Integrand size = 92, antiderivative size = 21 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=\log \left (\frac {15 \left (9+\frac {8}{x}\right )}{\left (e^4-x\right )^2}+x\right ) \] Output:
ln(x+15*(8/x+9)/(exp(2)^2-x)^2)
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=-2 \log \left (e^4-x\right )-\log (x)+\log \left (120+135 x+e^8 x^2-2 e^4 x^3+x^4\right ) \] Input:
Integrate[(360*x + 270*x^2 + E^12*x^2 - 3*E^8*x^3 - x^5 + E^4*(-120 + 3*x^ 4))/(-120*x^2 - 135*x^3 + E^12*x^3 - 3*E^8*x^4 - x^6 + E^4*(120*x + 135*x^ 2 + 3*x^5)),x]
Output:
-2*Log[E^4 - x] - Log[x] + Log[120 + 135*x + E^8*x^2 - 2*E^4*x^3 + x^4]
Time = 0.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6, 6, 2026, 2462, 2009}
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.
\(\displaystyle \int \frac {-x^5+e^4 \left (3 x^4-120\right )-3 e^8 x^3+e^{12} x^2+270 x^2+360 x}{-x^6-3 e^8 x^4+e^{12} x^3-135 x^3-120 x^2+e^4 \left (3 x^5+135 x^2+120 x\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-x^5+e^4 \left (3 x^4-120\right )-3 e^8 x^3+\left (270+e^{12}\right ) x^2+360 x}{-x^6-3 e^8 x^4+e^{12} x^3-135 x^3-120 x^2+e^4 \left (3 x^5+135 x^2+120 x\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-x^5+e^4 \left (3 x^4-120\right )-3 e^8 x^3+\left (270+e^{12}\right ) x^2+360 x}{-x^6-3 e^8 x^4+\left (e^{12}-135\right ) x^3-120 x^2+e^4 \left (3 x^5+135 x^2+120 x\right )}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-x^5+e^4 \left (3 x^4-120\right )-3 e^8 x^3+\left (270+e^{12}\right ) x^2+360 x}{x \left (-x^5+3 e^4 x^4-3 e^8 x^3-\left (135-e^{12}\right ) x^2-15 \left (8-9 e^4\right ) x+120 e^4\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {4 x^3-6 e^4 x^2+2 e^8 x+135}{x^4-2 e^4 x^3+e^8 x^2+135 x+120}+\frac {2}{e^4-x}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (x^4-2 e^4 x^3+e^8 x^2+135 x+120\right )-2 \log \left (e^4-x\right )-\log (x)\) |
Input:
Int[(360*x + 270*x^2 + E^12*x^2 - 3*E^8*x^3 - x^5 + E^4*(-120 + 3*x^4))/(- 120*x^2 - 135*x^3 + E^12*x^3 - 3*E^8*x^4 - x^6 + E^4*(120*x + 135*x^2 + 3* x^5)),x]
Output:
-2*Log[E^4 - x] - Log[x] + Log[120 + 135*x + E^8*x^2 - 2*E^4*x^3 + x^4]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.57 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86
method | result | size |
risch | \(-\ln \left (-x \right )-2 \ln \left (x -{\mathrm e}^{4}\right )+\ln \left (x^{2} {\mathrm e}^{8}-2 x^{3} {\mathrm e}^{4}+x^{4}+135 x +120\right )\) | \(39\) |
norman | \(-\ln \left (x \right )-2 \ln \left ({\mathrm e}^{4}-x \right )+\ln \left (x^{2} {\mathrm e}^{8}-2 x^{3} {\mathrm e}^{4}+x^{4}+135 x +120\right )\) | \(43\) |
parallelrisch | \(-\ln \left (x \right )-2 \ln \left (x -{\mathrm e}^{4}\right )+\ln \left (x^{2} {\mathrm e}^{8}-2 x^{3} {\mathrm e}^{4}+x^{4}+135 x +120\right )\) | \(43\) |
default | \(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{5}-3 \textit {\_Z}^{4} {\mathrm e}^{4}+3 \textit {\_Z}^{3} {\mathrm e}^{8}+\left (-{\mathrm e}^{12}+135\right ) \textit {\_Z}^{2}+\left (-135 \,{\mathrm e}^{4}+120\right ) \textit {\_Z} -120 \,{\mathrm e}^{4}\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-6 \textit {\_R}^{3} {\mathrm e}^{4}+6 \textit {\_R}^{2} {\mathrm e}^{8}+\left (-2 \,{\mathrm e}^{12}-135\right ) \textit {\_R} -135 \,{\mathrm e}^{4}-240\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}^{12}-9 \textit {\_R}^{2} {\mathrm e}^{8}+12 \textit {\_R}^{3} {\mathrm e}^{4}-5 \textit {\_R}^{4}+135 \,{\mathrm e}^{4}-270 \textit {\_R} -120}\right )-\ln \left (x \right )\) | \(127\) |
Input:
int((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+360*x)/( x^3*exp(2)^6-3*x^4*exp(2)^4+(3*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x^3-120 *x^2),x,method=_RETURNVERBOSE)
Output:
-ln(-x)-2*ln(x-exp(4))+ln(x^2*exp(8)-2*x^3*exp(4)+x^4+135*x+120)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=\log \left (x^{4} - 2 \, x^{3} e^{4} + x^{2} e^{8} + 135 \, x + 120\right ) - 2 \, \log \left (x - e^{4}\right ) - \log \left (x\right ) \] Input:
integrate((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+36 0*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x ^3-120*x^2),x, algorithm="fricas")
Output:
log(x^4 - 2*x^3*e^4 + x^2*e^8 + 135*x + 120) - 2*log(x - e^4) - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
Time = 5.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=- \log {\left (x \right )} - 2 \log {\left (x - e^{4} \right )} + \log {\left (x^{4} - 2 x^{3} e^{4} + x^{2} e^{8} + 135 x + 120 \right )} \] Input:
integrate((x**2*exp(2)**6-3*x**3*exp(2)**4+(3*x**4-120)*exp(2)**2-x**5+270 *x**2+360*x)/(x**3*exp(2)**6-3*x**4*exp(2)**4+(3*x**5+135*x**2+120*x)*exp( 2)**2-x**6-135*x**3-120*x**2),x)
Output:
-log(x) - 2*log(x - exp(4)) + log(x**4 - 2*x**3*exp(4) + x**2*exp(8) + 135 *x + 120)
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=\log \left (x^{4} - 2 \, x^{3} e^{4} + x^{2} e^{8} + 135 \, x + 120\right ) - 2 \, \log \left (x - e^{4}\right ) - \log \left (x\right ) \] Input:
integrate((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+36 0*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x ^3-120*x^2),x, algorithm="maxima")
Output:
log(x^4 - 2*x^3*e^4 + x^2*e^8 + 135*x + 120) - 2*log(x - e^4) - log(x)
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=\log \left (x^{4} - 2 \, x^{3} e^{4} + x^{2} e^{8} + 135 \, x + 120\right ) - 2 \, \log \left ({\left | x - e^{4} \right |}\right ) - \log \left ({\left | x \right |}\right ) \] Input:
integrate((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+36 0*x)/(x^3*exp(2)^6-3*x^4*exp(2)^4+(3*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x ^3-120*x^2),x, algorithm="giac")
Output:
log(x^4 - 2*x^3*e^4 + x^2*e^8 + 135*x + 120) - 2*log(abs(x - e^4)) - log(a bs(x))
Time = 3.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=\ln \left (x^4-2\,{\mathrm {e}}^4\,x^3+{\mathrm {e}}^8\,x^2+135\,x+120\right )-2\,\ln \left (x-{\mathrm {e}}^4\right )-\ln \left (x\right ) \] Input:
int(-(360*x + exp(4)*(3*x^4 - 120) - 3*x^3*exp(8) + x^2*exp(12) + 270*x^2 - x^5)/(3*x^4*exp(8) - exp(4)*(120*x + 135*x^2 + 3*x^5) - x^3*exp(12) + 12 0*x^2 + 135*x^3 + x^6),x)
Output:
log(135*x - 2*x^3*exp(4) + x^2*exp(8) + x^4 + 120) - 2*log(x - exp(4)) - l og(x)
\[ \int \frac {360 x+270 x^2+e^{12} x^2-3 e^8 x^3-x^5+e^4 \left (-120+3 x^4\right )}{-120 x^2-135 x^3+e^{12} x^3-3 e^8 x^4-x^6+e^4 \left (120 x+135 x^2+3 x^5\right )} \, dx=405 \left (\int \frac {x}{e^{12} x^{2}-3 e^{8} x^{3}+3 e^{4} x^{4}+135 e^{4} x -x^{5}+120 e^{4}-135 x^{2}-120 x}d x \right )-240 \left (\int \frac {1}{e^{12} x^{3}-3 e^{8} x^{4}+3 e^{4} x^{5}+135 e^{4} x^{2}-x^{6}+120 e^{4} x -135 x^{3}-120 x^{2}}d x \right ) e^{4}-135 \left (\int \frac {1}{e^{12} x^{2}-3 e^{8} x^{3}+3 e^{4} x^{4}+135 e^{4} x -x^{5}+120 e^{4}-135 x^{2}-120 x}d x \right ) e^{4}+480 \left (\int \frac {1}{e^{12} x^{2}-3 e^{8} x^{3}+3 e^{4} x^{4}+135 e^{4} x -x^{5}+120 e^{4}-135 x^{2}-120 x}d x \right )+\mathrm {log}\left (x \right ) \] Input:
int((x^2*exp(2)^6-3*x^3*exp(2)^4+(3*x^4-120)*exp(2)^2-x^5+270*x^2+360*x)/( x^3*exp(2)^6-3*x^4*exp(2)^4+(3*x^5+135*x^2+120*x)*exp(2)^2-x^6-135*x^3-120 *x^2),x)
Output:
405*int(x/(e**12*x**2 - 3*e**8*x**3 + 3*e**4*x**4 + 135*e**4*x + 120*e**4 - x**5 - 135*x**2 - 120*x),x) - 240*int(1/(e**12*x**3 - 3*e**8*x**4 + 3*e* *4*x**5 + 135*e**4*x**2 + 120*e**4*x - x**6 - 135*x**3 - 120*x**2),x)*e**4 - 135*int(1/(e**12*x**2 - 3*e**8*x**3 + 3*e**4*x**4 + 135*e**4*x + 120*e* *4 - x**5 - 135*x**2 - 120*x),x)*e**4 + 480*int(1/(e**12*x**2 - 3*e**8*x** 3 + 3*e**4*x**4 + 135*e**4*x + 120*e**4 - x**5 - 135*x**2 - 120*x),x) + lo g(x)