*x^4 + x^5)) ,x]
Leaf count is larger than twice the leaf count of optimal. \(295\) vs. \(2(28)=56\).
Time = 1.74 (sec) , antiderivative size = 295, normalized size of antiderivative = 10.54, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2026, 2463, 6, 7292, 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.
| \(\displaystyle \int \frac {1464843750 x^2+e^8 \left (-187500 x^3+187500000 x^2-46875000000 x\right )+e^x \left (-244140625 x^2+e^8 \left (15625 x^3-15656250 x^2+3921875000 x\right )+244140625 x\right )+e^{16} \left (6 x^4-12000 x^3+9000000 x^2-3000000000 x+375000000000\right )}{244140625 x^3+e^8 \left (-31250 x^4+31250000 x^3-7812500000 x^2\right )+e^{16} \left (x^5-2000 x^4+1500000 x^3-500000000 x^2+62500000000 x\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {1464843750 x^2+e^8 \left (-187500 x^3+187500000 x^2-46875000000 x\right )+e^x \left (-244140625 x^2+e^8 \left (15625 x^3-15656250 x^2+3921875000 x\right )+244140625 x\right )+e^{16} \left (6 x^4-12000 x^3+9000000 x^2-3000000000 x+375000000000\right )}{x \left (e^{16} x^4-250 e^8 \left (125+8 e^8\right ) x^3+15625 \left (15625+2000 e^8+96 e^{16}\right ) x^2-62500000 e^8 \left (125+8 e^8\right ) x+62500000000 e^{16}\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {1464843750 x^2+e^8 \left (-187500 x^3+187500000 x^2-46875000000 x\right )+e^x \left (-244140625 x^2+e^8 \left (15625 x^3-15656250 x^2+3921875000 x\right )+244140625 x\right )+e^{16} \left (6 x^4-12000 x^3+9000000 x^2-3000000000 x+375000000000\right )}{x \left (e^8 x^2-1000 e^8 x-15625 x+250000 e^8\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {1464843750 x^2+e^8 \left (-187500 x^3+187500000 x^2-46875000000 x\right )+e^x \left (-244140625 x^2+e^8 \left (15625 x^3-15656250 x^2+3921875000 x\right )+244140625 x\right )+e^{16} \left (6 x^4-12000 x^3+9000000 x^2-3000000000 x+375000000000\right )}{x \left (e^8 x^2+\left (-15625-1000 e^8\right ) x+250000 e^8\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {1464843750 x^2+e^8 \left (-187500 x^3+187500000 x^2-46875000000 x\right )+e^x \left (-244140625 x^2+e^8 \left (15625 x^3-15656250 x^2+3921875000 x\right )+244140625 x\right )+e^{16} \left (6 x^4-12000 x^3+9000000 x^2-3000000000 x+375000000000\right )}{x \left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 e^{16} (500-x)^4}{x \left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )^2}-\frac {187500 e^8 (500-x)^2}{\left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )^2}+\frac {15625 e^x \left (e^8 x^2-\left (15625+1002 e^8\right ) x+125 \left (125+2008 e^8\right )\right )}{\left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )^2}+\frac {1464843750 x}{\left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {60 \sqrt {5} \left (125+24 e^8\right ) \text {arctanh}\left (\frac {2 e^8 (500-x)+15625}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {60 \sqrt {5} \left (125+8 e^8\right ) \text {arctanh}\left (\frac {2 e^8 (500-x)+15625}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {960 \sqrt {5} e^8 \text {arctanh}\left (\frac {2 e^8 (500-x)+15625}{625 \sqrt {5 \left (125+16 e^8\right )}}\right )}{\left (125+16 e^8\right )^{3/2}}-\frac {1500 e^8 (500-x) (x+500)}{\left (125+16 e^8\right ) \left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )}+\frac {15625 e^x}{e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8}+\frac {187500 \left (4000 e^8-\left (125+8 e^8\right ) x\right )}{\left (125+16 e^8\right ) \left (e^8 x^2-125 \left (125+8 e^8\right ) x+250000 e^8\right )}+6 \log (x)\) |
