\(\int \frac {-3 \log (3)-3 \log (3) \log (x)+(-x+x^2) \log ^2(x)}{x^2 \log ^2(x)} \, dx\) [244]

Optimal result

Integrand size = 31, antiderivative size = 20 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=e^3+x+\frac {3 \log (3)}{x \log (x)}-\log (x) \]

[Out]

x+3*ln(3)/ln(x)/x+exp(3)-ln(x)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6874, 45, 2343, 2346, 2209} \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x-\log (x)+\frac {3 \log (3)}{x \log (x)} \]

[In]

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

[Out]

x + (3*Log[3])/(x*Log[x]) - Log[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])

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2343

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log
[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

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 {-1+x}{x}-\frac {3 \log (3)}{x^2 \log ^2(x)}-\frac {3 \log (3)}{x^2 \log (x)}\right ) \, dx \\ & = -\left ((3 \log (3)) \int \frac {1}{x^2 \log ^2(x)} \, dx\right )-(3 \log (3)) \int \frac {1}{x^2 \log (x)} \, dx+\int \frac {-1+x}{x} \, dx \\ & = \frac {3 \log (3)}{x \log (x)}+(3 \log (3)) \int \frac {1}{x^2 \log (x)} \, dx-(3 \log (3)) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+\int \left (1-\frac {1}{x}\right ) \, dx \\ & = x-3 \operatorname {ExpIntegralEi}(-\log (x)) \log (3)+\frac {3 \log (3)}{x \log (x)}-\log (x)+(3 \log (3)) \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right ) \\ & = x+\frac {3 \log (3)}{x \log (x)}-\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x+\frac {3 \log (3)}{x \log (x)}-\log (x) \]

[In]

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

[Out]

x + (3*Log[3])/(x*Log[x]) - Log[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

method	result	size
risch	\(x -\ln \left (x \right )+\frac {3 \ln \left (3\right )}{\ln \left (x \right ) x}\)	\(18\)
norman	\(\frac {x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+3 \ln \left (3\right )}{x \ln \left (x \right )}\)	\(27\)
parallelrisch	\(\frac {x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+3 \ln \left (3\right )}{x \ln \left (x \right )}\)	\(27\)
default	\(x -\ln \left (x \right )+3 \ln \left (3\right ) \operatorname {Ei}_{1}\left (\ln \left (x \right )\right )-3 \ln \left (3\right ) \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )\)	\(33\)
parts	\(x -\ln \left (x \right )+3 \ln \left (3\right ) \operatorname {Ei}_{1}\left (\ln \left (x \right )\right )-3 \ln \left (3\right ) \left (-\frac {1}{x \ln \left (x \right )}+\operatorname {Ei}_{1}\left (\ln \left (x \right )\right )\right )\)	\(33\)

[In]

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

[Out]

x-ln(x)+3*ln(3)/ln(x)/x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\frac {x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + 3 \, \log \left (3\right )}{x \log \left (x\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x - \log {\left (x \right )} + \frac {3 \log {\left (3 \right )}}{x \log {\left (x \right )}} \]

[In]

integrate(((x**2-x)*ln(x)**2-3*ln(3)*ln(x)-3*ln(3))/x**2/ln(x)**2,x)

[Out]

x - log(x) + 3*log(3)/(x*log(x))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=-3 \, {\rm Ei}\left (-\log \left (x\right )\right ) \log \left (3\right ) + 3 \, \Gamma \left (-1, \log \left (x\right )\right ) \log \left (3\right ) + x - \log \left (x\right ) \]

[In]

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

[Out]

-3*Ei(-log(x))*log(3) + 3*gamma(-1, log(x))*log(3) + x - log(x)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x + \frac {3 \, \log \left (3\right )}{x \log \left (x\right )} - \log \left (x\right ) \]

[In]

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

[Out]

x + 3*log(3)/(x*log(x)) - log(x)

Mupad [B] (verification not implemented)

Time = 7.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-3 \log (3)-3 \log (3) \log (x)+\left (-x+x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=x-\ln \left (x\right )+\frac {3\,\ln \left (3\right )}{x\,\ln \left (x\right )} \]

[In]

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

[Out]

x - log(x) + (3*log(3))/(x*log(x))