Integrand size = 83, antiderivative size = 26 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=\log \left (-4-e^{2 x-2 e^5 x}+\left (e^{4 x}+x\right )^2\right ) \] Output:
ln((exp(4*x)+x)^2-4-exp(x)^2/exp(x*exp(5))^2)
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).
Time = 11.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=-2 e^5 x+\log \left (-e^{2 x}-4 e^{2 e^5 x}+e^{8 x+2 e^5 x}+2 e^{4 x+2 e^5 x} x+e^{2 e^5 x} x^2\right ) \] Input:
Integrate[(E^(2*x)*(-2 + 2*E^5) + E^(2*E^5*x)*(8*E^(8*x) + 2*x + E^(4*x)*( 2 + 8*x)))/(-E^(2*x) + E^(2*E^5*x)*(-4 + E^(8*x) + 2*E^(4*x)*x + x^2)),x]
Output:
-2*E^5*x + Log[-E^(2*x) - 4*E^(2*E^5*x) + E^(8*x + 2*E^5*x) + 2*E^(4*x + 2 *E^5*x)*x + E^(2*E^5*x)*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.
\(\displaystyle \int \frac {e^{2 e^5 x} \left (2 x+8 e^{8 x}+e^{4 x} (8 x+2)\right )+\left (2 e^5-2\right ) e^{2 x}}{e^{2 e^5 x} \left (x^2+2 e^{4 x} x+e^{8 x}-4\right )-e^{2 x}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (4 e^{4 x}+1\right ) \left (x+e^{4 x}\right )}{x^2+2 e^{4 x} x+e^{8 x}-4}+\frac {2 e^{2 x} \left (-\left (\left (1-e^5\right ) x^2\right )+2 \left (1+e^5\right ) e^{4 x} x+x+e^{4 x}+3 \left (1+\frac {e^5}{3}\right ) e^{8 x}+4 \left (1-e^5\right )\right )}{\left (-x^2-2 e^{4 x} x-e^{8 x}+4\right ) \left (-e^{2 e^5 x} x^2-2 e^{2 \left (2+e^5\right ) x} x+e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (-e^{2 e^5 x} x-4 e^{2 \left (4+e^5\right ) x}-e^{2 \left (2+e^5\right ) x} (4 x+1)+\left (1-e^5\right ) e^{2 x}\right )}{-e^{2 e^5 x} \left (x^2-4\right )-2 e^{2 \left (2+e^5\right ) x} x+e^{2 x}-e^{2 \left (4+e^5\right ) x}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {e^{2 e^5 x} x+4 e^{2 \left (4+e^5\right ) x}+e^{2 \left (2+e^5\right ) x} (4 x+1)-e^{2 x} \left (1-e^5\right )}{-2 e^{2 \left (2+e^5\right ) x} x+e^{2 x}-e^{2 \left (4+e^5\right ) x}+e^{2 e^5 x} \left (4-x^2\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {e^{2 e^5 x} x+4 e^{2 \left (4+e^5\right ) x}+e^{2 \left (2+e^5\right ) x} (4 x+1)-e^{2 x} \left (1-e^5\right )}{-2 e^{2 \left (2+e^5\right ) x} x+e^{2 x}-e^{2 \left (4+e^5\right ) x}+e^{2 e^5 x} \left (4-x^2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {-4 e^{2 e^5 x} x^2+e^{2 e^5 x} x-4 e^{2 \left (2+e^5\right ) x} x+16 e^{2 e^5 x}+e^{2 \left (2+e^5\right ) x}+3 e^{2 x} \left (1+\frac {e^5}{3}\right )}{-e^{2 e^5 x} x^2-2 e^{2 \left (2+e^5\right ) x} x+e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}}-4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\left (3+e^5\right ) \int \frac {e^{2 x}}{-e^{2 e^5 x} x^2-2 e^{2 \left (2+e^5\right ) x} x+e^{2 x}+4 e^{2 e^5 x}-e^{2 \left (4+e^5\right ) x}}dx-16 \int \frac {e^{2 e^5 x}}{e^{2 e^5 x} x^2+2 e^{2 \left (2+e^5\right ) x} x-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}}dx-\int \frac {e^{2 \left (2+e^5\right ) x}}{e^{2 e^5 x} x^2+2 e^{2 \left (2+e^5\right ) x} x-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}}dx-\int \frac {e^{2 e^5 x} x}{e^{2 e^5 x} x^2+2 e^{2 \left (2+e^5\right ) x} x-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}}dx+4 \int \frac {e^{2 \left (2+e^5\right ) x} x}{e^{2 e^5 x} x^2+2 e^{2 \left (2+e^5\right ) x} x-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}}dx+4 \int \frac {e^{2 e^5 x} x^2}{e^{2 e^5 x} x^2+2 e^{2 \left (2+e^5\right ) x} x-e^{2 x}-4 e^{2 e^5 x}+e^{2 \left (4+e^5\right ) x}}dx-4 x\right )\) |
