Integrand size = 144, antiderivative size = 23 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=\left (-x+\log \left (-1+x+\frac {\left (2+e^4\right )^2 \log (5)}{x^2}\right )\right )^2 \] Output:
(ln(ln(5)/x^2*(2+exp(4))^2+x-1)-x)^2
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=\left (x-\log \left (\frac {-x^2+x^3+\left (2+e^4\right )^2 \log (5)}{x^2}\right )\right )^2 \] Input:
Integrate[(-4*x^4 + 2*x^5 + (16*x + 8*x^2 + E^8*(4*x + 2*x^2) + E^4*(16*x + 8*x^2))*Log[5] + (4*x^3 - 2*x^4 + (-16 + E^4*(-16 - 8*x) + E^8*(-4 - 2*x ) - 8*x)*Log[5])*Log[(-x^2 + x^3 + (4 + 4*E^4 + E^8)*Log[5])/x^2])/(-x^3 + x^4 + (4*x + 4*E^4*x + E^8*x)*Log[5]),x]
Output:
(x - Log[(-x^2 + x^3 + (2 + E^4)^2*Log[5])/x^2])^2
Time = 0.96 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2026, 7292, 27, 7237}
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.
\(\displaystyle \int \frac {2 x^5-4 x^4+\left (8 x^2+e^8 \left (2 x^2+4 x\right )+e^4 \left (8 x^2+16 x\right )+16 x\right ) \log (5)+\left (-2 x^4+4 x^3+\left (e^4 (-8 x-16)+e^8 (-2 x-4)-8 x-16\right ) \log (5)\right ) \log \left (\frac {x^3-x^2+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{x^4-x^3+\left (e^8 x+4 e^4 x+4 x\right ) \log (5)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^5-4 x^4+\left (8 x^2+e^8 \left (2 x^2+4 x\right )+e^4 \left (8 x^2+16 x\right )+16 x\right ) \log (5)+\left (-2 x^4+4 x^3+\left (e^4 (-8 x-16)+e^8 (-2 x-4)-8 x-16\right ) \log (5)\right ) \log \left (\frac {x^3-x^2+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{x \left (x^3-x^2+\left (2+e^4\right )^2 \log (5)\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (-x^4+2 x^3-\left (2+e^4\right )^2 x \log (5)-2 \left (2+e^4\right )^2 \log (5)\right ) \left (x-\log \left (\frac {x^3-x^2+\left (2+e^4\right )^2 \log (5)}{x^2}\right )\right )}{x \left (-x^3+x^2-\left (2+e^4\right )^2 \log (5)\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\left (-x^4+2 x^3-\left (2+e^4\right )^2 \log (5) x-2 \left (2+e^4\right )^2 \log (5)\right ) \left (x-\log \left (-\frac {-x^3+x^2-\left (2+e^4\right )^2 \log (5)}{x^2}\right )\right )}{x \left (-x^3+x^2-\left (2+e^4\right )^2 \log (5)\right )}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (x-\log \left (-\frac {-x^3+x^2-\left (2+e^4\right )^2 \log (5)}{x^2}\right )\right )^2\) |
Input:
Int[(-4*x^4 + 2*x^5 + (16*x + 8*x^2 + E^8*(4*x + 2*x^2) + E^4*(16*x + 8*x^ 2))*Log[5] + (4*x^3 - 2*x^4 + (-16 + E^4*(-16 - 8*x) + E^8*(-4 - 2*x) - 8* x)*Log[5])*Log[(-x^2 + x^3 + (4 + 4*E^4 + E^8)*Log[5])/x^2])/(-x^3 + x^4 + (4*x + 4*E^4*x + E^8*x)*Log[5]),x]
Output:
(x - Log[-((x^2 - x^3 - (2 + E^4)^2*Log[5])/x^2)])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(22)=44\).
