3.45.92 \(\int \frac {2 \log (\frac {1}{120} (x+240 e^{e^{10}} x))}{x} \, dx\)

Optimal. Leaf size=17 \[ \log ^2\left (\frac {x}{120}+2 e^{e^{10}} x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2421, 2301} \begin {gather*} \log ^2\left (\frac {1}{120} \left (1+240 e^{e^{10}}\right ) x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*Log[(x + 240*E^E^10*x)/120])/x,x]

[Out]

Log[((1 + 240*E^E^10)*x)/120]^2

Rule 12

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2421

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*(a + b*Log[c*
ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && LinearQ[v, x] &&  !(Binomial
MatchQ[u, x] && LinearMatchQ[v, x])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 \int \frac {\log \left (\frac {1}{120} \left (x+240 e^{e^{10}} x\right )\right )}{x} \, dx\\ &=2 \int \frac {\log \left (\frac {1}{120} \left (1+240 e^{e^{10}}\right ) x\right )}{x} \, dx\\ &=\log ^2\left (\frac {1}{120} \left (1+240 e^{e^{10}}\right ) x\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} \log ^2\left (\left (\frac {1}{120}+2 e^{e^{10}}\right ) x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*Log[(x + 240*E^E^10*x)/120])/x,x]

[Out]

Log[(1/120 + 2*E^E^10)*x]^2

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 13, normalized size = 0.76 \begin {gather*} \log \left (2 \, x e^{\left (e^{10}\right )} + \frac {1}{120} \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(2*x*exp(exp(10))+1/120*x)/x,x, algorithm="fricas")

[Out]

log(2*x*e^(e^10) + 1/120*x)^2

________________________________________________________________________________________

giac [A]  time = 0.24, size = 13, normalized size = 0.76 \begin {gather*} \log \left (2 \, x e^{\left (e^{10}\right )} + \frac {1}{120} \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(2*x*exp(exp(10))+1/120*x)/x,x, algorithm="giac")

[Out]

log(2*x*e^(e^10) + 1/120*x)^2

________________________________________________________________________________________

maple [A]  time = 0.06, size = 13, normalized size = 0.76

method	result	size
derivativedivides	\(\ln \left (\left (2 \,{\mathrm e}^{{\mathrm e}^{10}}+\frac {1}{120}\right ) x \right )^{2}\)	\(13\)
default	\(\ln \left (\left (2 \,{\mathrm e}^{{\mathrm e}^{10}}+\frac {1}{120}\right ) x \right )^{2}\)	\(13\)
norman	\(\ln \left (2 x \,{\mathrm e}^{{\mathrm e}^{10}}+\frac {x}{120}\right )^{2}\)	\(14\)
risch	\(\ln \left (2 x \,{\mathrm e}^{{\mathrm e}^{10}}+\frac {x}{120}\right )^{2}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*ln(2*x*exp(exp(10))+1/120*x)/x,x,method=_RETURNVERBOSE)

[Out]

ln((2*exp(exp(10))+1/120)*x)^2

________________________________________________________________________________________

maxima [B]  time = 0.44, size = 28, normalized size = 1.65 \begin {gather*} -2 \, {\left (\log \relax (5) + \log \relax (3) + 3 \, \log \relax (2) - \log \left (240 \, e^{\left (e^{10}\right )} + 1\right )\right )} \log \relax (x) + \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*log(2*x*exp(exp(10))+1/120*x)/x,x, algorithm="maxima")

[Out]

-2*(log(5) + log(3) + 3*log(2) - log(240*e^(e^10) + 1))*log(x) + log(x)^2

________________________________________________________________________________________

mupad [B]  time = 4.17, size = 13, normalized size = 0.76 \begin {gather*} {\ln \left (\frac {x}{120}+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{10}}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(x/120 + 2*x*exp(exp(10))))/x,x)

[Out]

log(x/120 + 2*x*exp(exp(10)))^2

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 14, normalized size = 0.82 \begin {gather*} \log {\left (\frac {x}{120} + 2 x e^{e^{10}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*ln(2*x*exp(exp(10))+1/120*x)/x,x)

[Out]

log(x/120 + 2*x*exp(exp(10)))**2

________________________________________________________________________________________