Input:
Int[(E^(2*x)*(-2 + 2*E^5) + E^(2*E^5*x)*(8*E^(8*x) + 2*x + E^(4*x)*(2 + 8* x)))/(-E^(2*x) + E^(2*E^5*x)*(-4 + E^(8*x) + 2*E^(4*x)*x + x^2)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 0.76 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15
method | result | size |
risch | \(\ln \left ({\mathrm e}^{8 x}+2 x \,{\mathrm e}^{4 x}+x^{2}-4\right )-2 x \,{\mathrm e}^{5}+\ln \left ({\mathrm e}^{2 x \,{\mathrm e}^{5}}-\frac {{\mathrm e}^{2 x}}{{\mathrm e}^{8 x}+2 x \,{\mathrm e}^{4 x}+x^{2}-4}\right )\) | \(56\) |
parallelrisch | \(\ln \left (2 \,{\mathrm e}^{4 x} {\mathrm e}^{2 x \,{\mathrm e}^{5}} x +{\mathrm e}^{2 x \,{\mathrm e}^{5}} x^{2}+{\mathrm e}^{8 x} {\mathrm e}^{2 x \,{\mathrm e}^{5}}-4 \,{\mathrm e}^{2 x \,{\mathrm e}^{5}}-{\mathrm e}^{2 x}\right )-2 x \,{\mathrm e}^{5}\) | \(63\) |
Input:
int(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp( x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x,method= _RETURNVERBOSE)
Output:
ln(exp(8*x)+2*x*exp(4*x)+x^2-4)-2*x*exp(5)+ln(exp(2*x*exp(5))-exp(2*x)/(ex p(8*x)+2*x*exp(4*x)+x^2-4))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=-2 \, x e^{5} + \log \left (x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4\right ) + \log \left (\frac {{\left (x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4\right )} e^{\left (2 \, x e^{5}\right )} - e^{\left (2 \, x\right )}}{x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4}\right ) \] Input:
integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2 )*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="fricas")
Output:
-2*x*e^5 + log(x^2 + 2*x*e^(4*x) + e^(8*x) - 4) + log(((x^2 + 2*x*e^(4*x) + e^(8*x) - 4)*e^(2*x*e^5) - e^(2*x))/(x^2 + 2*x*e^(4*x) + e^(8*x) - 4))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.67 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=- 2 x e^{5} + \log {\left (x^{2} e^{2 x e^{5}} + 2 x e^{4 x} e^{2 x e^{5}} + e^{8 x} e^{2 x e^{5}} - e^{2 x} - 4 e^{2 x e^{5}} \right )} \] Input:
integrate(((8*exp(4*x)**2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))**2+(2*exp(5) -2)*exp(x)**2)/((exp(4*x)**2+2*x*exp(4*x)+x**2-4)*exp(x*exp(5))**2-exp(x)* *2),x)
Output:
-2*x*exp(5) + log(x**2*exp(2*x*exp(5)) + 2*x*exp(4*x)*exp(2*x*exp(5)) + ex p(8*x)*exp(2*x*exp(5)) - exp(2*x) - 4*exp(2*x*exp(5)))
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=-2 \, x e^{5} + \log \left (x + e^{\left (4 \, x\right )} + 2\right ) + \log \left (x + e^{\left (4 \, x\right )} - 2\right ) + \log \left (\frac {{\left (x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4\right )} e^{\left (2 \, x e^{5}\right )} - e^{\left (2 \, x\right )}}{x^{2} + 2 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 4}\right ) \] Input:
integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2 )*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="maxima")
Output:
-2*x*e^5 + log(x + e^(4*x) + 2) + log(x + e^(4*x) - 2) + log(((x^2 + 2*x*e ^(4*x) + e^(8*x) - 4)*e^(2*x*e^5) - e^(2*x))/(x^2 + 2*x*e^(4*x) + e^(8*x) - 4))
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (23) = 46\).
