3.4.34 \(\int \frac {-1+6 x^2+x^2 \log (x^2)}{2 x} \, dx\)

Optimal. Leaf size=17 \[ -2+\frac {1}{4} \left (-1+x^2\right ) \left (5+\log \left (x^2\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2304} \begin {gather*} \frac {5 x^2}{4}+\frac {1}{4} x^2 \log \left (x^2\right )-\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 6*x^2 + x^2*Log[x^2])/(2*x),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-1+6 x^2+x^2 \log \left (x^2\right )}{x} \, dx\\ &=\frac {1}{2} \int \left (\frac {-1+6 x^2}{x}+x \log \left (x^2\right )\right ) \, dx\\ &=\frac {1}{2} \int \frac {-1+6 x^2}{x} \, dx+\frac {1}{2} \int x \log \left (x^2\right ) \, dx\\ &=-\frac {x^2}{4}+\frac {1}{4} x^2 \log \left (x^2\right )+\frac {1}{2} \int \left (-\frac {1}{x}+6 x\right ) \, dx\\ &=\frac {5 x^2}{4}-\frac {\log (x)}{2}+\frac {1}{4} x^2 \log \left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.47 \begin {gather*} \frac {5 x^2}{4}-\frac {\log (x)}{2}+\frac {1}{4} x^2 \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 17, normalized size = 1.00 \begin {gather*} \frac {5}{4} \, x^{2} + \frac {1}{4} \, {\left (x^{2} - 1\right )} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*x^2 + 1/4*(x^2 - 1)*log(x^2)

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giac [A]  time = 0.40, size = 21, normalized size = 1.24 \begin {gather*} \frac {1}{4} \, x^{2} \log \left (x^{2}\right ) + \frac {5}{4} \, x^{2} - \frac {1}{4} \, \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2*log(x^2) + 5/4*x^2 - 1/4*log(x^2)

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maple [A]  time = 0.03, size = 20, normalized size = 1.18

method	result	size
norman	\(\frac {5 x^{2}}{4}+\frac {x^{2} \ln \left (x^{2}\right )}{4}-\frac {\ln \relax (x )}{2}\)	\(20\)
risch	\(\frac {5 x^{2}}{4}+\frac {x^{2} \ln \left (x^{2}\right )}{4}-\frac {\ln \relax (x )}{2}\)	\(20\)
derivativedivides	\(\frac {x^{2} \ln \left (x^{2}\right )}{4}+\frac {5 x^{2}}{4}-\frac {\ln \left (x^{2}\right )}{4}\)	\(22\)
default	\(\frac {x^{2} \ln \left (x^{2}\right )}{4}+\frac {5 x^{2}}{4}-\frac {\ln \left (x^{2}\right )}{4}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*x^2+1/4*x^2*ln(x^2)-1/2*ln(x)

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maxima [A]  time = 0.45, size = 19, normalized size = 1.12 \begin {gather*} \frac {1}{4} \, x^{2} \log \left (x^{2}\right ) + \frac {5}{4} \, x^{2} - \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2*log(x^2) + 5/4*x^2 - 1/2*log(x)

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mupad [B]  time = 0.36, size = 21, normalized size = 1.24 \begin {gather*} \frac {x^2\,\ln \left (x^2\right )}{4}-\frac {\ln \left (x^2\right )}{4}+\frac {5\,x^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*log(x^2))/4 - log(x^2)/4 + (5*x^2)/4

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sympy [A]  time = 0.13, size = 20, normalized size = 1.18 \begin {gather*} \frac {x^{2} \log {\left (x^{2} \right )}}{4} + \frac {5 x^{2}}{4} - \frac {\log {\relax (x )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*log(x**2)/4 + 5*x**2/4 - log(x)/2

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