3.76.55 \(\int \frac {365-146 x-73 \log (9)}{-365 x+7573 x^2-3000 x^3+300 x^4+(73 x-3000 x^2+600 x^3) \log (9)+300 x^2 \log ^2(9)} \, dx\)

Optimal. Leaf size=17 \[ \log \left (-5-\frac {73}{60 x (-5+x+\log (9))}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 28, normalized size of antiderivative = 1.65, number of steps used = 7, number of rules used = 6, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6, 1680, 12, 1107, 616, 31} \begin {gather*} \log \left (300 x^2-300 x (5-\log (9))+73\right )-\log (x (x-5+\log (9))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(365 - 146*x - 73*Log[9])/(-365*x + 7573*x^2 - 3000*x^3 + 300*x^4 + (73*x - 3000*x^2 + 600*x^3)*Log[9] + 3
00*x^2*Log[9]^2),x]

[Out]

Log[73 + 300*x^2 - 300*x*(5 - Log[9])] - Log[x*(-5 + x + Log[9])]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 616

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 1107

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 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {365-146 x-73 \log (9)}{-365 x-3000 x^3+300 x^4+\left (73 x-3000 x^2+600 x^3\right ) \log (9)+x^2 \left (7573+300 \log ^2(9)\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {584 x}{-1200 x^4-(-5+\log (9))^2 \left (1802-750 \log (9)+75 \log ^2(9)\right )+4 x^2 \left (3677-1500 \log (9)+150 \log ^2(9)\right )} \, dx,x,x+\frac {-3000+600 \log (9)}{1200}\right )\\ &=584 \operatorname {Subst}\left (\int \frac {x}{-1200 x^4-(-5+\log (9))^2 \left (1802-750 \log (9)+75 \log ^2(9)\right )+4 x^2 \left (3677-1500 \log (9)+150 \log ^2(9)\right )} \, dx,x,x+\frac {-3000+600 \log (9)}{1200}\right )\\ &=292 \operatorname {Subst}\left (\int \frac {1}{-1200 x^2-(-5+\log (9))^2 \left (1802-750 \log (9)+75 \log ^2(9)\right )+4 x \left (3677-1500 \log (9)+150 \log ^2(9)\right )} \, dx,x,\left (x+\frac {-3000+600 \log (9)}{1200}\right )^2\right )\\ &=1200 \operatorname {Subst}\left (\int \frac {1}{-1200 x+300 (5-\log (9))^2} \, dx,x,\left (x+\frac {-3000+600 \log (9)}{1200}\right )^2\right )-1200 \operatorname {Subst}\left (\int \frac {1}{-1200 x+4 \left (1802-750 \log (9)+75 \log ^2(9)\right )} \, dx,x,\left (x+\frac {-3000+600 \log (9)}{1200}\right )^2\right )\\ &=\log \left (73+300 x^2-300 x (5-\log (9))\right )-\log (-x (5-x-\log (9)))\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 43, normalized size = 2.53 \begin {gather*} -73 \left (\frac {\log (x)}{73}+\frac {1}{73} \log (5-x-\log (9))-\frac {1}{73} \log \left (73-1500 x+300 x^2+300 x \log (9)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(365 - 146*x - 73*Log[9])/(-365*x + 7573*x^2 - 3000*x^3 + 300*x^4 + (73*x - 3000*x^2 + 600*x^3)*Log[
9] + 300*x^2*Log[9]^2),x]

[Out]

-73*(Log[x]/73 + Log[5 - x - Log[9]]/73 - Log[73 - 1500*x + 300*x^2 + 300*x*Log[9]]/73)

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fricas [A]  time = 0.64, size = 32, normalized size = 1.88 \begin {gather*} \log \left (300 \, x^{2} + 600 \, x \log \relax (3) - 1500 \, x + 73\right ) - \log \left (x^{2} + 2 \, x \log \relax (3) - 5 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-146*log(3)-146*x+365)/(1200*x^2*log(3)^2+2*(600*x^3-3000*x^2+73*x)*log(3)+300*x^4-3000*x^3+7573*x^
2-365*x),x, algorithm="fricas")

[Out]

log(300*x^2 + 600*x*log(3) - 1500*x + 73) - log(x^2 + 2*x*log(3) - 5*x)

