3.91.40 \(\int \frac {-45+54 x+7 x^2+x^3+(-18+18 x+6 x^2) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+(18 x+6 x^2) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx\)

Optimal. Leaf size=27 \[ x+\log \left (\frac {5+\left (-1-\frac {x}{3}-\log (3)\right )^2+\log (x)}{26 x}\right ) \]

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Rubi [A]  time = 0.37, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 4, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6, 6742, 43, 6684} \begin {gather*} \log \left (x^2+6 x (1+\log (3))+9 \log (x)+9 \left (6+\log ^2(3)+\log (9)\right )\right )+x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-45 + 54*x + 7*x^2 + x^3 + (-18 + 18*x + 6*x^2)*Log[3] + (-9 + 9*x)*Log[3]^2 + (-9 + 9*x)*Log[x])/(54*x +
6*x^2 + x^3 + (18*x + 6*x^2)*Log[3] + 9*x*Log[3]^2 + 9*x*Log[x]),x]

[Out]

x - Log[x] + Log[x^2 + 6*x*(1 + Log[3]) + 9*(6 + Log[3]^2 + Log[9]) + 9*Log[x]]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+x \left (54+9 \log ^2(3)\right )+9 x \log (x)} \, dx\\ &=\int \left (\frac {-1+x}{x}+\frac {9+2 x^2+6 x (1+\log (3))}{x \left (x^2+6 x (1+\log (3))+54 \left (1+\frac {1}{6} \left (\log ^2(3)+\log (9)\right )\right )+9 \log (x)\right )}\right ) \, dx\\ &=\int \frac {-1+x}{x} \, dx+\int \frac {9+2 x^2+6 x (1+\log (3))}{x \left (x^2+6 x (1+\log (3))+54 \left (1+\frac {1}{6} \left (\log ^2(3)+\log (9)\right )\right )+9 \log (x)\right )} \, dx\\ &=\log \left (x^2+6 x (1+\log (3))+9 \left (6+\log ^2(3)+\log (9)\right )+9 \log (x)\right )+\int \left (1-\frac {1}{x}\right ) \, dx\\ &=x-\log (x)+\log \left (x^2+6 x (1+\log (3))+9 \left (6+\log ^2(3)+\log (9)\right )+9 \log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-45+54 x+7 x^2+x^3+\left (-18+18 x+6 x^2\right ) \log (3)+(-9+9 x) \log ^2(3)+(-9+9 x) \log (x)}{54 x+6 x^2+x^3+\left (18 x+6 x^2\right ) \log (3)+9 x \log ^2(3)+9 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-45 + 54*x + 7*x^2 + x^3 + (-18 + 18*x + 6*x^2)*Log[3] + (-9 + 9*x)*Log[3]^2 + (-9 + 9*x)*Log[x])/(
54*x + 6*x^2 + x^3 + (18*x + 6*x^2)*Log[3] + 9*x*Log[3]^2 + 9*x*Log[x]),x]

[Out]

Integrate[(-45 + 54*x + 7*x^2 + x^3 + (-18 + 18*x + 6*x^2)*Log[3] + (-9 + 9*x)*Log[3]^2 + (-9 + 9*x)*Log[x])/(
54*x + 6*x^2 + x^3 + (18*x + 6*x^2)*Log[3] + 9*x*Log[3]^2 + 9*x*Log[x]), x]

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fricas [A]  time = 0.46, size = 32, normalized size = 1.19 \begin {gather*} x + \log \left (x^{2} + 6 \, {\left (x + 3\right )} \log \relax (3) + 9 \, \log \relax (3)^{2} + 6 \, x + 9 \, \log \relax (x) + 54\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^2+54*x-45)/(9*x*log(x)+9*x*log(3)^2+
(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x, algorithm="fricas")

[Out]

x + log(x^2 + 6*(x + 3)*log(3) + 9*log(3)^2 + 6*x + 9*log(x) + 54) - log(x)