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=-2 \, x e^{5} + \log \left (4 \, x^{2} e^{\left (2 \, x e^{5}\right )} + 8 \, x e^{\left (2 \, x e^{5} + 4 \, x\right )} - 16 \, e^{\left (2 \, x e^{5}\right )} + 4 \, e^{\left (2 \, x e^{5} + 8 \, x\right )} - 4 \, e^{\left (2 \, x\right )}\right ) \] Input:
integrate(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2 )*exp(x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x, algorithm="giac")
Output:
-2*x*e^5 + log(4*x^2*e^(2*x*e^5) + 8*x*e^(2*x*e^5 + 4*x) - 16*e^(2*x*e^5) + 4*e^(2*x*e^5 + 8*x) - 4*e^(2*x))
Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54 \[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=\ln \left (\frac {{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}-4\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}-{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}+2\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}}{{\mathrm {e}}^{8\,x}+2\,x\,{\mathrm {e}}^{4\,x}+x^2-4}\right )+\ln \left ({\mathrm {e}}^{8\,x}+2\,x\,{\mathrm {e}}^{4\,x}+x^2-4\right )-2\,x\,{\mathrm {e}}^5 \] Input:
int(-(exp(2*x*exp(5))*(2*x + 8*exp(8*x) + exp(4*x)*(8*x + 2)) + exp(2*x)*( 2*exp(5) - 2))/(exp(2*x) - exp(2*x*exp(5))*(exp(8*x) + 2*x*exp(4*x) + x^2 - 4)),x)
Output:
log((exp(8*x)*exp(2*x*exp(5)) - 4*exp(2*x*exp(5)) - exp(2*x) + x^2*exp(2*x *exp(5)) + 2*x*exp(4*x)*exp(2*x*exp(5)))/(exp(8*x) + 2*x*exp(4*x) + x^2 - 4)) + log(exp(8*x) + 2*x*exp(4*x) + x^2 - 4) - 2*x*exp(5)
\[ \int \frac {e^{2 x} \left (-2+2 e^5\right )+e^{2 e^5 x} \left (8 e^{8 x}+2 x+e^{4 x} (2+8 x)\right )}{-e^{2 x}+e^{2 e^5 x} \left (-4+e^{8 x}+2 e^{4 x} x+x^2\right )} \, dx=8 \left (\int \frac {e^{2 e^{5} x +8 x}}{e^{2 e^{5} x +8 x}+2 e^{2 e^{5} x +4 x} x +e^{2 e^{5} x} x^{2}-4 e^{2 e^{5} x}-e^{2 x}}d x \right )+2 \left (\int \frac {e^{2 e^{5} x +4 x}}{e^{2 e^{5} x +8 x}+2 e^{2 e^{5} x +4 x} x +e^{2 e^{5} x} x^{2}-4 e^{2 e^{5} x}-e^{2 x}}d x \right )+2 \left (\int \frac {e^{2 x}}{e^{2 e^{5} x +8 x}+2 e^{2 e^{5} x +4 x} x +e^{2 e^{5} x} x^{2}-4 e^{2 e^{5} x}-e^{2 x}}d x \right ) e^{5}-2 \left (\int \frac {e^{2 x}}{e^{2 e^{5} x +8 x}+2 e^{2 e^{5} x +4 x} x +e^{2 e^{5} x} x^{2}-4 e^{2 e^{5} x}-e^{2 x}}d x \right )+8 \left (\int \frac {e^{2 e^{5} x +4 x} x}{e^{2 e^{5} x +8 x}+2 e^{2 e^{5} x +4 x} x +e^{2 e^{5} x} x^{2}-4 e^{2 e^{5} x}-e^{2 x}}d x \right )+2 \left (\int \frac {e^{2 e^{5} x} x}{e^{2 e^{5} x +8 x}+2 e^{2 e^{5} x +4 x} x +e^{2 e^{5} x} x^{2}-4 e^{2 e^{5} x}-e^{2 x}}d x \right ) \] Input:
int(((8*exp(4*x)^2+(8*x+2)*exp(4*x)+2*x)*exp(x*exp(5))^2+(2*exp(5)-2)*exp( x)^2)/((exp(4*x)^2+2*x*exp(4*x)+x^2-4)*exp(x*exp(5))^2-exp(x)^2),x)
Output:
2*(4*int(e**(2*e**5*x + 8*x)/(e**(2*e**5*x + 8*x) + 2*e**(2*e**5*x + 4*x)* x + e**(2*e**5*x)*x**2 - 4*e**(2*e**5*x) - e**(2*x)),x) + int(e**(2*e**5*x + 4*x)/(e**(2*e**5*x + 8*x) + 2*e**(2*e**5*x + 4*x)*x + e**(2*e**5*x)*x** 2 - 4*e**(2*e**5*x) - e**(2*x)),x) + int(e**(2*x)/(e**(2*e**5*x + 8*x) + 2 *e**(2*e**5*x + 4*x)*x + e**(2*e**5*x)*x**2 - 4*e**(2*e**5*x) - e**(2*x)), x)*e**5 - int(e**(2*x)/(e**(2*e**5*x + 8*x) + 2*e**(2*e**5*x + 4*x)*x + e* *(2*e**5*x)*x**2 - 4*e**(2*e**5*x) - e**(2*x)),x) + 4*int((e**(2*e**5*x + 4*x)*x)/(e**(2*e**5*x + 8*x) + 2*e**(2*e**5*x + 4*x)*x + e**(2*e**5*x)*x** 2 - 4*e**(2*e**5*x) - e**(2*x)),x) + int((e**(2*e**5*x)*x)/(e**(2*e**5*x + 8*x) + 2*e**(2*e**5*x + 4*x)*x + e**(2*e**5*x)*x**2 - 4*e**(2*e**5*x) - e **(2*x)),x))