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giac [A]  time = 0.15, size = 34, normalized size = 2.00 \begin {gather*} \log \left ({\left | 300 \, x^{2} + 600 \, x \log \relax (3) - 1500 \, x + 73 \right |}\right ) - \log \left ({\left | x^{2} + 2 \, x \log \relax (3) - 5 \, x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-146*log(3)-146*x+365)/(1200*x^2*log(3)^2+2*(600*x^3-3000*x^2+73*x)*log(3)+300*x^4-3000*x^3+7573*x^
2-365*x),x, algorithm="giac")

[Out]

log(abs(300*x^2 + 600*x*log(3) - 1500*x + 73)) - log(abs(x^2 + 2*x*log(3) - 5*x))

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maple [A]  time = 0.07, size = 32, normalized size = 1.88

method	result	size
default	\(-\ln \relax (x )+\ln \left (600 x \ln \relax (3)+300 x^{2}-1500 x +73\right )-\ln \left (2 \ln \relax (3)+x -5\right )\)	\(32\)
norman	\(-\ln \relax (x )+\ln \left (600 x \ln \relax (3)+300 x^{2}-1500 x +73\right )-\ln \left (2 \ln \relax (3)+x -5\right )\)	\(32\)
risch	\(-\ln \left (x^{2}+\left (2 \ln \relax (3)-5\right ) x \right )+\ln \left (-300 x^{2}+\left (-600 \ln \relax (3)+1500\right ) x -73\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-146*ln(3)-146*x+365)/(1200*x^2*ln(3)^2+2*(600*x^3-3000*x^2+73*x)*ln(3)+300*x^4-3000*x^3+7573*x^2-365*x),
x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(600*x*ln(3)+300*x^2-1500*x+73)-ln(2*ln(3)+x-5)

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maxima [A]  time = 0.36, size = 32, normalized size = 1.88 \begin {gather*} \log \left (300 \, x^{2} + 300 \, x {\left (2 \, \log \relax (3) - 5\right )} + 73\right ) - \log \left (x + 2 \, \log \relax (3) - 5\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-146*log(3)-146*x+365)/(1200*x^2*log(3)^2+2*(600*x^3-3000*x^2+73*x)*log(3)+300*x^4-3000*x^3+7573*x^
2-365*x),x, algorithm="maxima")

[Out]

log(300*x^2 + 300*x*(2*log(3) - 5) + 73) - log(x + 2*log(3) - 5) - log(x)

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mupad [B]  time = 0.34, size = 88, normalized size = 5.18 \begin {gather*} -\mathrm {atan}\left (\frac {-\ln \relax (3)\,525600000{}\mathrm {i}+x\,\left (105120000\,\ln \relax (3)-262800000\right )\,2{}\mathrm {i}+{\ln \relax (3)}^2\,105120000{}\mathrm {i}+x^2\,105120000{}\mathrm {i}[phone]{}\mathrm {i}}{2\,x^2\,\left (432000000\,{\ln \relax (3)}^2-2160000000\,\ln \relax (3)[phone]\right )-525600000\,\ln \relax (3)[phone]\,{\ln \relax (3)}^2+2\,x\,\left (16094880000\,\ln \relax (3)-6480000000\,{\ln \relax (3)}^2+864000000\,{\ln \relax (3)}^3-13237200000\right )[phone]}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(146*x + 146*log(3) - 365)/(1200*x^2*log(3)^2 - 365*x + 2*log(3)*(73*x - 3000*x^2 + 600*x^3) + 7573*x^2 -
3000*x^3 + 300*x^4),x)

[Out]

-atan((x*(105120000*log(3) - 262800000)*2i - log(3)*525600000i + log(3)^2*105120000i + x^2*105120000i + 657000
000i)/(2*x^2*(432000000*log(3)^2 - 2160000000*log(3) + 2647440000) - 525600000*log(3) + 105120000*log(3)^2 + 2
*x*(16094880000*log(3) - 6480000000*log(3)^2 + 864000000*log(3)^3 - 13237200000) + 657000000))*2i

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sympy [A]  time = 0.84, size = 29, normalized size = 1.71 \begin {gather*} - \log {\left (x^{2} + x \left (-5 + 2 \log {\relax (3 )}\right ) \right )} + \log {\left (x^{2} + x \left (-5 + 2 \log {\relax (3 )}\right ) + \frac {73}{300} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-146*ln(3)-146*x+365)/(1200*x**2*ln(3)**2+2*(600*x**3-3000*x**2+73*x)*ln(3)+300*x**4-3000*x**3+7573
*x**2-365*x),x)

[Out]

-log(x**2 + x*(-5 + 2*log(3))) + log(x**2 + x*(-5 + 2*log(3)) + 73/300)

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