Integrand size = 93, antiderivative size = 33 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=\frac {(4-x) x^2}{\left (2+\frac {e^{2 x}}{9}\right ) \left (-1+\frac {x^2}{9}\right )} \] Output:
x^2/(1/9*x^2-1)/(1/9*exp(x)^2+2)*(4-x)
Time = 2.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=-\frac {81 (-4+x) x^2}{\left (18+e^{2 x}\right ) \left (-9+x^2\right )} \] Input:
Integrate[(-104976*x + 39366*x^2 - 1458*x^4 + E^(2*x)*(-5832*x + 8019*x^2 - 1458*x^3 - 729*x^4 + 162*x^5))/(26244 - 5832*x^2 + 324*x^4 + E^(4*x)*(81 - 18*x^2 + x^4) + E^(2*x)*(2916 - 648*x^2 + 36*x^4)),x]
Output:
(-81*(-4 + x)*x^2)/((18 + E^(2*x))*(-9 + 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.
\(\displaystyle \int \frac {-1458 x^4+39366 x^2+e^{2 x} \left (162 x^5-729 x^4-1458 x^3+8019 x^2-5832 x\right )-104976 x}{324 x^4-5832 x^2+e^{4 x} \left (x^4-18 x^2+81\right )+e^{2 x} \left (36 x^4-648 x^2+2916\right )+26244} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-1458 x^4+39366 x^2+e^{2 x} \left (162 x^5-729 x^4-1458 x^3+8019 x^2-5832 x\right )-104976 x}{\left (e^{2 x}+18\right )^2 \left (9-x^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {81 x \left (2 x^4-9 x^3-18 x^2+99 x-72\right )}{\left (e^{2 x}+18\right ) \left (x^2-9\right )^2}-\frac {2916 (x-4) x^2}{\left (e^{2 x}+18\right )^2 \left (x^2-9\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {243}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (x-3)^2}dx+4374 \int \frac {1}{\left (18+e^{2 x}\right )^2 (x-3)}dx-243 \int \frac {1}{\left (18+e^{2 x}\right ) (x-3)}dx+\frac {1701}{2} \int \frac {1}{\left (18+e^{2 x}\right ) (x+3)^2}dx-30618 \int \frac {1}{\left (18+e^{2 x}\right )^2 (x+3)}dx+1701 \int \frac {1}{\left (18+e^{2 x}\right ) (x+3)}dx-\frac {81 x}{e^{2 x}+18}+\frac {324}{e^{2 x}+18}\) |
Input:
Int[(-104976*x + 39366*x^2 - 1458*x^4 + E^(2*x)*(-5832*x + 8019*x^2 - 1458 *x^3 - 729*x^4 + 162*x^5))/(26244 - 5832*x^2 + 324*x^4 + E^(4*x)*(81 - 18* x^2 + x^4) + E^(2*x)*(2916 - 648*x^2 + 36*x^4)),x]
Output:
$Aborted
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {81 \left (x -4\right ) x^{2}}{\left (x^{2}-9\right ) \left ({\mathrm e}^{2 x}+18\right )}\) | \(24\) |
norman | \(\frac {-81 x^{3}+324 x^{2}}{\left ({\mathrm e}^{2 x}+18\right ) \left (x^{2}-9\right )}\) | \(28\) |
parallelrisch | \(-\frac {81 x^{3}-324 x^{2}}{\left ({\mathrm e}^{2 x}+18\right ) \left (x^{2}-9\right )}\) | \(29\) |
Input:
int(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39366*x^ 2-104976*x)/((x^4-18*x^2+81)*exp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2+324*x ^4-5832*x^2+26244),x,method=_RETURNVERBOSE)
Output:
-81*(x-4)*x^2/(x^2-9)/(exp(2*x)+18)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=-\frac {81 \, {\left (x^{3} - 4 \, x^{2}\right )}}{18 \, x^{2} + {\left (x^{2} - 9\right )} e^{\left (2 \, x\right )} - 162} \] Input:
integrate(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39 366*x^2-104976*x)/((x^4-18*x^2+81)*exp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2 +324*x^4-5832*x^2+26244),x, algorithm="fricas")
Output:
-81*(x^3 - 4*x^2)/(18*x^2 + (x^2 - 9)*e^(2*x) - 162)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=\frac {- 81 x^{3} + 324 x^{2}}{18 x^{2} + \left (x^{2} - 9\right ) e^{2 x} - 162} \] Input:
integrate(((162*x**5-729*x**4-1458*x**3+8019*x**2-5832*x)*exp(x)**2-1458*x **4+39366*x**2-104976*x)/((x**4-18*x**2+81)*exp(x)**4+(36*x**4-648*x**2+29 16)*exp(x)**2+324*x**4-5832*x**2+26244),x)
Output:
(-81*x**3 + 324*x**2)/(18*x**2 + (x**2 - 9)*exp(2*x) - 162)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=-\frac {81 \, {\left (x^{3} - 4 \, x^{2}\right )}}{18 \, x^{2} + {\left (x^{2} - 9\right )} e^{\left (2 \, x\right )} - 162} \] Input:
integrate(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39 366*x^2-104976*x)/((x^4-18*x^2+81)*exp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2 +324*x^4-5832*x^2+26244),x, algorithm="maxima")
Output:
-81*(x^3 - 4*x^2)/(18*x^2 + (x^2 - 9)*e^(2*x) - 162)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=-\frac {81 \, {\left (x^{3} - 4 \, x^{2}\right )}}{x^{2} e^{\left (2 \, x\right )} + 18 \, x^{2} - 9 \, e^{\left (2 \, x\right )} - 162} \] Input:
integrate(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39 366*x^2-104976*x)/((x^4-18*x^2+81)*exp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2 +324*x^4-5832*x^2+26244),x, algorithm="giac")
Output:
-81*(x^3 - 4*x^2)/(x^2*e^(2*x) + 18*x^2 - 9*e^(2*x) - 162)
Time = 2.82 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=-\frac {81\,\left (x^5-4\,x^4-9\,x^3+36\,x^2\right )}{{\left (x^2-9\right )}^2\,\left ({\mathrm {e}}^{2\,x}+18\right )} \] Input:
int(-(104976*x + exp(2*x)*(5832*x - 8019*x^2 + 1458*x^3 + 729*x^4 - 162*x^ 5) - 39366*x^2 + 1458*x^4)/(exp(4*x)*(x^4 - 18*x^2 + 81) + exp(2*x)*(36*x^ 4 - 648*x^2 + 2916) - 5832*x^2 + 324*x^4 + 26244),x)
Output:
-(81*(36*x^2 - 9*x^3 - 4*x^4 + x^5))/((x^2 - 9)^2*(exp(2*x) + 18))
Time = 0.46 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-104976 x+39366 x^2-1458 x^4+e^{2 x} \left (-5832 x+8019 x^2-1458 x^3-729 x^4+162 x^5\right )}{26244-5832 x^2+324 x^4+e^{4 x} \left (81-18 x^2+x^4\right )+e^{2 x} \left (2916-648 x^2+36 x^4\right )} \, dx=\frac {-18 e^{2 x} x^{2}+162 e^{2 x}-81 x^{3}+2916}{e^{2 x} x^{2}-9 e^{2 x}+18 x^{2}-162} \] Input:
int(((162*x^5-729*x^4-1458*x^3+8019*x^2-5832*x)*exp(x)^2-1458*x^4+39366*x^ 2-104976*x)/((x^4-18*x^2+81)*exp(x)^4+(36*x^4-648*x^2+2916)*exp(x)^2+324*x ^4-5832*x^2+26244),x)
Output:
(9*( - 2*e**(2*x)*x**2 + 18*e**(2*x) - 9*x**3 + 324))/(e**(2*x)*x**2 - 9*e **(2*x) + 18*x**2 - 162)