3.13.55 \(\int \frac {-6+4 e^3-2 e^6+(4+2 e^6) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+(6-4 e^3+2 e^6+(2-4 e^3) \log (x)+2 \log ^2(x)) \log (4+e^6+(-1-2 e^3) \log (x)+\log ^2(x))}{4 x+e^6 x+(-x-2 e^3 x) \log (x)+x \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \left (-1+\log (x)+\log \left (4-\log (x)+\left (-e^3+\log (x)\right )^2\right )\right )^2 \]

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Rubi [A]  time = 0.34, antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 4, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 12, 6688, 6686} \begin {gather*} \left (-\log \left (\log ^2(x)-2 e^3 \log (x)-\log (x)+e^6+4\right )-\log (x)+1\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 + 4*E^3 - 2*E^6 + (4 + 2*E^6)*Log[x] - 4*E^3*Log[x]^2 + 2*Log[x]^3 + (6 - 4*E^3 + 2*E^6 + (2 - 4*E^3)*
Log[x] + 2*Log[x]^2)*Log[4 + E^6 + (-1 - 2*E^3)*Log[x] + Log[x]^2])/(4*x + E^6*x + (-x - 2*E^3*x)*Log[x] + x*L
og[x]^2),x]

[Out]

(1 - Log[x] - Log[4 + E^6 - Log[x] - 2*E^3*Log[x] + Log[x]^2])^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6+4 e^3-2 e^6+\left (4+2 e^6\right ) \log (x)-4 e^3 \log ^2(x)+2 \log ^3(x)+\left (6-4 e^3+2 e^6+\left (2-4 e^3\right ) \log (x)+2 \log ^2(x)\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) \log (x)+\log ^2(x)\right )}{\left (4+e^6\right ) x+\left (-x-2 e^3 x\right ) \log (x)+x \log ^2(x)} \, dx\\ &=\operatorname {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+2 x+e^6 x-2 e^3 x^2+x^3+3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6-x-2 e^3 x+x^2} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+2 x+e^6 x-2 e^3 x^2+x^3+3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+e^6 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+\left (3-2 e^3\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+\left (3-2 e^3+e^6\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )-2 e^3 x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {2 \left (-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+\left (3-2 e^3+e^6\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+\left (1-2 e^3\right ) x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {-3+2 e^3-e^6+\left (2+e^6\right ) x-2 e^3 x^2+x^3+\left (3-2 e^3+e^6\right ) \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+\left (1-2 e^3\right ) x \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )+x^2 \log \left (4+e^6+\left (-1-2 e^3\right ) x+x^2\right )}{4+e^6+\left (-1-2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {\left (3-2 e^3+e^6+\left (1-2 e^3\right ) x+x^2\right ) \left (-1+x+\log \left (4+e^6-x-2 e^3 x+x^2\right )\right )}{4+e^6-\left (1+2 e^3\right ) x+x^2} \, dx,x,\log (x)\right )\\ &=\left (1-\log (x)-\log \left (4+e^6-\log (x)-2 e^3 \log (x)+\log ^2(x)\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 27, normalized size = 1.17 \begin {gather*} \left (-1+\log (x)+\log \left (4+e^6-\left (1+2 e^3\right ) \log (x)+\log ^2(x)\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 4*E^3 - 2*E^6 + (4 + 2*E^6)*Log[x] - 4*E^3*Log[x]^2 + 2*Log[x]^3 + (6 - 4*E^3 + 2*E^6 + (2 - 4
*E^3)*Log[x] + 2*Log[x]^2)*Log[4 + E^6 + (-1 - 2*E^3)*Log[x] + Log[x]^2])/(4*x + E^6*x + (-x - 2*E^3*x)*Log[x]
+ x*Log[x]^2),x]

[Out]

