Integrand size = 127, antiderivative size = 32 \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\frac {27}{2 x-\frac {x \left (e^{4/x}-\log (2)\right )}{2-x^2}} \] Output:
27/(2*x-(exp(4/x)-ln(2))*x/(-x^2+2))
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(32)=64\).
Time = 1.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\frac {27 \left (32-8 x^3+8 x^4+4 x^5-4 x^2 (8+\log (2))+\log (256)\right )}{4 x \left (-4+e^{4/x}+2 x^2-\log (2)\right ) \left (-4+2 x^2+x^3-\log (2)\right )} \] Input:
Integrate[(-216*x + 216*x^3 - 54*x^5 + E^(4/x)*(-216 + 54*x + 108*x^2 + 27 *x^3) + (-54*x - 27*x^3)*Log[2])/(16*x^3 + E^(8/x)*x^3 - 16*x^5 + 4*x^7 + (8*x^3 - 4*x^5)*Log[2] + x^3*Log[2]^2 + E^(4/x)*(-8*x^3 + 4*x^5 - 2*x^3*Lo g[2])),x]
Output:
(27*(32 - 8*x^3 + 8*x^4 + 4*x^5 - 4*x^2*(8 + Log[2]) + Log[256]))/(4*x*(-4 + E^(4/x) + 2*x^2 - Log[2])*(-4 + 2*x^2 + x^3 - Log[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.
| \(\displaystyle \int \frac {-54 x^5+216 x^3+\left (-27 x^3-54 x\right ) \log (2)+e^{4/x} \left (27 x^3+108 x^2+54 x-216\right )-216 x}{4 x^7-16 x^5+e^{8/x} x^3+16 x^3+x^3 \log ^2(2)+e^{4/x} \left (4 x^5-8 x^3-2 x^3 \log (2)\right )+\left (8 x^3-4 x^5\right ) \log (2)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-54 x^5+216 x^3+\left (-27 x^3-54 x\right ) \log (2)+e^{4/x} \left (27 x^3+108 x^2+54 x-216\right )-216 x}{4 x^7-16 x^5+e^{8/x} x^3+x^3 \left (16+\log ^2(2)\right )+e^{4/x} \left (4 x^5-8 x^3-2 x^3 \log (2)\right )+\left (8 x^3-4 x^5\right ) \log (2)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {27 \left (e^{4/x} \left (x^3+4 x^2+2 x-8\right )-x \left (2 x^4+x^2 (\log (2)-8)+8+\log (4)\right )\right )}{x^3 \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 27 \int -\frac {e^{4/x} \left (-x^3-4 x^2-2 x+8\right )+x \left (2 x^4-(8-\log (2)) x^2+\log (4)+8\right )}{x^3 \left (-2 x^2-e^{4/x}+\log (2)+4\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -27 \int \frac {e^{4/x} \left (-x^3-4 x^2-2 x+8\right )+x \left (2 x^4-(8-\log (2)) x^2+\log (4)+8\right )}{x^3 \left (-2 x^2-e^{4/x}+\log (2)+4\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -27 \int \frac {e^{4/x} \left (-x^3-4 x^2-2 x+8\right )+x \left (2 x^4-(8-\log (2)) x^2+\log (4)+8\right )}{x^3 \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -27 \int \left (\frac {-x^3-4 x^2-2 x+8}{x^3 \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )}+\frac {4 x^5+8 x^4-8 x^3-32 \left (1+\frac {\log (2)}{8}\right ) x^2+32 \left (1+\frac {\log (2)}{4}\right )}{x^3 \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -27 \left (-8 \int \frac {1}{\left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx-4 (8+\log (2)) \int \frac {1}{x \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx+8 \int \frac {x}{\left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx+4 \int \frac {x^2}{\left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx+\int \frac {1}{-2 x^2-e^{4/x}+4 \left (1+\frac {\log (2)}{4}\right )}dx+2 \int \frac {1}{x^2 \left (-2 x^2-e^{4/x}+4 \left (1+\frac {\log (2)}{4}\right )\right )}dx+4 \int \frac {1}{x \left (-2 x^2-e^{4/x}+4 \left (1+\frac {\log (2)}{4}\right )\right )}dx+4 (8+\log (4)) \int \frac {1}{x^3 \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}dx+8 \int \frac {1}{x^3 \left (2 x^2+e^{4/x}-4 \left (1+\frac {\log (2)}{4}\right )\right )}dx\right )\) |
