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

Optimal. Leaf size=21 \[ 8-x+\log (5) (-x+\log (5) \log (4-x)) \]

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {186, 43} \begin {gather*} \log ^2(5) \log (4-x)-x (1+\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(1 + Log[5])) + Log[5]^2*Log[4 - 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 186

Int[(u_)^(m_.)*(v_)^(n_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^n, x] /; FreeQ[{m, n}, x] &&
LinearQ[{u, v}, x] &&  !LinearMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x (1+\log (5))+(2+\log (5))^2}{-4+x} \, dx\\ &=\int \left (-1-\log (5)+\frac {\log ^2(5)}{-4+x}\right ) \, dx\\ &=-x (1+\log (5))+\log ^2(5) \log (4-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.95 \begin {gather*} (-4+x) (-1-\log (5))+\log ^2(5) \log (-4+x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4 + x)*(-1 - Log[5]) + Log[5]^2*Log[-4 + x]

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fricas [A]  time = 0.58, size = 18, normalized size = 0.86 \begin {gather*} \log \relax (5)^{2} \log \left (x - 4\right ) - x \log \relax (5) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(5)^2*log(x - 4) - x*log(5) - x

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giac [A]  time = 0.12, size = 19, normalized size = 0.90 \begin {gather*} \log \relax (5)^{2} \log \left ({\left | x - 4 \right |}\right ) - x \log \relax (5) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.47, size = 19, normalized size = 0.90

method	result	size
default	\(-x \ln \relax (5)-x +\ln \relax (5)^{2} \ln \left (x -4\right )\)	\(19\)
norman	\(\left (-\ln \relax (5)-1\right ) x +\ln \relax (5)^{2} \ln \left (x -4\right )\)	\(19\)
risch	\(-x \ln \relax (5)-x +\ln \relax (5)^{2} \ln \left (x -4\right )\)	\(19\)
meijerg	\(4 \ln \left (-\frac {x}{4}+1\right )-4 \left (-\ln \relax (5)-1\right ) \left (-\frac {x}{4}-\ln \left (-\frac {x}{4}+1\right )\right )+\ln \relax (5)^{2} \ln \left (-\frac {x}{4}+1\right )+4 \ln \relax (5) \ln \left (-\frac {x}{4}+1\right )\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 17, normalized size = 0.81 \begin {gather*} \log \relax (5)^{2} \log \left (x - 4\right ) - x {\left (\log \relax (5) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(5)^2*log(x - 4) - x*(log(5) + 1)

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mupad [B]  time = 5.91, size = 17, normalized size = 0.81 \begin {gather*} \ln \left (x-4\right )\,{\ln \relax (5)}^2-x\,\left (\ln \relax (5)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - 4)*log(5)^2 - x*(log(5) + 1)

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sympy [A]  time = 0.12, size = 15, normalized size = 0.71 \begin {gather*} - x \left (1 + \log {\relax (5 )}\right ) + \log {\relax (5 )}^{2} \log {\left (x - 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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