Integrand size = 99, antiderivative size = 26 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=-2+\left (\frac {2}{2-e}+x+25 x^4 \log ^2\left (x^2\right )\right )^2 \] Output:
(25*x^4*ln(x^2)^2+x+2/(2-exp(1)))^2-2
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=\frac {\left (-2+(-2+e) x+25 (-2+e) x^4 \log ^2\left (x^2\right )\right )^2}{(-2+e)^2} \] Input:
Integrate[(-4 - 4*x + 2*E*x + (-400*x^3 - 400*x^4 + 200*E*x^4)*Log[x^2] + (-400*x^3 - 500*x^4 + 250*E*x^4)*Log[x^2]^2 + (-10000*x^7 + 5000*E*x^7)*Lo g[x^2]^3 + (-10000*x^7 + 5000*E*x^7)*Log[x^2]^4)/(-2 + E),x]
Output:
(-2 + (-2 + E)*x + 25*(-2 + E)*x^4*Log[x^2]^2)^2/(-2 + E)^2
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(26)=52\).
Time = 0.49 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.27, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6, 27, 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.
\(\displaystyle \int \frac {\left (5000 e x^7-10000 x^7\right ) \log ^4\left (x^2\right )+\left (5000 e x^7-10000 x^7\right ) \log ^3\left (x^2\right )+\left (250 e x^4-500 x^4-400 x^3\right ) \log ^2\left (x^2\right )+\left (200 e x^4-400 x^4-400 x^3\right ) \log \left (x^2\right )+2 e x-4 x-4}{e-2} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (5000 e x^7-10000 x^7\right ) \log ^4\left (x^2\right )+\left (5000 e x^7-10000 x^7\right ) \log ^3\left (x^2\right )+\left (250 e x^4-500 x^4-400 x^3\right ) \log ^2\left (x^2\right )+\left (200 e x^4-400 x^4-400 x^3\right ) \log \left (x^2\right )+(2 e-4) x-4}{e-2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \left (-5000 \left (2 x^7-e x^7\right ) \log ^4\left (x^2\right )-5000 \left (2 x^7-e x^7\right ) \log ^3\left (x^2\right )-50 \left (-5 e x^4+10 x^4+8 x^3\right ) \log ^2\left (x^2\right )-200 \left (-e x^4+2 x^4+2 x^3\right ) \log \left (x^2\right )-2 (2-e) x-4\right )dx}{2-e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-(2-e) x^2-625 (2-e) x^8 \log ^4\left (x^2\right )-50 (2-e) x^5 \log ^2\left (x^2\right )+40 (2-e) x^5 \log \left (x^2\right )-100 x^4 \log ^2\left (x^2\right )+100 x^4 \log \left (x^2\right )-20 \left (2 (2-e) x^5+5 x^4\right ) \log \left (x^2\right )-4 x}{2-e}\) |
Input:
Int[(-4 - 4*x + 2*E*x + (-400*x^3 - 400*x^4 + 200*E*x^4)*Log[x^2] + (-400* x^3 - 500*x^4 + 250*E*x^4)*Log[x^2]^2 + (-10000*x^7 + 5000*E*x^7)*Log[x^2] ^3 + (-10000*x^7 + 5000*E*x^7)*Log[x^2]^4)/(-2 + E),x]
Output:
-((-4*x - (2 - E)*x^2 + 100*x^4*Log[x^2] + 40*(2 - E)*x^5*Log[x^2] - 20*(5 *x^4 + 2*(2 - E)*x^5)*Log[x^2] - 100*x^4*Log[x^2]^2 - 50*(2 - E)*x^5*Log[x ^2]^2 - 625*(2 - E)*x^8*Log[x^2]^4)/(2 - E))
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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(27)=54\).