(-1 + Log[x] + Log[4 + E^6 - (1 + 2*E^3)*Log[x] + Log[x]^2])^2

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fricas [B]  time = 0.67, size = 55, normalized size = 2.39 \begin {gather*} 2 \, {\left (\log \relax (x) - 1\right )} \log \left (-{\left (2 \, e^{3} + 1\right )} \log \relax (x) + \log \relax (x)^{2} + e^{6} + 4\right ) + \log \left (-{\left (2 \, e^{3} + 1\right )} \log \relax (x) + \log \relax (x)^{2} + e^{6} + 4\right )^{2} + \log \relax (x)^{2} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log(x)^2+(-2*exp(3)-1)*log(x)+exp(3)^2+
4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp(3)^2+4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x
)+x*exp(3)^2+4*x),x, algorithm="fricas")

[Out]

2*(log(x) - 1)*log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4) + log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4)^2
+ log(x)^2 - 2*log(x)

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giac [B]  time = 0.94, size = 74, normalized size = 3.22 \begin {gather*} \log \left (-2 \, e^{3} \log \relax (x) + \log \relax (x)^{2} + e^{6} - \log \relax (x) + 4\right )^{2} + 2 \, \log \left (-2 \, e^{3} \log \relax (x) + \log \relax (x)^{2} + e^{6} - \log \relax (x) + 4\right ) \log \relax (x) + \log \relax (x)^{2} - 2 \, \log \left (-2 \, e^{3} \log \relax (x) + \log \relax (x)^{2} + e^{6} - \log \relax (x) + 4\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log(x)^2+(-2*exp(3)-1)*log(x)+exp(3)^2+
4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp(3)^2+4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x
)+x*exp(3)^2+4*x),x, algorithm="giac")

[Out]

log(-2*e^3*log(x) + log(x)^2 + e^6 - log(x) + 4)^2 + 2*log(-2*e^3*log(x) + log(x)^2 + e^6 - log(x) + 4)*log(x)
+ log(x)^2 - 2*log(-2*e^3*log(x) + log(x)^2 + e^6 - log(x) + 4) - 2*log(x)

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maple [B]  time = 0.09, size = 72, normalized size = 3.13

method	result	size
risch	\(\ln \left (\ln \relax (x )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \relax (x )+{\mathrm e}^{6}+4\right )^{2}+2 \ln \relax (x ) \ln \left (\ln \relax (x )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \relax (x )+{\mathrm e}^{6}+4\right )+\ln \relax (x )^{2}-2 \ln \relax (x )-2 \ln \left (\ln \relax (x )^{2}+\left (-2 \,{\mathrm e}^{3}-1\right ) \ln \relax (x )+{\mathrm e}^{6}+4\right )\)	\(72\)
default	error in gcdex: invalid arguments\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(x)^2+(-4*exp(3)+2)*ln(x)+2*exp(3)^2-4*exp(3)+6)*ln(ln(x)^2+(-2*exp(3)-1)*ln(x)+exp(3)^2+4)+2*ln(x)^
3-4*exp(3)*ln(x)^2+(2*exp(3)^2+4)*ln(x)-2*exp(3)^2+4*exp(3)-6)/(x*ln(x)^2+(-2*x*exp(3)-x)*ln(x)+x*exp(3)^2+4*x
),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x)^2+(-2*exp(3)-1)*ln(x)+exp(6)+4)^2+2*ln(x)*ln(ln(x)^2+(-2*exp(3)-1)*ln(x)+exp(6)+4)+ln(x)^2-2*ln(x)-2*
ln(ln(x)^2+(-2*exp(3)-1)*ln(x)+exp(6)+4)

