3.79 \(\int (\frac {1}{x}+x) \log (x) \, dx\)

Optimal. Leaf size=25 \[ -\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)+\frac {\log ^2(x)}{2} \]

[Out]

-1/4*x^2+1/2*x^2*ln(x)+1/2*ln(x)^2

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1593, 14, 2351, 2301, 2304} \[ -\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)+\frac {\log ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1) + x)*Log[x],x]

[Out]

-x^2/4 + (x^2*Log[x])/2 + Log[x]^2/2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rubi steps

\begin {align*} \int \left (\frac {1}{x}+x\right ) \log (x) \, dx &=\int \frac {\left (1+x^2\right ) \log (x)}{x} \, dx\\ &=\int \left (\frac {\log (x)}{x}+x \log (x)\right ) \, dx\\ &=\int \frac {\log (x)}{x} \, dx+\int x \log (x) \, dx\\ &=-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)+\frac {\log ^2(x)}{2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \[ -\frac {x^2}{4}+\frac {1}{2} x^2 \log (x)+\frac {\log ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1) + x)*Log[x],x]

[Out]

-1/4*x^2 + (x^2*Log[x])/2 + Log[x]^2/2

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fricas [A]  time = 0.43, size = 19, normalized size = 0.76 \[ \frac {1}{2} \, x^{2} \log \relax (x) - \frac {1}{4} \, x^{2} + \frac {1}{2} \, \log \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/x+x)*log(x),x, algorithm="fricas")

[Out]

1/2*x^2*log(x) - 1/4*x^2 + 1/2*log(x)^2

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giac [A]  time = 0.96, size = 19, normalized size = 0.76 \[ \frac {1}{2} \, x^{2} \log \relax (x) - \frac {1}{4} \, x^{2} + \frac {1}{2} \, \log \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/x+x)*log(x),x, algorithm="giac")

[Out]

1/2*x^2*log(x) - 1/4*x^2 + 1/2*log(x)^2

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \[ \frac {x^{2} \ln \relax (x )}{2}-\frac {x^{2}}{4}+\frac {\ln \relax (x )^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/x+x)*ln(x),x)

[Out]

-1/4*x^2+1/2*x^2*ln(x)+1/2*ln(x)^2

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maxima [A]  time = 0.57, size = 24, normalized size = 0.96 \[ -\frac {1}{4} \, x^{2} + \frac {1}{2} \, {\left (x^{2} + 2 \, \log \relax (x)\right )} \log \relax (x) - \frac {1}{2} \, \log \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/x+x)*log(x),x, algorithm="maxima")

[Out]

-1/4*x^2 + 1/2*(x^2 + 2*log(x))*log(x) - 1/2*log(x)^2

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mupad [B]  time = 0.23, size = 19, normalized size = 0.76 \[ \frac {x^2\,\ln \relax (x)}{2}-\frac {x^2}{4}+\frac {{\ln \relax (x)}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)*(x + 1/x),x)

[Out]

(x^2*log(x))/2 + log(x)^2/2 - x^2/4

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sympy [A]  time = 0.10, size = 19, normalized size = 0.76 \[ \frac {x^{2} \log {\relax (x )}}{2} - \frac {x^{2}}{4} + \frac {\log {\relax (x )}^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/x+x)*ln(x),x)

[Out]

x**2*log(x)/2 - x**2/4 + log(x)**2/2

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