Integrand size = 166, antiderivative size = 36 \[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=4 e^{2 \left (e^{3 x}-x^2\right )^2} \log ^2\left (\frac {3+x^2}{x}-\log (2)\right ) \]
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=4 e^{2 \left (e^{3 x}-x^2\right )^2} \log ^2\left (\frac {3}{x}+x-\log (2)\right ) \]
Integrate[(E^(2*E^(6*x) - 4*E^(3*x)*x^2 + 2*x^4)*((24 - 8*x^2)*Log[(3 + x^ 2 - x*Log[2])/x] + (-96*x^4 - 32*x^6 + 32*x^5*Log[2] + E^(6*x)*(-144*x - 4 8*x^3 + 48*x^2*Log[2]) + E^(3*x)*(96*x^2 + 144*x^3 + 32*x^4 + 48*x^5 + (-3 2*x^3 - 48*x^4)*Log[2]))*Log[(3 + x^2 - x*Log[2])/x]^2))/(-3*x - x^3 + x^2 *Log[2]),x]
Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(36)=72\).
Time = 1.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 4.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {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.
| \(\displaystyle \int \frac {e^{2 x^4-4 e^{3 x} x^2+2 e^{6 x}} \left (\left (24-8 x^2\right ) \log \left (\frac {x^2-x \log (2)+3}{x}\right )+\left (-32 x^6+32 x^5 \log (2)-96 x^4+e^{6 x} \left (-48 x^3+48 x^2 \log (2)-144 x\right )+e^{3 x} \left (48 x^5+32 x^4+144 x^3+96 x^2+\left (-48 x^4-32 x^3\right ) \log (2)\right )\right ) \log ^2\left (\frac {x^2-x \log (2)+3}{x}\right )\right )}{-x^3+x^2 \log (2)-3 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{2 x^4-4 e^{3 x} x^2+2 e^{6 x}} \left (\left (24-8 x^2\right ) \log \left (\frac {x^2-x \log (2)+3}{x}\right )+\left (-32 x^6+32 x^5 \log (2)-96 x^4+e^{6 x} \left (-48 x^3+48 x^2 \log (2)-144 x\right )+e^{3 x} \left (48 x^5+32 x^4+144 x^3+96 x^2+\left (-48 x^4-32 x^3\right ) \log (2)\right )\right ) \log ^2\left (\frac {x^2-x \log (2)+3}{x}\right )\right )}{x \left (-x^2+x \log (2)-3\right )}dx\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {4 e^{2 x^4-4 e^{3 x} x^2+2 e^{6 x}} \left (2 x^6-2 x^5 \log (2)+6 x^4+3 e^{6 x} \left (x^3-x^2 \log (2)+3 x\right )-e^{3 x} \left (3 x^5+2 x^4+9 x^3+6 x^2-\left (3 x^4+2 x^3\right ) \log (2)\right )\right ) \log ^2\left (\frac {x^2-x \log (2)+3}{x}\right )}{x \left (2 x^3-3 e^{3 x} x^2-2 e^{3 x} x+3 e^{6 x}\right ) \left (x^2-x \log (2)+3\right )}\) |
Int[(E^(2*E^(6*x) - 4*E^(3*x)*x^2 + 2*x^4)*((24 - 8*x^2)*Log[(3 + x^2 - x* Log[2])/x] + (-96*x^4 - 32*x^6 + 32*x^5*Log[2] + E^(6*x)*(-144*x - 48*x^3 + 48*x^2*Log[2]) + E^(3*x)*(96*x^2 + 144*x^3 + 32*x^4 + 48*x^5 + (-32*x^3 - 48*x^4)*Log[2]))*Log[(3 + x^2 - x*Log[2])/x]^2))/(-3*x - x^3 + x^2*Log[2 ]),x]
(4*E^(2*E^(6*x) - 4*E^(3*x)*x^2 + 2*x^4)*(6*x^4 + 2*x^6 - 2*x^5*Log[2] + 3 *E^(6*x)*(3*x + x^3 - x^2*Log[2]) - E^(3*x)*(6*x^2 + 9*x^3 + 2*x^4 + 3*x^5 - (2*x^3 + 3*x^4)*Log[2]))*Log[(3 + x^2 - x*Log[2])/x]^2)/(x*(3*E^(6*x) - 2*E^(3*x)*x - 3*E^(3*x)*x^2 + 2*x^3)*(3 + x^2 - x*Log[2]))
3.16.84.3.1 Defintions of rubi rules used