Time = 3.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92
method | result | size |
risch | \(625 x^{8} \ln \left (x^{2}\right )^{4}+\frac {\left (50 x^{5} {\mathrm e}-100 x^{5}-100 x^{4}\right ) \ln \left (x^{2}\right )^{2}}{{\mathrm e}-2}+\frac {x^{2} {\mathrm e}}{{\mathrm e}-2}-\frac {2 x^{2}}{{\mathrm e}-2}-\frac {4 x}{{\mathrm e}-2}\) | \(76\) |
parallelrisch | \(\frac {625 \ln \left (x^{2}\right )^{4} x^{8} {\mathrm e}-1250 x^{8} \ln \left (x^{2}\right )^{4}+50 \ln \left (x^{2}\right )^{2} x^{5} {\mathrm e}-100 x^{5} \ln \left (x^{2}\right )^{2}-100 x^{4} \ln \left (x^{2}\right )^{2}+x^{2} {\mathrm e}-2 x^{2}-4 x}{{\mathrm e}-2}\) | \(82\) |
Input:
int(((5000*x^7*exp(1)-10000*x^7)*ln(x^2)^4+(5000*x^7*exp(1)-10000*x^7)*ln( x^2)^3+(250*x^4*exp(1)-500*x^4-400*x^3)*ln(x^2)^2+(200*x^4*exp(1)-400*x^4- 400*x^3)*ln(x^2)+2*x*exp(1)-4*x-4)/(exp(1)-2),x,method=_RETURNVERBOSE)
Output:
625*x^8*ln(x^2)^4+1/(exp(1)-2)*(50*x^5*exp(1)-100*x^5-100*x^4)*ln(x^2)^2+1 /(exp(1)-2)*x^2*exp(1)-2/(exp(1)-2)*x^2-4/(exp(1)-2)*x
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=\frac {625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{4} + x^{2} e + 50 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} - 2 \, x^{2} - 4 \, x}{e - 2} \] Input:
integrate(((5000*exp(1)*x^7-10000*x^7)*log(x^2)^4+(5000*exp(1)*x^7-10000*x ^7)*log(x^2)^3+(250*x^4*exp(1)-500*x^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1) -400*x^4-400*x^3)*log(x^2)+2*exp(1)*x-4*x-4)/(exp(1)-2),x, algorithm="fric as")
Output:
(625*(x^8*e - 2*x^8)*log(x^2)^4 + x^2*e + 50*(x^5*e - 2*x^5 - 2*x^4)*log(x ^2)^2 - 2*x^2 - 4*x)/(e - 2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=625 x^{8} \log {\left (x^{2} \right )}^{4} + x^{2} - \frac {4 x}{-2 + e} + \frac {\left (- 100 x^{5} + 50 e x^{5} - 100 x^{4}\right ) \log {\left (x^{2} \right )}^{2}}{-2 + e} \] Input:
integrate(((5000*exp(1)*x**7-10000*x**7)*ln(x**2)**4+(5000*exp(1)*x**7-100 00*x**7)*ln(x**2)**3+(250*x**4*exp(1)-500*x**4-400*x**3)*ln(x**2)**2+(200* x**4*exp(1)-400*x**4-400*x**3)*ln(x**2)+2*exp(1)*x-4*x-4)/(exp(1)-2),x)
Output:
625*x**8*log(x**2)**4 + x**2 - 4*x/(-2 + E) + (-100*x**5 + 50*E*x**5 - 100 *x**4)*log(x**2)**2/(-2 + E)
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (25) = 50\).
Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 5.73 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=-\frac {625 \, x^{8} {\left (e - 2\right )} \log \left (x^{2}\right )^{3} - 625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{4} - 625 \, {\left (x^{8} e - 2 \, x^{8}\right )} \log \left (x^{2}\right )^{3} - x^{2} e - 50 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} + 2 \, x^{2} + 20 \, {\left (2 \, x^{5} {\left (e - 2\right )} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) - 20 \, {\left (2 \, x^{5} e - 4 \, x^{5} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) + 4 \, x}{e - 2} \] Input:
integrate(((5000*exp(1)*x^7-10000*x^7)*log(x^2)^4+(5000*exp(1)*x^7-10000*x ^7)*log(x^2)^3+(250*x^4*exp(1)-500*x^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1) -400*x^4-400*x^3)*log(x^2)+2*exp(1)*x-4*x-4)/(exp(1)-2),x, algorithm="maxi ma")
Output:
-(625*x^8*(e - 2)*log(x^2)^3 - 625*(x^8*e - 2*x^8)*log(x^2)^4 - 625*(x^8*e - 2*x^8)*log(x^2)^3 - x^2*e - 50*(x^5*e - 2*x^5 - 2*x^4)*log(x^2)^2 + 2*x ^2 + 20*(2*x^5*(e - 2) - 5*x^4)*log(x^2) - 20*(2*x^5*e - 4*x^5 - 5*x^4)*lo g(x^2) + 4*x)/(e - 2)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 8.50 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=-\frac {40000 \, x^{8} \log \left (x^{2}\right )^{4} + 512 \, x^{5} e - 2560 \, x^{5} \log \left (x^{2}\right ) - 3200 \, x^{4} \log \left (x^{2}\right ) - 32 \, x^{2} e - 1600 \, {\left (x^{5} e - 2 \, x^{5} - 2 \, x^{4}\right )} \log \left (x^{2}\right )^{2} + 64 \, x^{2} - 625 \, {\left (32 \, x^{8} \log \left (x^{2}\right )^{4} - 32 \, x^{8} \log \left (x^{2}\right )^{3} + 24 \, x^{8} \log \left (x^{2}\right )^{2} - 12 \, x^{8} \log \left (x^{2}\right ) + 3 \, x^{8}\right )} e - 625 \, {\left (32 \, x^{8} \log \left (x^{2}\right )^{3} - 24 \, x^{8} \log \left (x^{2}\right )^{2} + 12 \, x^{8} \log \left (x^{2}\right ) - 3 \, x^{8}\right )} e + 256 \, {\left (5 \, x^{5} \log \left (x^{2}\right ) - 2 \, x^{5}\right )} e - 640 \, {\left (2 \, x^{5} e - 4 \, x^{5} - 5 \, x^{4}\right )} \log \left (x^{2}\right ) + 128 \, x}{32 \, {\left (e - 2\right )}} \] Input:
integrate(((5000*exp(1)*x^7-10000*x^7)*log(x^2)^4+(5000*exp(1)*x^7-10000*x ^7)*log(x^2)^3+(250*x^4*exp(1)-500*x^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1) -400*x^4-400*x^3)*log(x^2)+2*exp(1)*x-4*x-4)/(exp(1)-2),x, algorithm="giac ")
Output:
-1/32*(40000*x^8*log(x^2)^4 + 512*x^5*e - 2560*x^5*log(x^2) - 3200*x^4*log (x^2) - 32*x^2*e - 1600*(x^5*e - 2*x^5 - 2*x^4)*log(x^2)^2 + 64*x^2 - 625* (32*x^8*log(x^2)^4 - 32*x^8*log(x^2)^3 + 24*x^8*log(x^2)^2 - 12*x^8*log(x^ 2) + 3*x^8)*e - 625*(32*x^8*log(x^2)^3 - 24*x^8*log(x^2)^2 + 12*x^8*log(x^ 2) - 3*x^8)*e + 256*(5*x^5*log(x^2) - 2*x^5)*e - 640*(2*x^5*e - 4*x^5 - 5* x^4)*log(x^2) + 128*x)/(e - 2)
Time = 4.56 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=-\frac {x\,\left (25\,x^3\,{\ln \left (x^2\right )}^2+1\right )\,\left (2\,x-x\,\mathrm {e}+50\,x^4\,{\ln \left (x^2\right )}^2-25\,x^4\,{\ln \left (x^2\right )}^2\,\mathrm {e}+4\right )}{\mathrm {e}-2} \] Input:
int(-(4*x + log(x^2)*(400*x^3 - 200*x^4*exp(1) + 400*x^4) - log(x^2)^3*(50 00*x^7*exp(1) - 10000*x^7) - log(x^2)^4*(5000*x^7*exp(1) - 10000*x^7) - 2* x*exp(1) + log(x^2)^2*(400*x^3 - 250*x^4*exp(1) + 500*x^4) + 4)/(exp(1) - 2),x)
Output:
-(x*(25*x^3*log(x^2)^2 + 1)*(2*x - x*exp(1) + 50*x^4*log(x^2)^2 - 25*x^4*l og(x^2)^2*exp(1) + 4))/(exp(1) - 2)
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {-4-4 x+2 e x+\left (-400 x^3-400 x^4+200 e x^4\right ) \log \left (x^2\right )+\left (-400 x^3-500 x^4+250 e x^4\right ) \log ^2\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^3\left (x^2\right )+\left (-10000 x^7+5000 e x^7\right ) \log ^4\left (x^2\right )}{-2+e} \, dx=\frac {x \left (625 \mathrm {log}\left (x^{2}\right )^{4} e \,x^{7}-1250 \mathrm {log}\left (x^{2}\right )^{4} x^{7}+50 \mathrm {log}\left (x^{2}\right )^{2} e \,x^{4}-100 \mathrm {log}\left (x^{2}\right )^{2} x^{4}-100 \mathrm {log}\left (x^{2}\right )^{2} x^{3}+e x -2 x -4\right )}{e -2} \] Input:
int(((5000*exp(1)*x^7-10000*x^7)*log(x^2)^4+(5000*exp(1)*x^7-10000*x^7)*lo g(x^2)^3+(250*x^4*exp(1)-500*x^4-400*x^3)*log(x^2)^2+(200*x^4*exp(1)-400*x ^4-400*x^3)*log(x^2)+2*exp(1)*x-4*x-4)/(exp(1)-2),x)
Output:
(x*(625*log(x**2)**4*e*x**7 - 1250*log(x**2)**4*x**7 + 50*log(x**2)**2*e*x **4 - 100*log(x**2)**2*x**4 - 100*log(x**2)**2*x**3 + e*x - 2*x - 4))/(e - 2)