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maxima [B]  time = 0.51, size = 55, normalized size = 2.39 \begin {gather*} 2 \, {\left (\log \relax (x) - 1\right )} \log \left (-{\left (2 \, e^{3} + 1\right )} \log \relax (x) + \log \relax (x)^{2} + e^{6} + 4\right ) + \log \left (-{\left (2 \, e^{3} + 1\right )} \log \relax (x) + \log \relax (x)^{2} + e^{6} + 4\right )^{2} + \log \relax (x)^{2} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)^2+(-4*exp(3)+2)*log(x)+2*exp(3)^2-4*exp(3)+6)*log(log(x)^2+(-2*exp(3)-1)*log(x)+exp(3)^2+
4)+2*log(x)^3-4*exp(3)*log(x)^2+(2*exp(3)^2+4)*log(x)-2*exp(3)^2+4*exp(3)-6)/(x*log(x)^2+(-2*x*exp(3)-x)*log(x
)+x*exp(3)^2+4*x),x, algorithm="maxima")

[Out]

2*(log(x) - 1)*log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4) + log(-(2*e^3 + 1)*log(x) + log(x)^2 + e^6 + 4)^2
+ log(x)^2 - 2*log(x)

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mupad [B]  time = 2.31, size = 74, normalized size = 3.22 \begin {gather*} {\ln \left ({\mathrm {e}}^6-\ln \relax (x)+{\ln \relax (x)}^2-2\,{\mathrm {e}}^3\,\ln \relax (x)+4\right )}^2+2\,\ln \left ({\mathrm {e}}^6-\ln \relax (x)+{\ln \relax (x)}^2-2\,{\mathrm {e}}^3\,\ln \relax (x)+4\right )\,\ln \relax (x)-2\,\ln \left ({\mathrm {e}}^6-\ln \relax (x)+{\ln \relax (x)}^2-2\,{\mathrm {e}}^3\,\ln \relax (x)+4\right )+{\ln \relax (x)}^2-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(3) - 2*exp(6) - 4*exp(3)*log(x)^2 + log(exp(6) + log(x)^2 - log(x)*(2*exp(3) + 1) + 4)*(2*exp(6) -
4*exp(3) + 2*log(x)^2 - log(x)*(4*exp(3) - 2) + 6) + 2*log(x)^3 + log(x)*(2*exp(6) + 4) - 6)/(4*x + x*log(x)^2
+ x*exp(6) - log(x)*(x + 2*x*exp(3))),x)

[Out]

log(x)^2 - 2*log(x) - 2*log(exp(6) - log(x) + log(x)^2 - 2*exp(3)*log(x) + 4) + 2*log(exp(6) - log(x) + log(x)
^2 - 2*exp(3)*log(x) + 4)*log(x) + log(exp(6) - log(x) + log(x)^2 - 2*exp(3)*log(x) + 4)^2

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sympy [B]  time = 0.54, size = 88, normalized size = 3.83 \begin {gather*} \log {\relax (x )}^{2} + 2 \log {\relax (x )} \log {\left (\log {\relax (x )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\relax (x )} + 4 + e^{6} \right )} - 2 \log {\relax (x )} + \log {\left (\log {\relax (x )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\relax (x )} + 4 + e^{6} \right )}^{2} - 2 \log {\left (\log {\relax (x )}^{2} + \left (- 2 e^{3} - 1\right ) \log {\relax (x )} + 4 + e^{6} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(x)**2+(-4*exp(3)+2)*ln(x)+2*exp(3)**2-4*exp(3)+6)*ln(ln(x)**2+(-2*exp(3)-1)*ln(x)+exp(3)**2+4
)+2*ln(x)**3-4*exp(3)*ln(x)**2+(2*exp(3)**2+4)*ln(x)-2*exp(3)**2+4*exp(3)-6)/(x*ln(x)**2+(-2*x*exp(3)-x)*ln(x)
+x*exp(3)**2+4*x),x)

[Out]

log(x)**2 + 2*log(x)*log(log(x)**2 + (-2*exp(3) - 1)*log(x) + 4 + exp(6)) - 2*log(x) + log(log(x)**2 + (-2*exp
(3) - 1)*log(x) + 4 + exp(6))**2 - 2*log(log(x)**2 + (-2*exp(3) - 1)*log(x) + 4 + exp(6))

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