3.55.1 \(\int \frac {198-120 x+12 x^2+(-18+12 x) \log (2 x)}{-10 x+x^2+x \log (2 x)} \, dx\)

Optimal. Leaf size=27 \[ 3 \left (x+3 \left (-4+x+\log \left (\frac {(10-x-\log (2 x))^2}{x^2}\right )\right )\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6742, 43, 6684} \begin {gather*} 12 x-18 \log (x)+18 \log (-x-\log (2 x)+10) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(198 - 120*x + 12*x^2 + (-18 + 12*x)*Log[2*x])/(-10*x + x^2 + x*Log[2*x]),x]

[Out]

12*x - 18*Log[x] + 18*Log[10 - x - Log[2*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 \left (\frac {6 (-3+2 x)}{x}+\frac {18 (1+x)}{x (-10+x+\log (2 x))}\right ) \, dx\\ &=6 \int \frac {-3+2 x}{x} \, dx+18 \int \frac {1+x}{x (-10+x+\log (2 x))} \, dx\\ &=18 \log (10-x-\log (2 x))+6 \int \left (2-\frac {3}{x}\right ) \, dx\\ &=12 x-18 \log (x)+18 \log (10-x-\log (2 x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 24, normalized size = 0.89 \begin {gather*} 6 (2 x-3 \log (x)+3 \log (10-x-\log (2 x))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(198 - 120*x + 12*x^2 + (-18 + 12*x)*Log[2*x])/(-10*x + x^2 + x*Log[2*x]),x]

[Out]

6*(2*x - 3*Log[x] + 3*Log[10 - x - Log[2*x]])

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fricas [A]  time = 0.62, size = 20, normalized size = 0.74 \begin {gather*} 12 \, x - 18 \, \log \left (2 \, x\right ) + 18 \, \log \left (x + \log \left (2 \, x\right ) - 10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-18)*log(2*x)+12*x^2-120*x+198)/(x*log(2*x)+x^2-10*x),x, algorithm="fricas")

[Out]

12*x - 18*log(2*x) + 18*log(x + log(2*x) - 10)

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giac [A]  time = 0.12, size = 18, normalized size = 0.67 \begin {gather*} 12 \, x + 18 \, \log \left (x + \log \left (2 \, x\right ) - 10\right ) - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-18)*log(2*x)+12*x^2-120*x+198)/(x*log(2*x)+x^2-10*x),x, algorithm="giac")

[Out]

12*x + 18*log(x + log(2*x) - 10) - 18*log(x)

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maple [A]  time = 0.06, size = 19, normalized size = 0.70

method	result	size
risch	\(12 x -18 \ln \relax (x )+18 \ln \left (\ln \left (2 x \right )+x -10\right )\)	\(19\)
norman	\(-18 \ln \left (2 x \right )+12 x +18 \ln \left (\ln \left (2 x \right )+x -10\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x-18)*ln(2*x)+12*x^2-120*x+198)/(x*ln(2*x)+x^2-10*x),x,method=_RETURNVERBOSE)

[Out]

12*x-18*ln(x)+18*ln(ln(2*x)+x-10)

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maxima [A]  time = 0.55, size = 18, normalized size = 0.67 \begin {gather*} 12 \, x + 18 \, \log \left (x + \log \relax (2) + \log \relax (x) - 10\right ) - 18 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-18)*log(2*x)+12*x^2-120*x+198)/(x*log(2*x)+x^2-10*x),x, algorithm="maxima")

[Out]

12*x + 18*log(x + log(2) + log(x) - 10) - 18*log(x)

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mupad [B]  time = 3.60, size = 18, normalized size = 0.67 \begin {gather*} 12\,x+18\,\ln \left (x+\ln \left (2\,x\right )-10\right )-18\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^2 - 120*x + log(2*x)*(12*x - 18) + 198)/(x*log(2*x) - 10*x + x^2),x)

[Out]

12*x + 18*log(x + log(2*x) - 10) - 18*log(x)

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sympy [A]  time = 0.14, size = 19, normalized size = 0.70 \begin {gather*} 12 x - 18 \log {\relax (x )} + 18 \log {\left (x + \log {\left (2 x \right )} - 10 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x-18)*ln(2*x)+12*x**2-120*x+198)/(x*ln(2*x)+x**2-10*x),x)

[Out]

12*x - 18*log(x) + 18*log(x + log(2*x) - 10)

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