3.41.8 \(\int \frac {e^{-6+3 e^3} (-1-\log (4))+e^{-6+3 e^3} \log (x)}{-x \log (4)+x \log (x)} \, dx\) [4008]

Optimal. Leaf size=23 \[ e^{3 \left (-2+e^3\right )} (\log (x)-\log (\log (4)-\log (x))) \]

[Out]

(ln(x)-ln(2*ln(2)-ln(x)))*exp(3*exp(3)-6)

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Rubi [A]
time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.39, number of steps used = 4, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 45} \begin {gather*} e^{3 e^3-6} \log (x)-e^{3 e^3-6} \log (\log (4)-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-6 + 3*E^3)*(-1 - Log[4]) + E^(-6 + 3*E^3)*Log[x])/(-(x*Log[4]) + x*Log[x]),x]

[Out]

E^(-6 + 3*E^3)*Log[x] - E^(-6 + 3*E^3)*Log[Log[4] - 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 45

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])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int \frac {e^{-6+3 e^3} (1-x+\log (4))}{-x+\log (4)} \, dx,x,\log (x)\right )\\ &=e^{-6+3 e^3} \text {Subst}\left (\int \frac {1-x+\log (4)}{-x+\log (4)} \, dx,x,\log (x)\right )\\ &=e^{-6+3 e^3} \text {Subst}\left (\int \left (1+\frac {1}{-x+\log (4)}\right ) \, dx,x,\log (x)\right )\\ &=e^{-6+3 e^3} \log (x)-e^{-6+3 e^3} \log (\log (4)-\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 26, normalized size = 1.13 \begin {gather*} e^{-6+3 e^3} \left (\log \left (\frac {x}{4}\right )-\log \left (\log \left (\frac {x}{4}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 + 3*E^3)*(-1 - Log[4]) + E^(-6 + 3*E^3)*Log[x])/(-(x*Log[4]) + x*Log[x]),x]

[Out]

E^(-6 + 3*E^3)*(Log[x/4] - Log[Log[x/4]])

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Maple [A]
time = 0.21, size = 22, normalized size = 0.96

method	result	size
default	\({\mathrm e}^{3 \,{\mathrm e}^{3}} {\mathrm e}^{-6} \left (\ln \left (x \right )-\ln \left (\ln \left (x \right )-2 \ln \left (2\right )\right )\right )\)	\(22\)
risch	\({\mathrm e}^{3 \,{\mathrm e}^{3}-6} \ln \left (x \right )-\ln \left (\ln \left (x \right )-2 \ln \left (2\right )\right ) {\mathrm e}^{3 \,{\mathrm e}^{3}-6}\)	\(29\)
norman	\({\mathrm e}^{3 \,{\mathrm e}^{3}} {\mathrm e}^{-6} \ln \left (x \right )-{\mathrm e}^{3 \,{\mathrm e}^{3}} {\mathrm e}^{-6} \ln \left (2 \ln \left (2\right )-\ln \left (x \right )\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*exp(3)-6)*ln(x)+(-1-2*ln(2))*exp(3*exp(3)-6))/(x*ln(x)-2*x*ln(2)),x,method=_RETURNVERBOSE)

[Out]

exp(exp(3))^3*exp(-6)*(ln(x)-ln(ln(x)-2*ln(2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (23) = 46\).
time = 0.49, size = 88, normalized size = 3.83 \begin {gather*} -2 \, e^{\left (3 \, e^{3} - 6\right )} \log \left (2\right ) \log \left (-2 \, \log \left (2\right ) + \log \left (x\right )\right ) + e^{\left (3 \, e^{3} - 6\right )} \log \left (x\right ) \log \left (-2 \, \log \left (2\right ) + \log \left (x\right )\right ) + {\left ({\left (2 \, \log \left (2\right ) - \log \left (x\right )\right )} \log \left (-2 \, \log \left (2\right ) + \log \left (x\right )\right ) - 2 \, \log \left (2\right ) + \log \left (x\right )\right )} e^{\left (3 \, e^{3} - 6\right )} - e^{\left (3 \, e^{3} - 6\right )} \log \left (-2 \, \log \left (2\right ) + \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3*exp(3)-6)*log(x)+(-1-2*log(2))*exp(3*exp(3)-6))/(x*log(x)-2*x*log(2)),x, algorithm="maxima")

[Out]

-2*e^(3*e^3 - 6)*log(2)*log(-2*log(2) + log(x)) + e^(3*e^3 - 6)*log(x)*log(-2*log(2) + log(x)) + ((2*log(2) -
log(x))*log(-2*log(2) + log(x)) - 2*log(2) + log(x))*e^(3*e^3 - 6) - e^(3*e^3 - 6)*log(-2*log(2) + log(x))

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Fricas [A]
time = 0.36, size = 28, normalized size = 1.22 \begin {gather*} e^{\left (3 \, e^{3} - 6\right )} \log \left (x\right ) - e^{\left (3 \, e^{3} - 6\right )} \log \left (-2 \, \log \left (2\right ) + \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3*exp(3)-6)*log(x)+(-1-2*log(2))*exp(3*exp(3)-6))/(x*log(x)-2*x*log(2)),x, algorithm="fricas")

[Out]

e^(3*e^3 - 6)*log(x) - e^(3*e^3 - 6)*log(-2*log(2) + log(x))

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Sympy [A]
time = 0.06, size = 32, normalized size = 1.39 \begin {gather*} \frac {e^{3 e^{3}} \log {\left (x \right )}}{e^{6}} - \frac {e^{3 e^{3}} \log {\left (\log {\left (x \right )} - 2 \log {\left (2 \right )} \right )}}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3*exp(3)-6)*ln(x)+(-1-2*ln(2))*exp(3*exp(3)-6))/(x*ln(x)-2*x*ln(2)),x)

[Out]

exp(-6)*exp(3*exp(3))*log(x) - exp(-6)*exp(3*exp(3))*log(log(x) - 2*log(2))

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Giac [A]
time = 0.40, size = 45, normalized size = 1.96 \begin {gather*} -\frac {1}{2} \, e^{\left (3 \, e^{3} - 6\right )} \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (2 \, \log \left (2\right ) - \log \left ({\left | x \right |}\right )\right )}^{2}\right ) + e^{\left (3 \, e^{3} - 6\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(3*exp(3)-6)*log(x)+(-1-2*log(2))*exp(3*exp(3)-6))/(x*log(x)-2*x*log(2)),x, algorithm="giac")

[Out]

-1/2*e^(3*e^3 - 6)*log(1/4*pi^2*(sgn(x) - 1)^2 + (2*log(2) - log(abs(x)))^2) + e^(3*e^3 - 6)*log(x)

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Mupad [B]
time = 2.59, size = 19, normalized size = 0.83 \begin {gather*} -{\mathrm {e}}^{3\,{\mathrm {e}}^3-6}\,\left (\ln \left (\ln \left (\frac {x}{4}\right )\right )-\ln \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*exp(3) - 6)*(2*log(2) + 1) - exp(3*exp(3) - 6)*log(x))/(2*x*log(2) - x*log(x)),x)

[Out]

-exp(3*exp(3) - 6)*(log(log(x/4)) - log(x))

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