Int[(1464843750*x^2 + E^8*(-46875000000*x + 187500000*x^2 - 187500*x^3) + E^16*(375000000000 - 3000000000*x + 9000000*x^2 - 12000*x^3 + 6*x^4) + E^x *(244140625*x - 244140625*x^2 + E^8*(3921875000*x - 15656250*x^2 + 15625*x ^3)))/(244140625*x^3 + E^8*(-7812500000*x^2 + 31250000*x^3 - 31250*x^4) + E^16*(62500000000*x - 500000000*x^2 + 1500000*x^3 - 2000*x^4 + x^5)),x]
(15625*E^x)/(250000*E^8 - 125*(125 + 8*E^8)*x + E^8*x^2) - (1500*E^8*(500 - x)*(500 + x))/((125 + 16*E^8)*(250000*E^8 - 125*(125 + 8*E^8)*x + E^8*x^ 2)) + (187500*(4000*E^8 - (125 + 8*E^8)*x))/((125 + 16*E^8)*(250000*E^8 - 125*(125 + 8*E^8)*x + E^8*x^2)) - (960*Sqrt[5]*E^8*ArcTanh[(15625 + 2*E^8* (500 - x))/(625*Sqrt[5*(125 + 16*E^8)])])/(125 + 16*E^8)^(3/2) - (60*Sqrt[ 5]*(125 + 8*E^8)*ArcTanh[(15625 + 2*E^8*(500 - x))/(625*Sqrt[5*(125 + 16*E ^8)])])/(125 + 16*E^8)^(3/2) + (60*Sqrt[5]*(125 + 24*E^8)*ArcTanh[(15625 + 2*E^8*(500 - x))/(625*Sqrt[5*(125 + 16*E^8)])])/(125 + 16*E^8)^(3/2) + 6* Log[x]
3.9.55.3.1 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]
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_.)*(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 = 2.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {15625 \,{\mathrm e}^{x}}{x^{2} {\mathrm e}^{8}-1000 x \,{\mathrm e}^{8}+250000 \,{\mathrm e}^{8}-15625 x}+6 \ln \left (x \right )\) | \(31\) |
| norman | \(\frac {15625 \,{\mathrm e}^{x}}{x^{2} {\mathrm e}^{8}-1000 x \,{\mathrm e}^{8}+250000 \,{\mathrm e}^{8}-15625 x}+6 \ln \left (x \right )\) | \(37\) |
| parallelrisch | \(\frac {\left (6 x^{2} {\mathrm e}^{16} \ln \left (x \right )-6000 \ln \left (x \right ) {\mathrm e}^{16} x +1500000 \,{\mathrm e}^{16} \ln \left (x \right )-93750 \ln \left (x \right ) {\mathrm e}^{8} x +15625 \,{\mathrm e}^{8} {\mathrm e}^{x}\right ) {\mathrm e}^{-8}}{x^{2} {\mathrm e}^{8}-1000 x \,{\mathrm e}^{8}+250000 \,{\mathrm e}^{8}-15625 x}\) | \(79\) |
| parts | \(\text {Expression too large to display}\) | \(3289\) |
| default | \(\text {Expression too large to display}\) | \(4348\) |
int((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+2441406 25*x)*exp(x)+(6*x^4-12000*x^3+9000000*x^2-3000000000*x+375000000000)*exp(4 )^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*x^2)/((x ^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-31250*x^4+ 31250000*x^3-7812500000*x^2)*exp(4)^2+244140625*x^3),x,method=_RETURNVERBO SE)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=\frac {6 \, {\left ({\left (x^{2} - 1000 \, x + 250000\right )} e^{8} - 15625 \, x\right )} \log \left (x\right ) + 15625 \, e^{x}}{{\left (x^{2} - 1000 \, x + 250000\right )} e^{8} - 15625 \, x} \]
integrate((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+2 44140625*x)*exp(x)+(6*x^4-12000*x^3+9000000*x^2-3000000000*x+375000000000) *exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*x^ 2)/((x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-3125 0*x^4+31250000*x^3-7812500000*x^2)*exp(4)^2+244140625*x^3),x, algorithm=\
(6*((x^2 - 1000*x + 250000)*e^8 - 15625*x)*log(x) + 15625*e^x)/((x^2 - 100 0*x + 250000)*e^8 - 15625*x)
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=6 \log {\left (x \right )} + \frac {15625 e^{x}}{x^{2} e^{8} - 1000 x e^{8} - 15625 x + 250000 e^{8}} \]
integrate((((15625*x**3-15656250*x**2+3921875000*x)*exp(4)**2-244140625*x* *2+244140625*x)*exp(x)+(6*x**4-12000*x**3+9000000*x**2-3000000000*x+375000 000000)*exp(4)**4+(-187500*x**3+187500000*x**2-46875000000*x)*exp(4)**2+14 64843750*x**2)/((x**5-2000*x**4+1500000*x**3-500000000*x**2+62500000000*x) *exp(4)**4+(-31250*x**4+31250000*x**3-7812500000*x**2)*exp(4)**2+244140625 *x**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 1272 vs. \(2 (23) = 46\).
