Integrand size = 154, antiderivative size = 28 \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=e^{-x} \log ^2\left (\left (3+\frac {\left (-1+\frac {e^2}{3}\right )^2}{x}\right )^2\right ) \] Output:
ln((3+1/x*(1/3*exp(2)-1)^2)^2)^2/exp(x)
\[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=\int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx \] Input:
Integrate[((-36 + 24*E^2 - 4*E^4)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324 *x) + 486*x + 729*x^2 + E^4*(54 + 54*x))/(81*x^2)] + (-9*x + 6*E^2*x - E^4 *x - 27*x^2)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4*(54 + 54*x))/(81*x^2)]^2)/(E^x*(9*x - 6*E^2*x + E^4*x + 27*x^2)),x]
Output:
Integrate[((-36 + 24*E^2 - 4*E^4)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324 *x) + 486*x + 729*x^2 + E^4*(54 + 54*x))/(81*x^2)] + (-9*x + 6*E^2*x - E^4 *x - 27*x^2)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4*(54 + 54*x))/(81*x^2)]^2)/(E^x*(9*x - 6*E^2*x + E^4*x + 27*x^2)), x ]
Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(28)=56\).
Time = 0.49 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6, 6, 2026, 2726}
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.
\(\displaystyle \int \frac {e^{-x} \left (\left (-27 x^2-e^4 x+6 e^2 x-9 x\right ) \log ^2\left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )+\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )\right )}{27 x^2+e^4 x-6 e^2 x+9 x} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{-x} \left (\left (-27 x^2-e^4 x+6 e^2 x-9 x\right ) \log ^2\left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )+\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )\right )}{27 x^2+\left (9-6 e^2\right ) x+e^4 x}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{-x} \left (\left (-27 x^2-e^4 x+6 e^2 x-9 x\right ) \log ^2\left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )+\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )\right )}{27 x^2+\left (9-6 e^2+e^4\right ) x}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{-x} \left (\left (-27 x^2-e^4 x+6 e^2 x-9 x\right ) \log ^2\left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )+\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {729 x^2+486 x+e^2 (-324 x-108)+e^4 (54 x+54)+e^8-12 e^6+81}{81 x^2}\right )\right )}{x \left (27 x+\left (e^2-3\right )^2\right )}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {e^{-x} \left (27 x^2+e^4 x-6 e^2 x+9 x\right ) \log ^2\left (\frac {729 x^2+486 x+54 e^4 (x+1)-108 e^2 (3 x+1)+e^8-12 e^6+81}{81 x^2}\right )}{x \left (27 x+\left (e^2-3\right )^2\right )}\) |
Input:
Int[((-36 + 24*E^2 - 4*E^4)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4*(54 + 54*x))/(81*x^2)] + (-9*x + 6*E^2*x - E^4*x - 2 7*x^2)*Log[(81 - 12*E^6 + E^8 + E^2*(-108 - 324*x) + 486*x + 729*x^2 + E^4 *(54 + 54*x))/(81*x^2)]^2)/(E^x*(9*x - 6*E^2*x + E^4*x + 27*x^2)),x]
Output:
((9*x - 6*E^2*x + E^4*x + 27*x^2)*Log[(81 - 12*E^6 + E^8 + 486*x + 729*x^2 + 54*E^4*(1 + x) - 108*E^2*(1 + 3*x))/(81*x^2)]^2)/(E^x*x*((-3 + E^2)^2 + 27*x))
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[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).
Time = 5.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
method | result | size |
norman | \(\ln \left (\frac {{\mathrm e}^{8}-12 \,{\mathrm e}^{6}+\left (54 x +54\right ) {\mathrm e}^{4}+\left (-324 x -108\right ) {\mathrm e}^{2}+729 x^{2}+486 x +81}{81 x^{2}}\right )^{2} {\mathrm e}^{-x}\) | \(52\) |
parallelrisch | \(\ln \left (\frac {{\mathrm e}^{8}-12 \,{\mathrm e}^{6}+\left (54 x +54\right ) {\mathrm e}^{4}+\left (-324 x -108\right ) {\mathrm e}^{2}+729 x^{2}+486 x +81}{81 x^{2}}\right )^{2} {\mathrm e}^{-x}\) | \(52\) |
risch | \(\text {Expression too large to display}\) | \(3672\) |
Input:
int(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*ln(1/81*(exp(2)^4-12*exp(2)^3+(54 *x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*exp(2)^2+ 24*exp(2)-36)*ln(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108 )*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x)/exp(x) ,x,method=_RETURNVERBOSE)
Output:
ln(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x ^2+486*x+81)/x^2)^2/exp(x)