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]
Time = 268.58 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22
| method | result | size |
| parallelrisch | \(4 \,{\mathrm e}^{2 \,{\mathrm e}^{6 x}-4 x^{2} {\mathrm e}^{3 x}+2 x^{4}} \ln \left (-\frac {x \ln \left (2\right )-x^{2}-3}{x}\right )^{2}\) | \(44\) |
| risch | \(\text {Expression too large to display}\) | \(1142\) |
int((((48*x^2*ln(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*ln(2)+48*x^ 5+32*x^4+144*x^3+96*x^2)*exp(3*x)+32*x^5*ln(2)-32*x^6-96*x^4)*ln((-x*ln(2) +x^2+3)/x)^2+(-8*x^2+24)*ln((-x*ln(2)+x^2+3)/x))*exp(exp(3*x)^2-2*x^2*exp( 3*x)+x^4)^2/(x^2*ln(2)-x^3-3*x),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=4 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (\frac {x^{2} - x \log \left (2\right ) + 3}{x}\right )^{2} \]
integrate((((48*x^2*log(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*log( 2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp(3*x)+32*x^5*log(2)-32*x^6-96*x^4)*log ((-x*log(2)+x^2+3)/x)^2+(-8*x^2+24)*log((-x*log(2)+x^2+3)/x))*exp(exp(3*x) ^2-2*x^2*exp(3*x)+x^4)^2/(x^2*log(2)-x^3-3*x),x, algorithm=\
Timed out. \[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=\text {Timed out} \]
integrate((((48*x**2*ln(2)-48*x**3-144*x)*exp(3*x)**2+((-48*x**4-32*x**3)* ln(2)+48*x**5+32*x**4+144*x**3+96*x**2)*exp(3*x)+32*x**5*ln(2)-32*x**6-96* x**4)*ln((-x*ln(2)+x**2+3)/x)**2+(-8*x**2+24)*ln((-x*ln(2)+x**2+3)/x))*exp (exp(3*x)**2-2*x**2*exp(3*x)+x**4)**2/(x**2*ln(2)-x**3-3*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (34) = 68\).
Time = 0.47 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86 \[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=4 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (x^{2} - x \log \left (2\right ) + 3\right )^{2} - 8 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (x^{2} - x \log \left (2\right ) + 3\right ) \log \left (x\right ) + 4 \, e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )} \log \left (x\right )^{2} \]
integrate((((48*x^2*log(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*log( 2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp(3*x)+32*x^5*log(2)-32*x^6-96*x^4)*log ((-x*log(2)+x^2+3)/x)^2+(-8*x^2+24)*log((-x*log(2)+x^2+3)/x))*exp(exp(3*x) ^2-2*x^2*exp(3*x)+x^4)^2/(x^2*log(2)-x^3-3*x),x, algorithm=\
4*e^(2*x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))*log(x^2 - x*log(2) + 3)^2 - 8*e^(2 *x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))*log(x^2 - x*log(2) + 3)*log(x) + 4*e^(2* x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))*log(x)^2
