Integrand size = 149, antiderivative size = 28 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=5+x \left (\frac {4}{x-\left (4+e^x-x\right ) x}+\log \left (-2+x^2\right )\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 2.36 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=-\frac {4}{3+e^x-x}+2 \sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )+\sqrt {2} \log \left (\sqrt {2}-x\right )-\sqrt {2} \log \left (\sqrt {2}+x\right )+x \log \left (-2+x^2\right ) \]
Integrate[(8 + 14*x^2 + 2*E^(2*x)*x^2 - 12*x^3 + 2*x^4 + E^x*(-8 + 16*x^2 - 4*x^3) + (-18 + 12*x + 7*x^2 - 6*x^3 + x^4 + E^(2*x)*(-2 + x^2) + E^x*(- 12 + 4*x + 6*x^2 - 2*x^3))*Log[-2 + x^2])/(-18 + 12*x + 7*x^2 - 6*x^3 + x^ 4 + E^(2*x)*(-2 + x^2) + E^x*(-12 + 4*x + 6*x^2 - 2*x^3)),x]
-4/(3 + E^x - x) + 2*Sqrt[2]*ArcTanh[x/Sqrt[2]] + Sqrt[2]*Log[Sqrt[2] - x] - Sqrt[2]*Log[Sqrt[2] + x] + x*Log[-2 + x^2]
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.
\(\displaystyle \int \frac {2 x^4-12 x^3+2 e^{2 x} x^2+14 x^2+e^x \left (-4 x^3+16 x^2-8\right )+\left (x^4-6 x^3+7 x^2+e^{2 x} \left (x^2-2\right )+e^x \left (-2 x^3+6 x^2+4 x-12\right )+12 x-18\right ) \log \left (x^2-2\right )+8}{x^4-6 x^3+7 x^2+e^{2 x} \left (x^2-2\right )+e^x \left (-2 x^3+6 x^2+4 x-12\right )+12 x-18} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\log \left (x^2-2\right )+\frac {2 \left (x^4-6 x^3+e^{2 x} x^2+7 x^2-2 e^x \left (x^3-4 x^2+2\right )+4\right )}{\left (-x+e^x+3\right )^2 \left (x^2-2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -16 \int \frac {1}{\left (-x+e^x+3\right )^2}dx+4 \int \frac {1}{-x+e^x+3}dx+4 \int \frac {x}{\left (-x+e^x+3\right )^2}dx+x \log \left (x^2-2\right )\) |
Int[(8 + 14*x^2 + 2*E^(2*x)*x^2 - 12*x^3 + 2*x^4 + E^x*(-8 + 16*x^2 - 4*x^ 3) + (-18 + 12*x + 7*x^2 - 6*x^3 + x^4 + E^(2*x)*(-2 + x^2) + E^x*(-12 + 4 *x + 6*x^2 - 2*x^3))*Log[-2 + x^2])/(-18 + 12*x + 7*x^2 - 6*x^3 + x^4 + E^ (2*x)*(-2 + x^2) + E^x*(-12 + 4*x + 6*x^2 - 2*x^3)),x]
3.12.53.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.67 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\ln \left (x^{2}-2\right ) x +\frac {4}{-{\mathrm e}^{x}+x -3}\) | \(21\) |
parallelrisch | \(\frac {\ln \left (x^{2}-2\right ) x^{2}-{\mathrm e}^{x} \ln \left (x^{2}-2\right ) x -3 \ln \left (x^{2}-2\right ) x +4}{-{\mathrm e}^{x}+x -3}\) | \(43\) |
int((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-1 8)*ln(x^2-2)+2*exp(x)^2*x^2+(-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+14*x^2+8 )/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12*x-18), x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=\frac {{\left (x^{2} - x e^{x} - 3 \, x\right )} \log \left (x^{2} - 2\right ) + 4}{x - e^{x} - 3} \]
integrate((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+ 12*x-18)*log(x^2-2)+2*exp(x)^2*x^2+(-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+1 4*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12 *x-18),x, algorithm=\
Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=x \log {\left (x^{2} - 2 \right )} - \frac {4}{- x + e^{x} + 3} \]
integrate((((x**2-2)*exp(x)**2+(-2*x**3+6*x**2+4*x-12)*exp(x)+x**4-6*x**3+ 7*x**2+12*x-18)*ln(x**2-2)+2*exp(x)**2*x**2+(-4*x**3+16*x**2-8)*exp(x)+2*x **4-12*x**3+14*x**2+8)/((x**2-2)*exp(x)**2+(-2*x**3+6*x**2+4*x-12)*exp(x)+ x**4-6*x**3+7*x**2+12*x-18),x)
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=\frac {{\left (x^{2} - x e^{x} - 3 \, x\right )} \log \left (x^{2} - 2\right ) + 4}{x - e^{x} - 3} \]
integrate((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+ 12*x-18)*log(x^2-2)+2*exp(x)^2*x^2+(-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+1 4*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12 *x-18),x, algorithm=\
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=\frac {x^{2} \log \left (x^{2} - 2\right ) - x e^{x} \log \left (x^{2} - 2\right ) - 3 \, x \log \left (x^{2} - 2\right ) + 4}{x - e^{x} - 3} \]
integrate((((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+ 12*x-18)*log(x^2-2)+2*exp(x)^2*x^2+(-4*x^3+16*x^2-8)*exp(x)+2*x^4-12*x^3+1 4*x^2+8)/((x^2-2)*exp(x)^2+(-2*x^3+6*x^2+4*x-12)*exp(x)+x^4-6*x^3+7*x^2+12 *x-18),x, algorithm=\
Time = 10.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {8+14 x^2+2 e^{2 x} x^2-12 x^3+2 x^4+e^x \left (-8+16 x^2-4 x^3\right )+\left (-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )\right ) \log \left (-2+x^2\right )}{-18+12 x+7 x^2-6 x^3+x^4+e^{2 x} \left (-2+x^2\right )+e^x \left (-12+4 x+6 x^2-2 x^3\right )} \, dx=x\,\ln \left (x^2-2\right )-\frac {4}{{\mathrm {e}}^x-x+3} \]