Input:
Int[(-216*x + 216*x^3 - 54*x^5 + E^(4/x)*(-216 + 54*x + 108*x^2 + 27*x^3) + (-54*x - 27*x^3)*Log[2])/(16*x^3 + E^(8/x)*x^3 - 16*x^5 + 4*x^7 + (8*x^3 - 4*x^5)*Log[2] + x^3*Log[2]^2 + E^(4/x)*(-8*x^3 + 4*x^5 - 2*x^3*Log[2])) ,x]
Output:
$Aborted
Time = 0.80 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {27 \left (x^{2}-2\right )}{x \left (-2 x^{2}+\ln \left (2\right )-{\mathrm e}^{\frac {4}{x}}+4\right )}\) | \(30\) |
| parallelrisch | \(-\frac {27 x^{2}-54}{x \left (-2 x^{2}+\ln \left (2\right )-{\mathrm e}^{\frac {4}{x}}+4\right )}\) | \(32\) |
| norman | \(\frac {-27 x^{3}+54 x}{x^{2} \left (-2 x^{2}+\ln \left (2\right )-{\mathrm e}^{\frac {4}{x}}+4\right )}\) | \(33\) |
Input:
int(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*ln(2)-54*x^5+216*x^ 3-216*x)/(x^3*exp(4/x)^2+(-2*x^3*ln(2)+4*x^5-8*x^3)*exp(4/x)+x^3*ln(2)^2+( -4*x^5+8*x^3)*ln(2)+4*x^7-16*x^5+16*x^3),x,method=_RETURNVERBOSE)
Output:
-27*(x^2-2)/x/(-2*x^2+ln(2)-exp(4/x)+4)
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\frac {27 \, {\left (x^{2} - 2\right )}}{2 \, x^{3} + x e^{\frac {4}{x}} - x \log \left (2\right ) - 4 \, x} \] Input:
integrate(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5 +216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*l og(2)^2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x, algorithm="fricas")
Output:
27*(x^2 - 2)/(2*x^3 + x*e^(4/x) - x*log(2) - 4*x)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\frac {27 x^{2} - 54}{2 x^{3} + x e^{\frac {4}{x}} - 4 x - x \log {\left (2 \right )}} \] Input:
integrate(((27*x**3+108*x**2+54*x-216)*exp(4/x)+(-27*x**3-54*x)*ln(2)-54*x **5+216*x**3-216*x)/(x**3*exp(4/x)**2+(-2*x**3*ln(2)+4*x**5-8*x**3)*exp(4/ x)+x**3*ln(2)**2+(-4*x**5+8*x**3)*ln(2)+4*x**7-16*x**5+16*x**3),x)
Output:
(27*x**2 - 54)/(2*x**3 + x*exp(4/x) - 4*x - x*log(2))
Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\frac {27 \, {\left (x^{2} - 2\right )}}{2 \, x^{3} - x {\left (\log \left (2\right ) + 4\right )} + x e^{\frac {4}{x}}} \] Input:
integrate(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5 +216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*l og(2)^2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x, algorithm="maxima")
Output:
27*(x^2 - 2)/(2*x^3 - x*(log(2) + 4) + x*e^(4/x))
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\frac {27 \, {\left (\frac {1}{x} - \frac {2}{x^{3}}\right )}}{\frac {e^{\frac {4}{x}}}{x^{2}} - \frac {\log \left (2\right )}{x^{2}} - \frac {4}{x^{2}} + 2} \] Input:
integrate(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5 +216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*l og(2)^2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x, algorithm="giac")
Output:
27*(1/x - 2/x^3)/(e^(4/x)/x^2 - log(2)/x^2 - 4/x^2 + 2)
Timed out. \[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx=\int -\frac {216\,x+\ln \left (2\right )\,\left (27\,x^3+54\,x\right )-{\mathrm {e}}^{4/x}\,\left (27\,x^3+108\,x^2+54\,x-216\right )-216\,x^3+54\,x^5}{x^3\,{\ln \left (2\right )}^2-{\mathrm {e}}^{4/x}\,\left (2\,x^3\,\ln \left (2\right )+8\,x^3-4\,x^5\right )+\ln \left (2\right )\,\left (8\,x^3-4\,x^5\right )+x^3\,{\mathrm {e}}^{8/x}+16\,x^3-16\,x^5+4\,x^7} \,d x \] Input:
int(-(216*x + log(2)*(54*x + 27*x^3) - exp(4/x)*(54*x + 108*x^2 + 27*x^3 - 216) - 216*x^3 + 54*x^5)/(x^3*log(2)^2 - exp(4/x)*(2*x^3*log(2) + 8*x^3 - 4*x^5) + log(2)*(8*x^3 - 4*x^5) + x^3*exp(8/x) + 16*x^3 - 16*x^5 + 4*x^7) ,x)
Output:
int(-(216*x + log(2)*(54*x + 27*x^3) - exp(4/x)*(54*x + 108*x^2 + 27*x^3 - 216) - 216*x^3 + 54*x^5)/(x^3*log(2)^2 - exp(4/x)*(2*x^3*log(2) + 8*x^3 - 4*x^5) + log(2)*(8*x^3 - 4*x^5) + x^3*exp(8/x) + 16*x^3 - 16*x^5 + 4*x^7) , x)
\[ \int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx =\text {Too large to display} \] Input:
int(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5+216*x ^3-216*x)/(x^3*exp(4/x)^2+(-2*x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*log(2)^ 2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x)
Output:
27*(int(e**(4/x)/(e**(8/x) - 2*e**(4/x)*log(2) + 4*e**(4/x)*x**2 - 8*e**(4 /x) + log(2)**2 - 4*log(2)*x**2 + 8*log(2) + 4*x**4 - 16*x**2 + 16),x) - 8 *int(e**(4/x)/(e**(8/x)*x**3 - 2*e**(4/x)*log(2)*x**3 + 4*e**(4/x)*x**5 - 8*e**(4/x)*x**3 + log(2)**2*x**3 - 4*log(2)*x**5 + 8*log(2)*x**3 + 4*x**7 - 16*x**5 + 16*x**3),x) + 2*int(e**(4/x)/(e**(8/x)*x**2 - 2*e**(4/x)*log(2 )*x**2 + 4*e**(4/x)*x**4 - 8*e**(4/x)*x**2 + log(2)**2*x**2 - 4*log(2)*x** 4 + 8*log(2)*x**2 + 4*x**6 - 16*x**4 + 16*x**2),x) + 4*int(e**(4/x)/(e**(8 /x)*x - 2*e**(4/x)*log(2)*x + 4*e**(4/x)*x**3 - 8*e**(4/x)*x + log(2)**2*x - 4*log(2)*x**3 + 8*log(2)*x + 4*x**5 - 16*x**3 + 16*x),x) - 2*int(x**2/( e**(8/x) - 2*e**(4/x)*log(2) + 4*e**(4/x)*x**2 - 8*e**(4/x) + log(2)**2 - 4*log(2)*x**2 + 8*log(2) + 4*x**4 - 16*x**2 + 16),x) - int(1/(e**(8/x) - 2 *e**(4/x)*log(2) + 4*e**(4/x)*x**2 - 8*e**(4/x) + log(2)**2 - 4*log(2)*x** 2 + 8*log(2) + 4*x**4 - 16*x**2 + 16),x)*log(2) + 8*int(1/(e**(8/x) - 2*e* *(4/x)*log(2) + 4*e**(4/x)*x**2 - 8*e**(4/x) + log(2)**2 - 4*log(2)*x**2 + 8*log(2) + 4*x**4 - 16*x**2 + 16),x) - 2*int(1/(e**(8/x)*x**2 - 2*e**(4/x )*log(2)*x**2 + 4*e**(4/x)*x**4 - 8*e**(4/x)*x**2 + log(2)**2*x**2 - 4*log (2)*x**4 + 8*log(2)*x**2 + 4*x**6 - 16*x**4 + 16*x**2),x)*log(2) - 8*int(1 /(e**(8/x)*x**2 - 2*e**(4/x)*log(2)*x**2 + 4*e**(4/x)*x**4 - 8*e**(4/x)*x* *2 + log(2)**2*x**2 - 4*log(2)*x**4 + 8*log(2)*x**2 + 4*x**6 - 16*x**4 + 1 6*x**2),x))