Time = 293.46 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78
method | result | size |
norman | \(x^{2}+{\ln \left (\frac {\left ({\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4\right ) \ln \left (5\right )+x^{3}-x^{2}}{x^{2}}\right )}^{2}-2 x \ln \left (\frac {\left ({\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4\right ) \ln \left (5\right )+x^{3}-x^{2}}{x^{2}}\right )\) | \(64\) |
parallelrisch | \(-1+x^{2}-2 x \ln \left (\frac {\left ({\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4\right ) \ln \left (5\right )+x^{3}-x^{2}}{x^{2}}\right )+{\ln \left (\frac {\left ({\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4\right ) \ln \left (5\right )+x^{3}-x^{2}}{x^{2}}\right )}^{2}\) | \(65\) |
default | \(\text {Expression too large to display}\) | \(1484\) |
parts | \(\text {Expression too large to display}\) | \(1484\) |
risch | \(\text {Expression too large to display}\) | \(3064\) |
Input:
int(((((-2*x-4)*exp(4)^2+(-8*x-16)*exp(4)-8*x-16)*ln(5)-2*x^4+4*x^3)*ln((( exp(4)^2+4*exp(4)+4)*ln(5)+x^3-x^2)/x^2)+((2*x^2+4*x)*exp(4)^2+(8*x^2+16*x )*exp(4)+8*x^2+16*x)*ln(5)+2*x^5-4*x^4)/((x*exp(4)^2+4*x*exp(4)+4*x)*ln(5) +x^4-x^3),x,method=_RETURNVERBOSE)
Output:
x^2+ln(((exp(4)^2+4*exp(4)+4)*ln(5)+x^3-x^2)/x^2)^2-2*x*ln(((exp(4)^2+4*ex p(4)+4)*ln(5)+x^3-x^2)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=x^{2} - 2 \, x \log \left (\frac {x^{3} - x^{2} + {\left (e^{8} + 4 \, e^{4} + 4\right )} \log \left (5\right )}{x^{2}}\right ) + \log \left (\frac {x^{3} - x^{2} + {\left (e^{8} + 4 \, e^{4} + 4\right )} \log \left (5\right )}{x^{2}}\right )^{2} \] Input:
integrate(((((-2*x-4)*exp(4)^2+(-8*x-16)*exp(4)-8*x-16)*log(5)-2*x^4+4*x^3 )*log(((exp(4)^2+4*exp(4)+4)*log(5)+x^3-x^2)/x^2)+((2*x^2+4*x)*exp(4)^2+(8 *x^2+16*x)*exp(4)+8*x^2+16*x)*log(5)+2*x^5-4*x^4)/((x*exp(4)^2+4*x*exp(4)+ 4*x)*log(5)+x^4-x^3),x, algorithm="fricas")
Output:
x^2 - 2*x*log((x^3 - x^2 + (e^8 + 4*e^4 + 4)*log(5))/x^2) + log((x^3 - x^2 + (e^8 + 4*e^4 + 4)*log(5))/x^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=x^{2} - 2 x \log {\left (\frac {x^{3} - x^{2} + \left (4 + 4 e^{4} + e^{8}\right ) \log {\left (5 \right )}}{x^{2}} \right )} + \log {\left (\frac {x^{3} - x^{2} + \left (4 + 4 e^{4} + e^{8}\right ) \log {\left (5 \right )}}{x^{2}} \right )}^{2} \] Input:
integrate(((((-2*x-4)*exp(4)**2+(-8*x-16)*exp(4)-8*x-16)*ln(5)-2*x**4+4*x* *3)*ln(((exp(4)**2+4*exp(4)+4)*ln(5)+x**3-x**2)/x**2)+((2*x**2+4*x)*exp(4) **2+(8*x**2+16*x)*exp(4)+8*x**2+16*x)*ln(5)+2*x**5-4*x**4)/((x*exp(4)**2+4 *x*exp(4)+4*x)*ln(5)+x**4-x**3),x)
Output:
x**2 - 2*x*log((x**3 - x**2 + (4 + 4*exp(4) + exp(8))*log(5))/x**2) + log( (x**3 - x**2 + (4 + 4*exp(4) + exp(8))*log(5))/x**2)**2
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=x^{2} - 2 \, {\left (x + 2 \, \log \left (x\right )\right )} \log \left (x^{3} - x^{2} + {\left (e^{8} + 4 \, e^{4} + 4\right )} \log \left (5\right )\right ) + \log \left (x^{3} - x^{2} + {\left (e^{8} + 4 \, e^{4} + 4\right )} \log \left (5\right )\right )^{2} + 4 \, x \log \left (x\right ) + 4 \, \log \left (x\right )^{2} \] Input:
integrate(((((-2*x-4)*exp(4)^2+(-8*x-16)*exp(4)-8*x-16)*log(5)-2*x^4+4*x^3 )*log(((exp(4)^2+4*exp(4)+4)*log(5)+x^3-x^2)/x^2)+((2*x^2+4*x)*exp(4)^2+(8 *x^2+16*x)*exp(4)+8*x^2+16*x)*log(5)+2*x^5-4*x^4)/((x*exp(4)^2+4*x*exp(4)+ 4*x)*log(5)+x^4-x^3),x, algorithm="maxima")
Output:
x^2 - 2*(x + 2*log(x))*log(x^3 - x^2 + (e^8 + 4*e^4 + 4)*log(5)) + log(x^3 - x^2 + (e^8 + 4*e^4 + 4)*log(5))^2 + 4*x*log(x) + 4*log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (22) = 44\).