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=e^{\left (-x\right )} \log \left (\frac {729 \, x^{2} + 54 \, {\left (x + 1\right )} e^{4} - 108 \, {\left (3 \, x + 1\right )} e^{2} + 486 \, x + e^{8} - 12 \, e^{6} + 81}{81 \, x^{2}}\right )^{2} \] Input:
integrate(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2 )^3+(54*x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*ex p(2)^2+24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-3 24*x-108)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x )/exp(x),x, algorithm="fricas")
Output:
e^(-x)*log(1/81*(729*x^2 + 54*(x + 1)*e^4 - 108*(3*x + 1)*e^2 + 486*x + e^ 8 - 12*e^6 + 81)/x^2)^2
Timed out. \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=\text {Timed out} \] Input:
integrate(((-x*exp(2)**2+6*exp(2)*x-27*x**2-9*x)*ln(1/81*(exp(2)**4-12*exp (2)**3+(54*x+54)*exp(2)**2+(-324*x-108)*exp(2)+729*x**2+486*x+81)/x**2)**2 +(-4*exp(2)**2+24*exp(2)-36)*ln(1/81*(exp(2)**4-12*exp(2)**3+(54*x+54)*exp (2)**2+(-324*x-108)*exp(2)+729*x**2+486*x+81)/x**2))/(x*exp(2)**2-6*exp(2) *x+27*x**2+9*x)/exp(x),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=4 \, {\left (4 \, \log \left (3\right )^{2} - 2 \, {\left (2 \, \log \left (3\right ) + \log \left (x\right )\right )} \log \left (27 \, x + e^{4} - 6 \, e^{2} + 9\right ) + \log \left (27 \, x + e^{4} - 6 \, e^{2} + 9\right )^{2} + 4 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (-x\right )} \] Input:
integrate(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2 )^3+(54*x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*ex p(2)^2+24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-3 24*x-108)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x )/exp(x),x, algorithm="maxima")
Output:
4*(4*log(3)^2 - 2*(2*log(3) + log(x))*log(27*x + e^4 - 6*e^2 + 9) + log(27 *x + e^4 - 6*e^2 + 9)^2 + 4*log(3)*log(x) + log(x)^2)*e^(-x)
Time = 1.35 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=e^{\left (-x\right )} \log \left (\frac {729 \, x^{2} + 54 \, x e^{4} - 324 \, x e^{2} + 486 \, x + e^{8} - 12 \, e^{6} + 54 \, e^{4} - 108 \, e^{2} + 81}{81 \, x^{2}}\right )^{2} \] Input:
integrate(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2 )^3+(54*x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*ex p(2)^2+24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-3 24*x-108)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x )/exp(x),x, algorithm="giac")
Output:
e^(-x)*log(1/81*(729*x^2 + 54*x*e^4 - 324*x*e^2 + 486*x + e^8 - 12*e^6 + 5 4*e^4 - 108*e^2 + 81)/x^2)^2
Time = 3.79 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx={\mathrm {e}}^{-x}\,{\ln \left (\frac {6\,x-\frac {4\,{\mathrm {e}}^6}{27}+\frac {{\mathrm {e}}^8}{81}+9\,x^2+\frac {{\mathrm {e}}^4\,\left (54\,x+54\right )}{81}-\frac {{\mathrm {e}}^2\,\left (324\,x+108\right )}{81}+1}{x^2}\right )}^2 \] Input:
int(-(exp(-x)*(log((6*x - (4*exp(6))/27 + exp(8)/81 + 9*x^2 + (exp(4)*(54* x + 54))/81 - (exp(2)*(324*x + 108))/81 + 1)/x^2)^2*(9*x - 6*x*exp(2) + x* exp(4) + 27*x^2) + log((6*x - (4*exp(6))/27 + exp(8)/81 + 9*x^2 + (exp(4)* (54*x + 54))/81 - (exp(2)*(324*x + 108))/81 + 1)/x^2)*(4*exp(4) - 24*exp(2 ) + 36)))/(9*x - 6*x*exp(2) + x*exp(4) + 27*x^2),x)
Output:
exp(-x)*log((6*x - (4*exp(6))/27 + exp(8)/81 + 9*x^2 + (exp(4)*(54*x + 54) )/81 - (exp(2)*(324*x + 108))/81 + 1)/x^2)^2
Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {e^{-x} \left (\left (-36+24 e^2-4 e^4\right ) \log \left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )+\left (-9 x+6 e^2 x-e^4 x-27 x^2\right ) \log ^2\left (\frac {81-12 e^6+e^8+e^2 (-108-324 x)+486 x+729 x^2+e^4 (54+54 x)}{81 x^2}\right )\right )}{9 x-6 e^2 x+e^4 x+27 x^2} \, dx=\frac {\mathrm {log}\left (\frac {e^{8}-12 e^{6}+54 e^{4} x +54 e^{4}-324 e^{2} x -108 e^{2}+729 x^{2}+486 x +81}{81 x^{2}}\right )^{2}}{e^{x}} \] Input:
int(((-x*exp(2)^2+6*exp(2)*x-27*x^2-9*x)*log(1/81*(exp(2)^4-12*exp(2)^3+(5 4*x+54)*exp(2)^2+(-324*x-108)*exp(2)+729*x^2+486*x+81)/x^2)^2+(-4*exp(2)^2 +24*exp(2)-36)*log(1/81*(exp(2)^4-12*exp(2)^3+(54*x+54)*exp(2)^2+(-324*x-1 08)*exp(2)+729*x^2+486*x+81)/x^2))/(x*exp(2)^2-6*exp(2)*x+27*x^2+9*x)/exp( x),x)
Output:
log((e**8 - 12*e**6 + 54*e**4*x + 54*e**4 - 324*e**2*x - 108*e**2 + 729*x* *2 + 486*x + 81)/(81*x**2))**2/e**x