\[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=\int { \frac {8 \, {\left (2 \, {\left (2 \, x^{6} - 2 \, x^{5} \log \left (2\right ) + 6 \, x^{4} + 3 \, {\left (x^{3} - x^{2} \log \left (2\right ) + 3 \, x\right )} e^{\left (6 \, x\right )} - {\left (3 \, x^{5} + 2 \, x^{4} + 9 \, x^{3} + 6 \, x^{2} - {\left (3 \, x^{4} + 2 \, x^{3}\right )} \log \left (2\right )\right )} e^{\left (3 \, x\right )}\right )} \log \left (\frac {x^{2} - x \log \left (2\right ) + 3}{x}\right )^{2} + {\left (x^{2} - 3\right )} \log \left (\frac {x^{2} - x \log \left (2\right ) + 3}{x}\right )\right )} e^{\left (2 \, x^{4} - 4 \, x^{2} e^{\left (3 \, x\right )} + 2 \, e^{\left (6 \, x\right )}\right )}}{x^{3} - x^{2} \log \left (2\right ) + 3 \, x} \,d x } \]
integrate((((48*x^2*log(2)-48*x^3-144*x)*exp(3*x)^2+((-48*x^4-32*x^3)*log( 2)+48*x^5+32*x^4+144*x^3+96*x^2)*exp(3*x)+32*x^5*log(2)-32*x^6-96*x^4)*log ((-x*log(2)+x^2+3)/x)^2+(-8*x^2+24)*log((-x*log(2)+x^2+3)/x))*exp(exp(3*x) ^2-2*x^2*exp(3*x)+x^4)^2/(x^2*log(2)-x^3-3*x),x, algorithm=\
integrate(8*(2*(2*x^6 - 2*x^5*log(2) + 6*x^4 + 3*(x^3 - x^2*log(2) + 3*x)* e^(6*x) - (3*x^5 + 2*x^4 + 9*x^3 + 6*x^2 - (3*x^4 + 2*x^3)*log(2))*e^(3*x) )*log((x^2 - x*log(2) + 3)/x)^2 + (x^2 - 3)*log((x^2 - x*log(2) + 3)/x))*e ^(2*x^4 - 4*x^2*e^(3*x) + 2*e^(6*x))/(x^3 - x^2*log(2) + 3*x), x)
Timed out. \[ \int \frac {e^{2 e^{6 x}-4 e^{3 x} x^2+2 x^4} \left (\left (24-8 x^2\right ) \log \left (\frac {3+x^2-x \log (2)}{x}\right )+\left (-96 x^4-32 x^6+32 x^5 \log (2)+e^{6 x} \left (-144 x-48 x^3+48 x^2 \log (2)\right )+e^{3 x} \left (96 x^2+144 x^3+32 x^4+48 x^5+\left (-32 x^3-48 x^4\right ) \log (2)\right )\right ) \log ^2\left (\frac {3+x^2-x \log (2)}{x}\right )\right )}{-3 x-x^3+x^2 \log (2)} \, dx=\int \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{6\,x}-4\,x^2\,{\mathrm {e}}^{3\,x}+2\,x^4}\,\left (\left ({\mathrm {e}}^{6\,x}\,\left (48\,x^3-48\,\ln \left (2\right )\,x^2+144\,x\right )-32\,x^5\,\ln \left (2\right )-{\mathrm {e}}^{3\,x}\,\left (96\,x^2-\ln \left (2\right )\,\left (48\,x^4+32\,x^3\right )+144\,x^3+32\,x^4+48\,x^5\right )+96\,x^4+32\,x^6\right )\,{\ln \left (\frac {x^2-\ln \left (2\right )\,x+3}{x}\right )}^2+\left (8\,x^2-24\right )\,\ln \left (\frac {x^2-\ln \left (2\right )\,x+3}{x}\right )\right )}{x^3-\ln \left (2\right )\,x^2+3\,x} \,d x \]
int((exp(2*exp(6*x) - 4*x^2*exp(3*x) + 2*x^4)*(log((x^2 - x*log(2) + 3)/x) *(8*x^2 - 24) + log((x^2 - x*log(2) + 3)/x)^2*(exp(6*x)*(144*x - 48*x^2*lo g(2) + 48*x^3) - 32*x^5*log(2) - exp(3*x)*(96*x^2 - log(2)*(32*x^3 + 48*x^ 4) + 144*x^3 + 32*x^4 + 48*x^5) + 96*x^4 + 32*x^6)))/(3*x - x^2*log(2) + x ^3),x)
int((exp(2*exp(6*x) - 4*x^2*exp(3*x) + 2*x^4)*(log((x^2 - x*log(2) + 3)/x) *(8*x^2 - 24) + log((x^2 - x*log(2) + 3)/x)^2*(exp(6*x)*(144*x - 48*x^2*lo g(2) + 48*x^3) - 32*x^5*log(2) - exp(3*x)*(96*x^2 - log(2)*(32*x^3 + 48*x^ 4) + 144*x^3 + 32*x^4 + 48*x^5) + 96*x^4 + 32*x^6)))/(3*x - x^2*log(2) + x ^3), x)