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=x^{2} - 2 \, x \log \left (x^{3} - x^{2} + e^{8} \log \left (5\right ) + 4 \, e^{4} \log \left (5\right ) + 4 \, \log \left (5\right )\right ) + 4 \, x \log \left (x\right ) - 4 \, \log \left (x^{3} - x^{2} + e^{8} \log \left (5\right ) + 4 \, e^{4} \log \left (5\right ) + 4 \, \log \left (5\right )\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2} \] Input:
integrate(((((-2*x-4)*exp(4)^2+(-8*x-16)*exp(4)-8*x-16)*log(5)-2*x^4+4*x^3 )*log(((exp(4)^2+4*exp(4)+4)*log(5)+x^3-x^2)/x^2)+((2*x^2+4*x)*exp(4)^2+(8 *x^2+16*x)*exp(4)+8*x^2+16*x)*log(5)+2*x^5-4*x^4)/((x*exp(4)^2+4*x*exp(4)+ 4*x)*log(5)+x^4-x^3),x, algorithm="giac")
Output:
x^2 - 2*x*log(x^3 - x^2 + e^8*log(5) + 4*e^4*log(5) + 4*log(5)) + 4*x*log( x) - 4*log(x^3 - x^2 + e^8*log(5) + 4*e^4*log(5) + 4*log(5))*log(x) + 4*lo g(x)^2
Time = 4.62 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx={\left (x-\ln \left (\frac {x^3-x^2+\ln \left (5\right )\,\left (4\,{\mathrm {e}}^4+{\mathrm {e}}^8+4\right )}{x^2}\right )\right )}^2 \] Input:
int(-(log((log(5)*(4*exp(4) + exp(8) + 4) - x^2 + x^3)/x^2)*(2*x^4 - 4*x^3 + log(5)*(8*x + exp(8)*(2*x + 4) + exp(4)*(8*x + 16) + 16)) - log(5)*(16* x + exp(8)*(4*x + 2*x^2) + exp(4)*(16*x + 8*x^2) + 8*x^2) + 4*x^4 - 2*x^5) /(log(5)*(4*x + 4*x*exp(4) + x*exp(8)) - x^3 + x^4),x)
Output:
(x - log((log(5)*(4*exp(4) + exp(8) + 4) - x^2 + x^3)/x^2))^2
Time = 0.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 5.96 \[ \int \frac {-4 x^4+2 x^5+\left (16 x+8 x^2+e^8 \left (4 x+2 x^2\right )+e^4 \left (16 x+8 x^2\right )\right ) \log (5)+\left (4 x^3-2 x^4+\left (-16+e^4 (-16-8 x)+e^8 (-4-2 x)-8 x\right ) \log (5)\right ) \log \left (\frac {-x^2+x^3+\left (4+4 e^4+e^8\right ) \log (5)}{x^2}\right )}{-x^3+x^4+\left (4 x+4 e^4 x+e^8 x\right ) \log (5)} \, dx=-2 \,\mathrm {log}\left (\mathrm {log}\left (5\right ) e^{8}+4 \,\mathrm {log}\left (5\right ) e^{4}+4 \,\mathrm {log}\left (5\right )+x^{3}-x^{2}\right )+\mathrm {log}\left (\frac {\mathrm {log}\left (5\right ) e^{8}+4 \,\mathrm {log}\left (5\right ) e^{4}+4 \,\mathrm {log}\left (5\right )+x^{3}-x^{2}}{x^{2}}\right )^{2}-2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (5\right ) e^{8}+4 \,\mathrm {log}\left (5\right ) e^{4}+4 \,\mathrm {log}\left (5\right )+x^{3}-x^{2}}{x^{2}}\right ) x +2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (5\right ) e^{8}+4 \,\mathrm {log}\left (5\right ) e^{4}+4 \,\mathrm {log}\left (5\right )+x^{3}-x^{2}}{x^{2}}\right )+4 \,\mathrm {log}\left (x \right )+x^{2} \] Input:
int(((((-2*x-4)*exp(4)^2+(-8*x-16)*exp(4)-8*x-16)*log(5)-2*x^4+4*x^3)*log( ((exp(4)^2+4*exp(4)+4)*log(5)+x^3-x^2)/x^2)+((2*x^2+4*x)*exp(4)^2+(8*x^2+1 6*x)*exp(4)+8*x^2+16*x)*log(5)+2*x^5-4*x^4)/((x*exp(4)^2+4*x*exp(4)+4*x)*l og(5)+x^4-x^3),x)
Output:
- 2*log(log(5)*e**8 + 4*log(5)*e**4 + 4*log(5) + x**3 - x**2) + log((log( 5)*e**8 + 4*log(5)*e**4 + 4*log(5) + x**3 - x**2)/x**2)**2 - 2*log((log(5) *e**8 + 4*log(5)*e**4 + 4*log(5) + x**3 - x**2)/x**2)*x + 2*log((log(5)*e* *8 + 4*log(5)*e**4 + 4*log(5) + x**3 - x**2)/x**2) + 4*log(x) + x**2