3.19.11 \(\int \frac {1}{9} (88+188 x+144 x^2+36 x^3+(22+108 x+54 x^2) \log (x^2)+18 x \log ^2(x^2)) \, dx\)

Optimal. Leaf size=16 \[ \left (-\frac {11}{9}-x \left (2+x+\log \left (x^2\right )\right )\right )^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 58, normalized size of antiderivative = 3.62, number of steps used = 9, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {12, 2356, 2295, 2304, 2305} \begin {gather*} x^4+4 x^3+\frac {58 x^2}{9}+x^2 \log ^2\left (x^2\right )+4 x^2 \log \left (x^2\right )+\frac {22}{9} x \log \left (x^2\right )+2 x^3 \log \left (x^2\right )+\frac {44 x}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(88 + 188*x + 144*x^2 + 36*x^3 + (22 + 108*x + 54*x^2)*Log[x^2] + 18*x*Log[x^2]^2)/9,x]

[Out]

(44*x)/9 + (58*x^2)/9 + 4*x^3 + x^4 + (22*x*Log[x^2])/9 + 4*x^2*Log[x^2] + 2*x^3*Log[x^2] + x^2*Log[x^2]^2

Rule 12

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{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 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]

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \left (88+188 x+144 x^2+36 x^3+\left (22+108 x+54 x^2\right ) \log \left (x^2\right )+18 x \log ^2\left (x^2\right )\right ) \, dx\\ &=\frac {88 x}{9}+\frac {94 x^2}{9}+\frac {16 x^3}{3}+x^4+\frac {1}{9} \int \left (22+108 x+54 x^2\right ) \log \left (x^2\right ) \, dx+2 \int x \log ^2\left (x^2\right ) \, dx\\ &=\frac {88 x}{9}+\frac {94 x^2}{9}+\frac {16 x^3}{3}+x^4+x^2 \log ^2\left (x^2\right )+\frac {1}{9} \int \left (22 \log \left (x^2\right )+108 x \log \left (x^2\right )+54 x^2 \log \left (x^2\right )\right ) \, dx-4 \int x \log \left (x^2\right ) \, dx\\ &=\frac {88 x}{9}+\frac {112 x^2}{9}+\frac {16 x^3}{3}+x^4-2 x^2 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )+\frac {22}{9} \int \log \left (x^2\right ) \, dx+6 \int x^2 \log \left (x^2\right ) \, dx+12 \int x \log \left (x^2\right ) \, dx\\ &=\frac {44 x}{9}+\frac {58 x^2}{9}+4 x^3+x^4+\frac {22}{9} x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )+2 x^3 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 58, normalized size = 3.62 \begin {gather*} \frac {44 x}{9}+\frac {58 x^2}{9}+4 x^3+x^4+\frac {22}{9} x \log \left (x^2\right )+4 x^2 \log \left (x^2\right )+2 x^3 \log \left (x^2\right )+x^2 \log ^2\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(88 + 188*x + 144*x^2 + 36*x^3 + (22 + 108*x + 54*x^2)*Log[x^2] + 18*x*Log[x^2]^2)/9,x]

[Out]

(44*x)/9 + (58*x^2)/9 + 4*x^3 + x^4 + (22*x*Log[x^2])/9 + 4*x^2*Log[x^2] + 2*x^3*Log[x^2] + x^2*Log[x^2]^2

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fricas [B]  time = 0.66, size = 47, normalized size = 2.94 \begin {gather*} x^{4} + x^{2} \log \left (x^{2}\right )^{2} + 4 \, x^{3} + \frac {58}{9} \, x^{2} + \frac {2}{9} \, {\left (9 \, x^{3} + 18 \, x^{2} + 11 \, x\right )} \log \left (x^{2}\right ) + \frac {44}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x^2)^2+1/9*(54*x^2+108*x+22)*log(x^2)+4*x^3+16*x^2+188/9*x+88/9,x, algorithm="fricas")

[Out]

x^4 + x^2*log(x^2)^2 + 4*x^3 + 58/9*x^2 + 2/9*(9*x^3 + 18*x^2 + 11*x)*log(x^2) + 44/9*x

