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

Optimal. Leaf size=20 \[ e^3+x+\frac {3 \log (3)}{x \log (x)}-\log (x) \]

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Rubi [A]  time = 0.20, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 43, 2306, 2309, 2178} \begin {gather*} x-\log (x)+\frac {3 \log (3)}{x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[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 43

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 2178

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

Rule 2306

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 2309

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 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 17, normalized size = 0.85 \begin {gather*} x+\frac {3 \log (3)}{x \log (x)}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.91, size = 26, normalized size = 1.30 \begin {gather*} \frac {x^{2} \log \relax (x) - x \log \relax (x)^{2} + 3 \, \log \relax (3)}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.29, size = 17, normalized size = 0.85 \begin {gather*} x + \frac {3 \, \log \relax (3)}{x \log \relax (x)} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.09, size = 18, normalized size = 0.90

method	result	size
risch	\(x -\ln \relax (x )+\frac {3 \ln \relax (3)}{\ln \relax (x ) x}\)	\(18\)
norman	\(\frac {x^{2} \ln \relax (x )-x \ln \relax (x )^{2}+3 \ln \relax (3)}{x \ln \relax (x )}\)	\(27\)
default	\(x -\ln \relax (x )+3 \ln \relax (3) \expIntegralEi \left (1, \ln \relax (x )\right )-3 \ln \relax (3) \left (-\frac {1}{x \ln \relax (x )}+\expIntegralEi \left (1, \ln \relax (x )\right )\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [C]  time = 0.39, size = 23, normalized size = 1.15 \begin {gather*} -3 \, {\rm Ei}\left (-\log \relax (x)\right ) \log \relax (3) + 3 \, \Gamma \left (-1, \log \relax (x)\right ) \log \relax (3) + x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.33, size = 17, normalized size = 0.85 \begin {gather*} x-\ln \relax (x)+\frac {3\,\ln \relax (3)}{x\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.10, size = 14, normalized size = 0.70 \begin {gather*} x - \log {\relax (x )} + \frac {3 \log {\relax (3 )}}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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