3.58.40 \(\int \frac {4+(-3-\log (5)) \log (x)}{3+\log (5)} \, dx\) [5740]

Optimal. Leaf size=18 \[ x+\frac {4 (1+x)}{3+\log (5)}-x \log (x) \]

[Out]

2/(1/2*ln(5)+3/2)*(1+x)+x-x*ln(x)

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2332} \begin {gather*} x+x (-\log (x))+\frac {4 x}{3+\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + (-3 - Log[5])*Log[x])/(3 + Log[5]),x]

[Out]

x + (4*x)/(3 + Log[5]) - x*Log[x]

Rule 12

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int (4+(-3-\log (5)) \log (x)) \, dx}{3+\log (5)}\\ &=\frac {4 x}{3+\log (5)}-\int \log (x) \, dx\\ &=x+\frac {4 x}{3+\log (5)}-x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.89 \begin {gather*} x+\frac {4 x}{3+\log (5)}-x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + (-3 - Log[5])*Log[x])/(3 + Log[5]),x]

[Out]

x + (4*x)/(3 + Log[5]) - x*Log[x]

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Maple [A]
time = 0.29, size = 28, normalized size = 1.56

method	result	size
norman	\(\frac {\left (7+\ln \left (5\right )\right ) x}{\ln \left (5\right )+3}-x \ln \left (x \right )\)	\(19\)
risch	\(\frac {7 x}{\ln \left (5\right )+3}+\frac {x \ln \left (5\right )}{\ln \left (5\right )+3}-x \ln \left (x \right )\)	\(26\)
default	\(\frac {7 x -x \ln \left (5\right ) \ln \left (x \right )-3 x \ln \left (x \right )+x \ln \left (5\right )}{\ln \left (5\right )+3}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(5)-3)*ln(x)+4)/(ln(5)+3),x,method=_RETURNVERBOSE)

[Out]

1/(ln(5)+3)*(7*x-x*ln(5)*ln(x)-3*x*ln(x)+x*ln(5))

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Maxima [A]
time = 0.35, size = 25, normalized size = 1.39 \begin {gather*} -\frac {{\left (x \log \left (x\right ) - x\right )} {\left (\log \left (5\right ) + 3\right )} - 4 \, x}{\log \left (5\right ) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)-3)*log(x)+4)/(log(5)+3),x, algorithm="maxima")

[Out]

-((x*log(x) - x)*(log(5) + 3) - 4*x)/(log(5) + 3)

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Fricas [A]
time = 0.38, size = 27, normalized size = 1.50 \begin {gather*} \frac {x \log \left (5\right ) - {\left (x \log \left (5\right ) + 3 \, x\right )} \log \left (x\right ) + 7 \, x}{\log \left (5\right ) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)-3)*log(x)+4)/(log(5)+3),x, algorithm="fricas")

[Out]

(x*log(5) - (x*log(5) + 3*x)*log(x) + 7*x)/(log(5) + 3)

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Sympy [A]
time = 0.04, size = 15, normalized size = 0.83 \begin {gather*} - x \log {\left (x \right )} + \frac {x \left (\log {\left (5 \right )} + 7\right )}{\log {\left (5 \right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(5)-3)*ln(x)+4)/(ln(5)+3),x)

[Out]

-x*log(x) + x*(log(5) + 7)/(log(5) + 3)

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Giac [A]
time = 0.40, size = 25, normalized size = 1.39 \begin {gather*} -\frac {{\left (x \log \left (x\right ) - x\right )} {\left (\log \left (5\right ) + 3\right )} - 4 \, x}{\log \left (5\right ) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)-3)*log(x)+4)/(log(5)+3),x, algorithm="giac")

[Out]

-((x*log(x) - x)*(log(5) + 3) - 4*x)/(log(5) + 3)

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Mupad [B]
time = 3.62, size = 18, normalized size = 1.00 \begin {gather*} \frac {x\,\left (\ln \left (5\right )+7\right )}{\ln \left (5\right )+3}-x\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(log(5) + 3) - 4)/(log(5) + 3),x)

[Out]

(x*(log(5) + 7))/(log(5) + 3) - x*log(x)

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