\(\int \frac {-153600+(320 x+800 x^2) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx\) [4737]

Optimal result

Integrand size = 43, antiderivative size = 27 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x+\log \left (5 \left (1-\frac {x}{3 \left (x-\frac {320}{x \log (3)}\right )}\right )\right ) \]

[Out]

ln(5-5/3*x/(x-320/x/ln(3)))-x

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {1687, 1600, 8, 12, 1121, 630, 31} \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-\log \left (320-x^2 \log (3)\right )+\log \left (480-x^2 \log (3)\right )-x \]

[In]

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]

[Out]

-x - Log[320 - x^2*Log[3]] + Log[480 - x^2*Log[3]]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1600

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]

Rule 1687

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {320 x \log (3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx+\int \frac {-153600+800 x^2 \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx \\ & = (320 \log (3)) \int \frac {x}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx+\int -1 \, dx \\ & = -x+(160 \log (3)) \text {Subst}\left (\int \frac {1}{153600-800 x \log (3)+x^2 \log ^2(3)} \, dx,x,x^2\right ) \\ & = -x+\log ^2(3) \text {Subst}\left (\int \frac {1}{-480 \log (3)+x \log ^2(3)} \, dx,x,x^2\right )-\log ^2(3) \text {Subst}\left (\int \frac {1}{-320 \log (3)+x \log ^2(3)} \, dx,x,x^2\right ) \\ & = -x-\log \left (320-x^2 \log (3)\right )+\log \left (480-x^2 \log (3)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x-\log \left (320-x^2 \log (3)\right )+\log \left (480-x^2 \log (3)\right ) \]

[In]

Integrate[(-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]

[Out]

-x - Log[320 - x^2*Log[3]] + Log[480 - x^2*Log[3]]

Maple [A] (verified)

Time = 0.08 (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	\(-x +\ln \left (\frac {x^{2} \ln \left (3\right )-480}{\ln \left (3\right )}\right )-\ln \left (\frac {x^{2} \ln \left (3\right )-320}{\ln \left (3\right )}\right )\)	\(35\)

[In]

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)

[Out]

-x+ln(x^2*ln(3)-480)-ln(x^2*ln(3)-320)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x - \log \left (x^{2} \log \left (3\right ) - 320\right ) + \log \left (x^{2} \log \left (3\right ) - 480\right ) \]

[In]

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="fri
cas")

[Out]

-x - log(x^2*log(3) - 320) + log(x^2*log(3) - 480)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=- x + \log {\left (x^{2} - \frac {480}{\log {\left (3 \right )}} \right )} - \log {\left (x^{2} - \frac {320}{\log {\left (3 \right )}} \right )} \]

[In]

integrate((-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)

[Out]

-x + log(x**2 - 480/log(3)) - log(x**2 - 320/log(3))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x - \log \left (x^{2} \log \left (3\right ) - 320\right ) + \log \left (x^{2} \log \left (3\right ) - 480\right ) \]

[In]

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="max
ima")

[Out]

-x - log(x^2*log(3) - 320) + log(x^2*log(3) - 480)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=-x - \log \left ({\left | x^{2} \log \left (3\right ) - 320 \right |}\right ) + \log \left ({\left | x^{2} \log \left (3\right ) - 480 \right |}\right ) \]

[In]

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="gia
c")

[Out]

-x - log(abs(x^2*log(3) - 320)) + log(abs(x^2*log(3) - 480))

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-153600+\left (320 x+800 x^2\right ) \log (3)-x^4 \log ^2(3)}{153600-800 x^2 \log (3)+x^4 \log ^2(3)} \, dx=2\,\mathrm {atanh}\left (-\frac {2560\,x^2\,{\ln \left (3\right )}^5}{12800\,x^2\,{\ln \left (3\right )}^5-4915200\,{\ln \left (3\right )}^4}\right )-x \]

[In]

int(-(x^4*log(3)^2 - log(3)*(320*x + 800*x^2) + 153600)/(x^4*log(3)^2 - 800*x^2*log(3) + 153600),x)

[Out]

2*atanh(-(2560*x^2*log(3)^5)/(12800*x^2*log(3)^5 - 4915200*log(3)^4)) - x