\(\int \frac {-2 x-\log (\frac {e^x}{6})}{x^2 \log ^3(\frac {e^x}{6})} \, dx\) [2355]

Optimal result

Integrand size = 28, antiderivative size = 16 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=6+\frac {1}{x \log ^2\left (\frac {e^x}{6}\right )} \]

[Out]

1/ln(1/6*exp(x))^2/x+6

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(16)=32\).

Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.44, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6874, 2194, 2191, 2188, 29, 2202} \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=\frac {1}{\left (x-\log \left (\frac {e^x}{6}\right )\right ) \log ^2\left (\frac {e^x}{6}\right )}-\frac {1}{x \left (x-\log \left (\frac {e^x}{6}\right )\right ) \log \left (\frac {e^x}{6}\right )} \]

[In]

Int[(-2*x - Log[E^x/6])/(x^2*Log[E^x/6]^3),x]

[Out]

1/((x - Log[E^x/6])*Log[E^x/6]^2) - 1/(x*(x - Log[E^x/6])*Log[E^x/6])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2188

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2191

Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Dist[b/(b*u - a*v), Int[1
/v, x], x] - Dist[a/(b*u - a*v), Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2194

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^(n + 1)/((n + 1)*
(b*u - a*v)), x] - Dist[a*((n + 1)/((n + 1)*(b*u - a*v))), Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Pi
ecewiseLinearQ[u, v, x] && LtQ[n, -1]

Rule 2202

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(-u^(m + 1))*(
v^(n + 1)/((m + 1)*(b*u - a*v))), x] + Dist[b*((m + n + 2)/((m + 1)*(b*u - a*v))), Int[u^(m + 1)*v^n, x], x] /
; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

Rule 6874

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(166\) vs. \(2(16)=32\).

Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 10.38 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=\frac {2 x^4-2 x^2 \log \left (\frac {e^x}{6}\right ) \left (4 x-3 \log (6)+3 \log \left (e^x\right )\right )+\log ^3\left (\frac {e^x}{6}\right ) \left (-11 x-2 \log (6)+2 \log \left (e^x\right )-6 x \log (x)+6 x \log \left (\log \left (\frac {e^x}{6}\right )\right )\right )+3 x \log ^2\left (\frac {e^x}{6}\right ) \left (6 x-\log (6)-2 \log (6) \log (x)+\log \left (e^x\right ) \left (1+2 \log (x)-2 \log \left (\log \left (\frac {e^x}{6}\right )\right )\right )+2 \log (6) \log \left (\log \left (\frac {e^x}{6}\right )\right )\right )}{2 x \log ^2\left (\frac {e^x}{6}\right ) \left (x+\log (6)-\log \left (e^x\right )\right )^4} \]

[In]

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

[Out]

(2*x^4 - 2*x^2*Log[E^x/6]*(4*x - 3*Log[6] + 3*Log[E^x]) + Log[E^x/6]^3*(-11*x - 2*Log[6] + 2*Log[E^x] - 6*x*Lo
g[x] + 6*x*Log[Log[E^x/6]]) + 3*x*Log[E^x/6]^2*(6*x - Log[6] - 2*Log[6]*Log[x] + Log[E^x]*(1 + 2*Log[x] - 2*Lo
g[Log[E^x/6]]) + 2*Log[6]*Log[Log[E^x/6]]))/(2*x*Log[E^x/6]^2*(x + Log[6] - Log[E^x])^4)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75

method	result	size
parallelrisch	\(\frac {1}{\ln \left (\frac {{\mathrm e}^{x}}{6}\right )^{2} x}\)	\(12\)
risch	\(-\frac {4}{x {\left (2 i \ln \left (3\right )+2 i \ln \left (2\right )-2 i \ln \left ({\mathrm e}^{x}\right )\right )}^{2}}\)	\(25\)
default	\(\frac {1}{{\left (\ln \left (\frac {{\mathrm e}^{x}}{6}\right )-x \right )}^{2} x}-\frac {1}{{\left (\ln \left (\frac {{\mathrm e}^{x}}{6}\right )-x \right )}^{2} \ln \left (\frac {{\mathrm e}^{x}}{6}\right )}-\frac {1}{\left (\ln \left (\frac {{\mathrm e}^{x}}{6}\right )-x \right ) \ln \left (\frac {{\mathrm e}^{x}}{6}\right )^{2}}\)	\(57\)
parts	\(\frac {1}{{\left (\ln \left (\frac {{\mathrm e}^{x}}{6}\right )-x \right )}^{2} x}-\frac {1}{{\left (\ln \left (\frac {{\mathrm e}^{x}}{6}\right )-x \right )}^{2} \ln \left (\frac {{\mathrm e}^{x}}{6}\right )}-\frac {1}{\left (\ln \left (\frac {{\mathrm e}^{x}}{6}\right )-x \right ) \ln \left (\frac {{\mathrm e}^{x}}{6}\right )^{2}}\)	\(57\)

[In]

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

[Out]

1/ln(1/6*exp(x))^2/x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=\frac {1}{x^{3} - 2 \, x^{2} \log \left (6\right ) + x \log \left (6\right )^{2}} \]

[In]

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

[Out]

1/(x^3 - 2*x^2*log(6) + x*log(6)^2)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=\frac {1}{x^{3} - 2 x^{2} \log {\left (6 \right )} + x \log {\left (6 \right )}^{2}} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (13) = 26\).

Time = 0.28 (sec) , antiderivative size = 162, normalized size of antiderivative = 10.12 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=\frac {2 \, x - \log \left (3\right ) - \log \left (2\right )}{{\left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (2\right ) + \log \left (2\right )^{2}\right )} x^{2} - {\left (\log \left (3\right )^{3} + 3 \, \log \left (3\right )^{2} \log \left (2\right ) + 3 \, \log \left (3\right ) \log \left (2\right )^{2} + \log \left (2\right )^{3}\right )} x} - \frac {2 \, x - 3 \, \log \left (3\right ) - 3 \, \log \left (2\right )}{\log \left (3\right )^{4} + 4 \, \log \left (3\right )^{3} \log \left (2\right ) + 6 \, \log \left (3\right )^{2} \log \left (2\right )^{2} + 4 \, \log \left (3\right ) \log \left (2\right )^{3} + \log \left (2\right )^{4} + {\left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) \log \left (2\right ) + \log \left (2\right )^{2}\right )} x^{2} - 2 \, {\left (\log \left (3\right )^{3} + 3 \, \log \left (3\right )^{2} \log \left (2\right ) + 3 \, \log \left (3\right ) \log \left (2\right )^{2} + \log \left (2\right )^{3}\right )} x} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (13) = 26\).

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=-\frac {x - 2 \, \log \left (6\right )}{{\left (x - \log \left (6\right )\right )}^{2} \log \left (6\right )^{2}} + \frac {1}{x \log \left (6\right )^{2}} \]

[In]

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

[Out]

-(x - 2*log(6))/((x - log(6))^2*log(6)^2) + 1/(x*log(6)^2)

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-2 x-\log \left (\frac {e^x}{6}\right )}{x^2 \log ^3\left (\frac {e^x}{6}\right )} \, dx=\frac {1}{x\,{\left (x-\ln \left (6\right )\right )}^2} \]

[In]

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

[Out]

1/(x*(x - log(6))^2)