3.11.100 \(\int \frac {72-18 x-10 x^3+3 x^4+(-72+4 x^2+12 x^3) \log (x)}{-4 x^3+x^4+4 x^3 \log (x)} \, dx\)

Optimal. Leaf size=22 \[ \frac {9}{x^2}+3 x+\log (x)+\log (x-4 (1-\log (x))) \]

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Rubi [A]  time = 0.21, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6742, 14, 6684} \begin {gather*} \frac {9}{x^2}+3 x+\log (x)+\log (-x-4 \log (x)+4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(72 - 18*x - 10*x^3 + 3*x^4 + (-72 + 4*x^2 + 12*x^3)*Log[x])/(-4*x^3 + x^4 + 4*x^3*Log[x]),x]

[Out]

9/x^2 + 3*x + Log[x] + Log[4 - x - 4*Log[x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 {-18+x^2+3 x^3}{x^3}+\frac {4+x}{x (-4+x+4 \log (x))}\right ) \, dx\\ &=\int \frac {-18+x^2+3 x^3}{x^3} \, dx+\int \frac {4+x}{x (-4+x+4 \log (x))} \, dx\\ &=\log (4-x-4 \log (x))+\int \left (3-\frac {18}{x^3}+\frac {1}{x}\right ) \, dx\\ &=\frac {9}{x^2}+3 x+\log (x)+\log (4-x-4 \log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 21, normalized size = 0.95 \begin {gather*} \frac {9}{x^2}+3 x+\log (x)+\log (4-x-4 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(72 - 18*x - 10*x^3 + 3*x^4 + (-72 + 4*x^2 + 12*x^3)*Log[x])/(-4*x^3 + x^4 + 4*x^3*Log[x]),x]

[Out]

9/x^2 + 3*x + Log[x] + Log[4 - x - 4*Log[x]]

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fricas [A]  time = 1.45, size = 29, normalized size = 1.32 \begin {gather*} \frac {3 \, x^{3} + x^{2} \log \left (x + 4 \, \log \relax (x) - 4\right ) + x^{2} \log \relax (x) + 9}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^3+4*x^2-72)*log(x)+3*x^4-10*x^3-18*x+72)/(4*x^3*log(x)+x^4-4*x^3),x, algorithm="fricas")

[Out]

(3*x^3 + x^2*log(x + 4*log(x) - 4) + x^2*log(x) + 9)/x^2

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giac [A]  time = 0.32, size = 19, normalized size = 0.86 \begin {gather*} 3 \, x + \frac {9}{x^{2}} + \log \left (x + 4 \, \log \relax (x) - 4\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^3+4*x^2-72)*log(x)+3*x^4-10*x^3-18*x+72)/(4*x^3*log(x)+x^4-4*x^3),x, algorithm="giac")

[Out]

3*x + 9/x^2 + log(x + 4*log(x) - 4) + log(x)

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maple [A]  time = 0.03, size = 27, normalized size = 1.23

method	result	size
norman	\(\frac {9+x^{2} \ln \relax (x )+3 x^{3}}{x^{2}}+\ln \left (x -4+4 \ln \relax (x )\right )\)	\(27\)
risch	\(\frac {9+x^{2} \ln \relax (x )+3 x^{3}}{x^{2}}+\ln \left (\frac {x}{4}+\ln \relax (x )-1\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^3+4*x^2-72)*ln(x)+3*x^4-10*x^3-18*x+72)/(4*x^3*ln(x)+x^4-4*x^3),x,method=_RETURNVERBOSE)

[Out]

(9+x^2*ln(x)+3*x^3)/x^2+ln(x-4+4*ln(x))

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maxima [A]  time = 0.51, size = 21, normalized size = 0.95 \begin {gather*} \frac {3 \, {\left (x^{3} + 3\right )}}{x^{2}} + \log \relax (x) + \log \left (\frac {1}{4} \, x + \log \relax (x) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^3+4*x^2-72)*log(x)+3*x^4-10*x^3-18*x+72)/(4*x^3*log(x)+x^4-4*x^3),x, algorithm="maxima")

[Out]

3*(x^3 + 3)/x^2 + log(x) + log(1/4*x + log(x) - 1)

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mupad [B]  time = 0.81, size = 19, normalized size = 0.86 \begin {gather*} 3\,x+\ln \left (\frac {x}{4}+\ln \relax (x)-1\right )+\ln \relax (x)+\frac {9}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(4*x^2 + 12*x^3 - 72) - 18*x - 10*x^3 + 3*x^4 + 72)/(4*x^3*log(x) - 4*x^3 + x^4),x)

[Out]

3*x + log(x/4 + log(x) - 1) + log(x) + 9/x^2

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sympy [A]  time = 0.17, size = 20, normalized size = 0.91 \begin {gather*} 3 x + \log {\relax (x )} + \log {\left (\frac {x}{4} + \log {\relax (x )} - 1 \right )} + \frac {9}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**3+4*x**2-72)*ln(x)+3*x**4-10*x**3-18*x+72)/(4*x**3*ln(x)+x**4-4*x**3),x)

[Out]

3*x + log(x) + log(x/4 + log(x) - 1) + 9/x**2

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