3.6.97 \(\int \frac {4+6 x+4 \log (-\frac {x^2}{\log (5)})}{x^2+x \log (-\frac {x^2}{\log (5)})} \, dx\)

Optimal. Leaf size=19 \[ \log \left (x^4 \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )^2\right ) \]

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Rubi [A]  time = 0.28, antiderivative size = 20, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2561, 6741, 12, 6742, 6684} \begin {gather*} 2 \log \left (\log \left (-\frac {x^2}{\log (5)}\right )+x\right )+4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 6*x + 4*Log[-(x^2/Log[5])])/(x^2 + x*Log[-(x^2/Log[5])]),x]

[Out]

4*Log[x] + 2*Log[x + Log[-(x^2/Log[5])]]

Rule 12

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

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6684

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

Rule 6741

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

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 {4+6 x+4 \log \left (-\frac {x^2}{\log (5)}\right )}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=\int \frac {2 \left (2+3 x+2 \log \left (-\frac {x^2}{\log (5)}\right )\right )}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=2 \int \frac {2+3 x+2 \log \left (-\frac {x^2}{\log (5)}\right )}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=2 \int \left (\frac {2}{x}+\frac {2+x}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )}\right ) \, dx\\ &=4 \log (x)+2 \int \frac {2+x}{x \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )} \, dx\\ &=4 \log (x)+2 \log \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 20, normalized size = 1.05 \begin {gather*} 4 \log (x)+2 \log \left (x+\log \left (-\frac {x^2}{\log (5)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 6*x + 4*Log[-(x^2/Log[5])])/(x^2 + x*Log[-(x^2/Log[5])]),x]

[Out]

4*Log[x] + 2*Log[x + Log[-(x^2/Log[5])]]

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fricas [A]  time = 0.59, size = 28, normalized size = 1.47 \begin {gather*} 2 \, \log \left (x + \log \left (-\frac {x^{2}}{\log \relax (5)}\right )\right ) + 2 \, \log \left (-\frac {x^{2}}{\log \relax (5)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(-x^2/log(5))+6*x+4)/(x*log(-x^2/log(5))+x^2),x, algorithm="fricas")

[Out]

2*log(x + log(-x^2/log(5))) + 2*log(-x^2/log(5))

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giac [A]  time = 0.39, size = 21, normalized size = 1.11 \begin {gather*} 2 \, \log \left (x + \log \left (-x^{2}\right ) - \log \left (\log \relax (5)\right )\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(-x^2/log(5))+6*x+4)/(x*log(-x^2/log(5))+x^2),x, algorithm="giac")

[Out]

2*log(x + log(-x^2) - log(log(5))) + 4*log(x)

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maple [A]  time = 0.02, size = 21, normalized size = 1.11

method	result	size
norman	\(4 \ln \relax (x )+2 \ln \left (\ln \left (-\frac {x^{2}}{\ln \relax (5)}\right )+x \right )\)	\(21\)
risch	\(4 \ln \relax (x )+2 \ln \left (\ln \left (-\frac {x^{2}}{\ln \relax (5)}\right )+x \right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(-x^2/ln(5))+6*x+4)/(x*ln(-x^2/ln(5))+x^2),x,method=_RETURNVERBOSE)

[Out]

4*ln(x)+2*ln(ln(-x^2/ln(5))+x)

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maxima [A]  time = 0.71, size = 21, normalized size = 1.11 \begin {gather*} 4 \, \log \relax (x) + 2 \, \log \left (\frac {1}{2} \, x + \log \relax (x) - \frac {1}{2} \, \log \left (-\log \relax (5)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(-x^2/log(5))+6*x+4)/(x*log(-x^2/log(5))+x^2),x, algorithm="maxima")

[Out]

4*log(x) + 2*log(1/2*x + log(x) - 1/2*log(-log(5)))

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mupad [B]  time = 0.76, size = 23, normalized size = 1.21 \begin {gather*} 2\,\ln \left (x+\ln \left (-x^2\right )-\ln \left (\ln \relax (5)\right )\right )+2\,\ln \left (x^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + 4*log(-x^2/log(5)) + 4)/(x*log(-x^2/log(5)) + x^2),x)

[Out]

2*log(x + log(-x^2) - log(log(5))) + 2*log(x^2)

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sympy [A]  time = 0.13, size = 19, normalized size = 1.00 \begin {gather*} 4 \log {\relax (x )} + 2 \log {\left (x + \log {\left (- \frac {x^{2}}{\log {\relax (5 )}} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(-x**2/ln(5))+6*x+4)/(x*ln(-x**2/ln(5))+x**2),x)

[Out]

4*log(x) + 2*log(x + log(-x**2/log(5)))

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