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giac [B]  time = 0.19, size = 56, normalized size = 3.50 \begin {gather*} x^{4} + x^{2} \log \left (x^{2}\right )^{2} + 4 \, x^{3} - 2 \, x^{2} \log \left (x^{2}\right ) + \frac {58}{9} \, x^{2} + \frac {2}{9} \, {\left (9 \, x^{3} + 27 \, x^{2} + 11 \, x\right )} \log \left (x^{2}\right ) + \frac {44}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x^2)^2+1/9*(54*x^2+108*x+22)*log(x^2)+4*x^3+16*x^2+188/9*x+88/9,x, algorithm="giac")

[Out]

x^4 + x^2*log(x^2)^2 + 4*x^3 - 2*x^2*log(x^2) + 58/9*x^2 + 2/9*(9*x^3 + 27*x^2 + 11*x)*log(x^2) + 44/9*x

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maple [B]  time = 0.02, size = 53, normalized size = 3.31

method	result	size
default	\(\frac {44 x}{9}+\frac {58 x^{2}}{9}+4 x^{3}+x^{4}+x^{2} \ln \left (x^{2}\right )^{2}+4 x^{2} \ln \left (x^{2}\right )+2 x^{3} \ln \left (x^{2}\right )+\frac {22 x \ln \left (x^{2}\right )}{9}\)	\(53\)
norman	\(\frac {44 x}{9}+\frac {58 x^{2}}{9}+4 x^{3}+x^{4}+x^{2} \ln \left (x^{2}\right )^{2}+4 x^{2} \ln \left (x^{2}\right )+2 x^{3} \ln \left (x^{2}\right )+\frac {22 x \ln \left (x^{2}\right )}{9}\)	\(53\)
risch	\(x^{2} \ln \left (x^{2}\right )^{2}-2 x^{2} \ln \left (x^{2}\right )+\frac {58 x^{2}}{9}+\frac {\left (18 x^{3}+54 x^{2}+22 x \right ) \ln \left (x^{2}\right )}{9}+4 x^{3}+\frac {44 x}{9}+x^{4}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*ln(x^2)^2+1/9*(54*x^2+108*x+22)*ln(x^2)+4*x^3+16*x^2+188/9*x+88/9,x,method=_RETURNVERBOSE)

[Out]

44/9*x+58/9*x^2+4*x^3+x^4+x^2*ln(x^2)^2+4*x^2*ln(x^2)+2*x^3*ln(x^2)+22/9*x*ln(x^2)

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maxima [B]  time = 0.41, size = 56, normalized size = 3.50 \begin {gather*} x^{4} + x^{2} \log \left (x^{2}\right )^{2} + 4 \, x^{3} - 2 \, x^{2} \log \left (x^{2}\right ) + \frac {58}{9} \, x^{2} + \frac {2}{9} \, {\left (9 \, x^{3} + 27 \, x^{2} + 11 \, x\right )} \log \left (x^{2}\right ) + \frac {44}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*log(x^2)^2+1/9*(54*x^2+108*x+22)*log(x^2)+4*x^3+16*x^2+188/9*x+88/9,x, algorithm="maxima")

[Out]

x^4 + x^2*log(x^2)^2 + 4*x^3 - 2*x^2*log(x^2) + 58/9*x^2 + 2/9*(9*x^3 + 27*x^2 + 11*x)*log(x^2) + 44/9*x

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mupad [B]  time = 1.21, size = 27, normalized size = 1.69 \begin {gather*} \frac {x\,\left (x+\ln \left (x^2\right )+2\right )\,\left (18\,x+9\,x\,\ln \left (x^2\right )+9\,x^2+22\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((188*x)/9 + (log(x^2)*(108*x + 54*x^2 + 22))/9 + 2*x*log(x^2)^2 + 16*x^2 + 4*x^3 + 88/9,x)

[Out]

(x*(x + log(x^2) + 2)*(18*x + 9*x*log(x^2) + 9*x^2 + 22))/9

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sympy [B]  time = 0.15, size = 49, normalized size = 3.06 \begin {gather*} x^{4} + 4 x^{3} + x^{2} \log {\left (x^{2} \right )}^{2} + \frac {58 x^{2}}{9} + \frac {44 x}{9} + \left (2 x^{3} + 4 x^{2} + \frac {22 x}{9}\right ) \log {\left (x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*ln(x**2)**2+1/9*(54*x**2+108*x+22)*ln(x**2)+4*x**3+16*x**2+188/9*x+88/9,x)

[Out]

x**4 + 4*x**3 + x**2*log(x**2)**2 + 58*x**2/9 + 44*x/9 + (2*x**3 + 4*x**2 + 22*x/9)*log(x**2)

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