### 3.230 $$\int x \log (x+x^3) \, dx$$

Optimal. Leaf size=31 $-\frac{3 x^2}{4}+\frac{1}{2} x^2 \log \left (x^3+x\right )+\frac{1}{2} \log \left (x^2+1\right )$

[Out]

(-3*x^2)/4 + Log[1 + x^2]/2 + (x^2*Log[x + x^3])/2

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Rubi [A]  time = 0.0293318, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {2525, 444, 43} $-\frac{3 x^2}{4}+\frac{1}{2} x^2 \log \left (x^3+x\right )+\frac{1}{2} \log \left (x^2+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Log[x + x^3],x]

[Out]

(-3*x^2)/4 + Log[1 + x^2]/2 + (x^2*Log[x + x^3])/2

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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])

Rubi steps

\begin{align*} \int x \log \left (x+x^3\right ) \, dx &=\frac{1}{2} x^2 \log \left (x+x^3\right )-\frac{1}{2} \int \frac{x \left (1+3 x^2\right )}{1+x^2} \, dx\\ &=\frac{1}{2} x^2 \log \left (x+x^3\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1+3 x}{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \log \left (x+x^3\right )-\frac{1}{4} \operatorname{Subst}\left (\int \left (3-\frac{2}{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 x^2}{4}+\frac{1}{2} \log \left (1+x^2\right )+\frac{1}{2} x^2 \log \left (x+x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.00809, size = 31, normalized size = 1. $-\frac{3 x^2}{4}+\frac{1}{2} x^2 \log \left (x^3+x\right )+\frac{1}{2} \log \left (x^2+1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Log[x + x^3],x]

[Out]

(-3*x^2)/4 + Log[1 + x^2]/2 + (x^2*Log[x + x^3])/2

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Maple [A]  time = 0.005, size = 26, normalized size = 0.8 \begin{align*} -{\frac{3\,{x}^{2}}{4}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{{x}^{2}\ln \left ({x}^{3}+x \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(x^3+x),x)

[Out]

-3/4*x^2+1/2*ln(x^2+1)+1/2*x^2*ln(x^3+x)

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Maxima [A]  time = 1.56524, size = 34, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac{3}{4} \, x^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x^3+x),x, algorithm="maxima")

[Out]

1/2*x^2*log(x^3 + x) - 3/4*x^2 + 1/2*log(x^2 + 1)

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Fricas [A]  time = 1.84842, size = 69, normalized size = 2.23 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac{3}{4} \, x^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x^3+x),x, algorithm="fricas")

[Out]

1/2*x^2*log(x^3 + x) - 3/4*x^2 + 1/2*log(x^2 + 1)

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Sympy [A]  time = 0.134358, size = 26, normalized size = 0.84 \begin{align*} \frac{x^{2} \log{\left (x^{3} + x \right )}}{2} - \frac{3 x^{2}}{4} + \frac{\log{\left (x^{2} + 1 \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(x**3+x),x)

[Out]

x**2*log(x**3 + x)/2 - 3*x**2/4 + log(x**2 + 1)/2

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Giac [A]  time = 1.34124, size = 34, normalized size = 1.1 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x^{3} + x\right ) - \frac{3}{4} \, x^{2} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x^3+x),x, algorithm="giac")

[Out]

1/2*x^2*log(x^3 + x) - 3/4*x^2 + 1/2*log(x^2 + 1)