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giac [A]  time = 0.14, size = 34, normalized size = 1.26 \begin {gather*} x + \log \left (x^{2} + 6 \, x \log \relax (3) + 9 \, \log \relax (3)^{2} + 6 \, x + 18 \, \log \relax (3) + 9 \, \log \relax (x) + 54\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^2+54*x-45)/(9*x*log(x)+9*x*log(3)^2+
(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x, algorithm="giac")

[Out]

x + log(x^2 + 6*x*log(3) + 9*log(3)^2 + 6*x + 18*log(3) + 9*log(x) + 54) - log(x)

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maple [A]  time = 0.07, size = 33, normalized size = 1.22

method	result	size
risch	\(x -\ln \relax (x )+\ln \left (\ln \relax (3)^{2}+\frac {2 x \ln \relax (3)}{3}+\frac {x^{2}}{9}+2 \ln \relax (3)+\frac {2 x}{3}+\ln \relax (x )+6\right )\)	\(33\)
norman	\(x -\ln \relax (x )+\ln \left (9 \ln \relax (3)^{2}+6 x \ln \relax (3)+x^{2}+9 \ln \relax (x )+18 \ln \relax (3)+6 x +54\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((9*x-9)*ln(x)+(9*x-9)*ln(3)^2+(6*x^2+18*x-18)*ln(3)+x^3+7*x^2+54*x-45)/(9*x*ln(x)+9*x*ln(3)^2+(6*x^2+18*x
)*ln(3)+x^3+6*x^2+54*x),x,method=_RETURNVERBOSE)

[Out]

x-ln(x)+ln(ln(3)^2+2/3*x*ln(3)+1/9*x^2+2*ln(3)+2/3*x+ln(x)+6)

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maxima [A]  time = 0.46, size = 31, normalized size = 1.15 \begin {gather*} x + \log \left (\frac {1}{9} \, x^{2} + \frac {2}{3} \, x {\left (\log \relax (3) + 1\right )} + \log \relax (3)^{2} + 2 \, \log \relax (3) + \log \relax (x) + 6\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-9)*log(x)+(9*x-9)*log(3)^2+(6*x^2+18*x-18)*log(3)+x^3+7*x^2+54*x-45)/(9*x*log(x)+9*x*log(3)^2+
(6*x^2+18*x)*log(3)+x^3+6*x^2+54*x),x, algorithm="maxima")

[Out]

x + log(1/9*x^2 + 2/3*x*(log(3) + 1) + log(3)^2 + 2*log(3) + log(x) + 6) - log(x)

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mupad [B]  time = 8.13, size = 30, normalized size = 1.11 \begin {gather*} x+\ln \left (\frac {2\,x}{3}+\ln \left (9\,x\right )+\frac {2\,x\,\ln \relax (3)}{3}+{\ln \relax (3)}^2+\frac {x^2}{9}+6\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((54*x + log(3)*(18*x + 6*x^2 - 18) + log(3)^2*(9*x - 9) + log(x)*(9*x - 9) + 7*x^2 + x^3 - 45)/(54*x + log
(3)*(18*x + 6*x^2) + 9*x*log(3)^2 + 9*x*log(x) + 6*x^2 + x^3),x)

[Out]

x + log((2*x)/3 + log(9*x) + (2*x*log(3))/3 + log(3)^2 + x^2/9 + 6) - log(x)

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sympy [A]  time = 0.25, size = 39, normalized size = 1.44 \begin {gather*} x - \log {\relax (x )} + \log {\left (\frac {x^{2}}{9} + \frac {2 x}{3} + \frac {2 x \log {\relax (3 )}}{3} + \log {\relax (x )} + \log {\relax (3 )}^{2} + 2 \log {\relax (3 )} + 6 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((9*x-9)*ln(x)+(9*x-9)*ln(3)**2+(6*x**2+18*x-18)*ln(3)+x**3+7*x**2+54*x-45)/(9*x*ln(x)+9*x*ln(3)**2+
(6*x**2+18*x)*ln(3)+x**3+6*x**2+54*x),x)

[Out]

x - log(x) + log(x**2/9 + 2*x/3 + 2*x*log(3)/3 + log(x) + log(3)**2 + 2*log(3) + 6)

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