3.72.70 \(\int \frac {-4-x^2+x^5 \log (x^2)+x^5 \log ^2(x^2)}{x^2} \, dx\)

Optimal. Leaf size=26 \[ 4+\frac {4-x^2+\frac {1}{4} x^5 \log ^2\left (x^2\right )}{x} \]

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {14, 2304, 2305} \begin {gather*} \frac {1}{4} x^4 \log ^2\left (x^2\right )-x+\frac {4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/x - x + (x^4*Log[x^2]^2)/4

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 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] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-4-x^2}{x^2}+x^3 \log \left (x^2\right )+x^3 \log ^2\left (x^2\right )\right ) \, dx\\ &=\int \frac {-4-x^2}{x^2} \, dx+\int x^3 \log \left (x^2\right ) \, dx+\int x^3 \log ^2\left (x^2\right ) \, dx\\ &=-\frac {x^4}{8}+\frac {1}{4} x^4 \log \left (x^2\right )+\frac {1}{4} x^4 \log ^2\left (x^2\right )+\int \left (-1-\frac {4}{x^2}\right ) \, dx-\int x^3 \log \left (x^2\right ) \, dx\\ &=\frac {4}{x}-x+\frac {1}{4} x^4 \log ^2\left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.85 \begin {gather*} \frac {4}{x}-x+\frac {1}{4} x^4 \log ^2\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/x - x + (x^4*Log[x^2]^2)/4

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fricas [A]  time = 0.93, size = 22, normalized size = 0.85 \begin {gather*} \frac {x^{5} \log \left (x^{2}\right )^{2} - 4 \, x^{2} + 16}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5*log(x^2)^2+x^5*log(x^2)-x^2-4)/x^2,x, algorithm="fricas")

[Out]

1/4*(x^5*log(x^2)^2 - 4*x^2 + 16)/x

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giac [A]  time = 0.20, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (x^{2}\right )^{2} - x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5*log(x^2)^2+x^5*log(x^2)-x^2-4)/x^2,x, algorithm="giac")

[Out]

1/4*x^4*log(x^2)^2 - x + 4/x

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maple [A]  time = 0.02, size = 21, normalized size = 0.81

method	result	size
default	\(\frac {x^{4} \ln \left (x^{2}\right )^{2}}{4}+\frac {4}{x}-x\)	\(21\)
norman	\(\frac {\frac {x^{5} \ln \left (x^{2}\right )^{2}}{4}-x^{2}+4}{x}\)	\(23\)
risch	\(\frac {x^{4} \ln \left (x^{2}\right )^{2}}{4}-\frac {x^{2}-4}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*ln(x^2)^2+x^5*ln(x^2)-x^2-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (x^{2}\right )^{2} - x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5*log(x^2)^2+x^5*log(x^2)-x^2-4)/x^2,x, algorithm="maxima")

[Out]

1/4*x^4*log(x^2)^2 - x + 4/x

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mupad [B]  time = 4.30, size = 20, normalized size = 0.77 \begin {gather*} \frac {4}{x}-x+\frac {x^4\,{\ln \left (x^2\right )}^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/x - x + (x^4*log(x^2)^2)/4

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sympy [A]  time = 0.11, size = 15, normalized size = 0.58 \begin {gather*} \frac {x^{4} \log {\left (x^{2} \right )}^{2}}{4} - x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5*ln(x**2)**2+x**5*ln(x**2)-x**2-4)/x**2,x)

[Out]

x**4*log(x**2)**2/4 - x + 4/x

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