Time = 0.35 (sec) , antiderivative size = 1272, normalized size of antiderivative = 45.43 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=\text {Too large to display} \]
integrate((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+2 44140625*x)*exp(x)+(6*x^4-12000*x^3+9000000*x^2-3000000000*x+375000000000) *exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*x^ 2)/((x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-3125 0*x^4+31250000*x^3-7812500000*x^2)*exp(4)^2+244140625*x^3),x, algorithm=\
-3/625*(625*e^(-16)*log(x^2*e^8 - 125*x*(8*e^8 + 125) + 250000*e^8) - 1250 *e^(-16)*log(x) + (256*e^24 - 12000*e^16 - 375000*e^8 - 1953125)*log((2*x* e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80*e^ 8 + 625) - 1000*e^8 - 15625))/((16*e^24 + 125*e^16)*sqrt(80*e^8 + 625)) + 20000*(x*(8*e^16 + 125*e^8) - 4000*e^16 - 250000*e^8 - 1953125)/(x^2*(16*e ^24 + 125*e^16) - 125*x*(128*e^24 + 3000*e^16 + 15625*e^8) + 4000000*e^24 + 31250000*e^16))*e^16 + 3/625*(625*e^(-16)*log(x^2*e^8 - 125*x*(8*e^8 + 1 25) + 250000*e^8) - (256*e^24 - 12000*e^16 - 375000*e^8 - 1953125)*log((2* x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x*e^8 + 625*sqrt(80* e^8 + 625) - 1000*e^8 - 15625))/((16*e^24 + 125*e^16)*sqrt(80*e^8 + 625)) - 1250*(x*(128*e^24 + 18000*e^16 + 375000*e^8 + 1953125) - 64000*e^24 - 40 00000*e^16 - 31250000*e^8)/(x^2*(16*e^32 + 125*e^24) - 125*x*(128*e^32 + 3 000*e^24 + 15625*e^16) + 4000000*e^32 + 31250000*e^24))*e^16 - 576/625*((8 *e^8 + 125)*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625)/(2*x *e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(sqrt(80*e^8 + 625)*(16 *e^8 + 125)) + 625*(x*(8*e^8 + 125) - 4000*e^8)/(x^2*(16*e^16 + 125*e^8) - 125*x*(128*e^16 + 3000*e^8 + 15625) + 4000000*e^16 + 31250000*e^8))*e^16 + 96/625*(32*e^8*log((2*x*e^8 - 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625) /(2*x*e^8 + 625*sqrt(80*e^8 + 625) - 1000*e^8 - 15625))/(sqrt(80*e^8 + 625 )*(16*e^8 + 125)) + 625*(x*(32*e^16 + 2000*e^8 + 15625) - 16000*e^16 - ...
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=\frac {2 \, {\left (3 \, x^{2} e^{8} \log \left (x\right ) - 3000 \, x e^{8} \log \left (x\right ) - 46875 \, x \log \left (x\right ) + 750000 \, e^{8} \log \left (x\right ) + 15625 \, e^{x}\right )}}{x^{2} e^{8} - 1000 \, x e^{8} - 15625 \, x + 250000 \, e^{8}} \]
integrate((((15625*x^3-15656250*x^2+3921875000*x)*exp(4)^2-244140625*x^2+2 44140625*x)*exp(x)+(6*x^4-12000*x^3+9000000*x^2-3000000000*x+375000000000) *exp(4)^4+(-187500*x^3+187500000*x^2-46875000000*x)*exp(4)^2+1464843750*x^ 2)/((x^5-2000*x^4+1500000*x^3-500000000*x^2+62500000000*x)*exp(4)^4+(-3125 0*x^4+31250000*x^3-7812500000*x^2)*exp(4)^2+244140625*x^3),x, algorithm=\
2*(3*x^2*e^8*log(x) - 3000*x*e^8*log(x) - 46875*x*log(x) + 750000*e^8*log( x) + 15625*e^x)/(x^2*e^8 - 1000*x*e^8 - 15625*x + 250000*e^8)
Time = 0.89 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1464843750 x^2+e^8 \left (-46875000000 x+187500000 x^2-187500 x^3\right )+e^{16} \left (375000000000-3000000000 x+9000000 x^2-12000 x^3+6 x^4\right )+e^x \left (244140625 x-244140625 x^2+e^8 \left (3921875000 x-15656250 x^2+15625 x^3\right )\right )}{244140625 x^3+e^8 \left (-7812500000 x^2+31250000 x^3-31250 x^4\right )+e^{16} \left (62500000000 x-500000000 x^2+1500000 x^3-2000 x^4+x^5\right )} \, dx=6\,\ln \left (x\right )+\frac {15625\,{\mathrm {e}}^{x-8}}{x^2-{\mathrm {e}}^{-8}\,\left (1000\,{\mathrm {e}}^8+15625\right )\,x+250000} \]
int((exp(x)*(244140625*x + exp(8)*(3921875000*x - 15656250*x^2 + 15625*x^3 ) - 244140625*x^2) - exp(8)*(46875000000*x - 187500000*x^2 + 187500*x^3) + exp(16)*(9000000*x^2 - 3000000000*x - 12000*x^3 + 6*x^4 + 375000000000) + 1464843750*x^2)/(exp(16)*(62500000000*x - 500000000*x^2 + 1500000*x^3 - 2 000*x^4 + x^5) - exp(8)*(7812500000*x^2 - 31250000*x^3 + 31250*x^4) + 2441 40625*x^3),x)