### 3.239 $$\int \log (x+x^3) \, dx$$

Optimal. Leaf size=16 $x \log \left (x^3+x\right )-3 x+2 \tan ^{-1}(x)$

[Out]

-3*x + 2*ArcTan[x] + x*Log[x + x^3]

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Rubi [A]  time = 0.0071385, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {2523, 388, 203} $x \log \left (x^3+x\right )-3 x+2 \tan ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x + x^3],x]

[Out]

-3*x + 2*ArcTan[x] + x*Log[x + x^3]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 388

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0023855, size = 16, normalized size = 1. $x \log \left (x^3+x\right )-3 x+2 \tan ^{-1}(x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x + x^3],x]

[Out]

-3*x + 2*ArcTan[x] + x*Log[x + x^3]

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Maple [A]  time = 0.004, size = 17, normalized size = 1.1 \begin{align*} -3\,x+2\,\arctan \left ( x \right ) +x\ln \left ({x}^{3}+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x+2*arctan(x)+x*ln(x^3+x)

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Maxima [A]  time = 1.49775, size = 22, normalized size = 1.38 \begin{align*} x \log \left (x^{3} + x\right ) - 3 \, x + 2 \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x^3 + x) - 3*x + 2*arctan(x)

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Fricas [A]  time = 1.80446, size = 49, normalized size = 3.06 \begin{align*} x \log \left (x^{3} + x\right ) - 3 \, x + 2 \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x^3 + x) - 3*x + 2*arctan(x)

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Sympy [A]  time = 0.133257, size = 15, normalized size = 0.94 \begin{align*} x \log{\left (x^{3} + x \right )} - 3 x + 2 \operatorname{atan}{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x**3 + x) - 3*x + 2*atan(x)

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Giac [A]  time = 1.23416, size = 22, normalized size = 1.38 \begin{align*} x \log \left (x^{3} + x\right ) - 3 \, x + 2 \, \arctan \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x^3 + x) - 3*x + 2*arctan(x)