Integrand size = 43, antiderivative size = 27 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x+\log \left (5 \left (1-\frac {x}{3 \left (x-\frac {320}{x \log (3)}\right )}\right )\right ) \] Output:
ln(5-5/3*x/(x-320/x/ln(3)))-x
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x-\log \left (320-x^2 \log (3)\right )+\log \left (480-x^2 \log (3)\right ) \] Input:
Integrate[(-153600 + (320*x + 800*x^2)*Log[3] - x^4*Log[3]^2)/(153600 - 80 0*x^2*Log[3] + x^4*Log[3]^2),x]
Output:
-x - Log[320 - x^2*Log[3]] + Log[480 - x^2*Log[3]]
Time = 0.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2202, 27, 1432, 1081, 2009, 2019, 24}
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.
\(\displaystyle \int \frac {x^4 \left (-\log ^2(3)\right )+\left (800 x^2+320 x\right ) \log (3)-153600}{x^4 \log ^2(3)-800 x^2 \log (3)+153600} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {320 x \log (3)}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx+\int \frac {-\log ^2(3) x^4+800 \log (3) x^2-153600}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 320 \log (3) \int \frac {x}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx+\int \frac {-\log ^2(3) x^4+800 \log (3) x^2-153600}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle 160 \log (3) \int \frac {1}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx^2+\int \frac {-\log ^2(3) x^4+800 \log (3) x^2-153600}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx\) |
\(\Big \downarrow \) 1081 |
\(\displaystyle 160 \log ^3(3) \int \left (\frac {1}{160 \log ^2(3) \left (320-x^2 \log (3)\right )}-\frac {1}{160 \log ^2(3) \left (480-x^2 \log (3)\right )}\right )dx^2+\int \frac {-\log ^2(3) x^4+800 \log (3) x^2-153600}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {-\log ^2(3) x^4+800 \log (3) x^2-153600}{\log ^2(3) x^4-800 \log (3) x^2+153600}dx+160 \log ^3(3) \left (\frac {\log \left (480-x^2 \log (3)\right )}{160 \log ^3(3)}-\frac {\log \left (320-x^2 \log (3)\right )}{160 \log ^3(3)}\right )\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int -1dx+160 \log ^3(3) \left (\frac {\log \left (480-x^2 \log (3)\right )}{160 \log ^3(3)}-\frac {\log \left (320-x^2 \log (3)\right )}{160 \log ^3(3)}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 160 \log ^3(3) \left (\frac {\log \left (480-x^2 \log (3)\right )}{160 \log ^3(3)}-\frac {\log \left (320-x^2 \log (3)\right )}{160 \log ^3(3)}\right )-x\) |
Input:
Int[(-153600 + (320*x + 800*x^2)*Log[3] - x^4*Log[3]^2)/(153600 - 800*x^2* Log[3] + x^4*Log[3]^2),x]
Output:
-x + 160*Log[3]^3*(-1/160*Log[320 - x^2*Log[3]]/Log[3]^3 + Log[480 - x^2*L og[3]]/(160*Log[3]^3))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
default | \(-x +\ln \left (x^{2} \ln \left (3\right )-480\right )-\ln \left (x^{2} \ln \left (3\right )-320\right )\) | \(25\) |
norman | \(-x +\ln \left (x^{2} \ln \left (3\right )-480\right )-\ln \left (x^{2} \ln \left (3\right )-320\right )\) | \(25\) |
risch | \(-x -\ln \left (x^{2} \ln \left (3\right )-320\right )+\ln \left (-x^{2} \ln \left (3\right )+480\right )\) | \(26\) |
parallelrisch | \(-x +\ln \left (\frac {x^{2} \ln \left (3\right )-480}{\ln \left (3\right )}\right )-\ln \left (\frac {x^{2} \ln \left (3\right )-320}{\ln \left (3\right )}\right )\) | \(35\) |
Input:
int((-x^4*ln(3)^2+(800*x^2+320*x)*ln(3)-153600)/(x^4*ln(3)^2-800*x^2*ln(3) +153600),x,method=_RETURNVERBOSE)
Output:
-x+ln(x^2*ln(3)-480)-ln(x^2*ln(3)-320)
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x - \log \left (x^{2} \log \left (3\right ) - 320\right ) + \log \left (x^{2} \log \left (3\right ) - 480\right ) \] Input:
integrate((-x^4*log(3)^2+(800*x^2+320*x)*log(3)-153600)/(x^4*log(3)^2-800* x^2*log(3)+153600),x, algorithm="fricas")
Output:
-x - log(x^2*log(3) - 320) + log(x^2*log(3) - 480)
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=- x + \log {\left (x^{2} - \frac {480}{\log {\left (3 \right )}} \right )} - \log {\left (x^{2} - \frac {320}{\log {\left (3 \right )}} \right )} \] Input:
integrate((-x**4*ln(3)**2+(800*x**2+320*x)*ln(3)-153600)/(x**4*ln(3)**2-80 0*x**2*ln(3)+153600),x)
Output:
-x + log(x**2 - 480/log(3)) - log(x**2 - 320/log(3))
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x - \log \left (x^{2} \log \left (3\right ) - 320\right ) + \log \left (x^{2} \log \left (3\right ) - 480\right ) \] Input:
integrate((-x^4*log(3)^2+(800*x^2+320*x)*log(3)-153600)/(x^4*log(3)^2-800* x^2*log(3)+153600),x, algorithm="maxima")
Output:
-x - log(x^2*log(3) - 320) + log(x^2*log(3) - 480)
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x - \log \left ({\left | x^{2} \log \left (3\right ) - 320 \right |}\right ) + \log \left ({\left | x^{2} \log \left (3\right ) - 480 \right |}\right ) \] Input:
integrate((-x^4*log(3)^2+(800*x^2+320*x)*log(3)-153600)/(x^4*log(3)^2-800* x^2*log(3)+153600),x, algorithm="giac")
Output:
-x - log(abs(x^2*log(3) - 320)) + log(abs(x^2*log(3) - 480))
Time = 3.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=2\,\mathrm {atanh}\left (-\frac {2560\,x^2\,{\ln \left (3\right )}^5}{12800\,x^2\,{\ln \left (3\right )}^5-4915200\,{\ln \left (3\right )}^4}\right )-x \] Input:
int(-(x^4*log(3)^2 - log(3)*(320*x + 800*x^2) + 153600)/(x^4*log(3)^2 - 80 0*x^2*log(3) + 153600),x)
Output:
2*atanh(-(2560*x^2*log(3)^5)/(12800*x^2*log(3)^5 - 4915200*log(3)^4)) - x
Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-\mathrm {log}\left (\sqrt {\mathrm {log}\left (3\right )}\, x -8 \sqrt {5}\right )+\mathrm {log}\left (\sqrt {\mathrm {log}\left (3\right )}\, x -4 \sqrt {30}\right )-\mathrm {log}\left (\sqrt {\mathrm {log}\left (3\right )}\, x +8 \sqrt {5}\right )+\mathrm {log}\left (\sqrt {\mathrm {log}\left (3\right )}\, x +4 \sqrt {30}\right )-x \] Input:
int((-x^4*log(3)^2+(800*x^2+320*x)*log(3)-153600)/(x^4*log(3)^2-800*x^2*lo g(3)+153600),x)
Output:
- log(sqrt(log(3))*x - 8*sqrt(5)) + log(sqrt(log(3))*x - 4*sqrt(30)) - lo g(sqrt(log(3))*x + 8*sqrt(5)) + log(sqrt(log(3))*x + 4*sqrt(30)) - x