### 3.43 $$\int e^{-2 \coth ^{-1}(a x)} x^2 \, dx$$

Optimal. Leaf size=33 $\frac{2 x}{a^2}-\frac{2 \log (a x+1)}{a^3}-\frac{x^2}{a}+\frac{x^3}{3}$

[Out]

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

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Rubi [A]  time = 0.050608, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6167, 6126, 77} $\frac{2 x}{a^2}-\frac{2 \log (a x+1)}{a^3}-\frac{x^2}{a}+\frac{x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/E^(2*ArcCoth[a*x]),x]

[Out]

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

Rule 6167

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rule 6126

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[(x^m*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0137995, size = 33, normalized size = 1. $\frac{2 x}{a^2}-\frac{2 \log (a x+1)}{a^3}-\frac{x^2}{a}+\frac{x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/E^(2*ArcCoth[a*x]),x]

[Out]

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

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Maple [A]  time = 0.039, size = 32, normalized size = 1. \begin{align*} 2\,{\frac{x}{{a}^{2}}}-{\frac{{x}^{2}}{a}}+{\frac{{x}^{3}}{3}}-2\,{\frac{\ln \left ( ax+1 \right ) }{{a}^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02015, size = 46, normalized size = 1.39 \begin{align*} \frac{a^{2} x^{3} - 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} - \frac{2 \, \log \left (a x + 1\right )}{a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^2*x^3 - 3*a*x^2 + 6*x)/a^2 - 2*log(a*x + 1)/a^3

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Fricas [A]  time = 1.58506, size = 76, normalized size = 2.3 \begin{align*} \frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x - 6 \, \log \left (a x + 1\right )}{3 \, a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^3*x^3 - 3*a^2*x^2 + 6*a*x - 6*log(a*x + 1))/a^3

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Sympy [A]  time = 0.270656, size = 27, normalized size = 0.82 \begin{align*} \frac{x^{3}}{3} - \frac{x^{2}}{a} + \frac{2 x}{a^{2}} - \frac{2 \log{\left (a x + 1 \right )}}{a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a*x-1)/(a*x+1),x)

[Out]

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

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Giac [A]  time = 1.14682, size = 51, normalized size = 1.55 \begin{align*} \frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x}{3 \, a^{3}} - \frac{2 \, \log \left ({\left | a x + 1 \right |}\right )}{a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^3*x^3 - 3*a^2*x^2 + 6*a*x)/a^3 - 2*log(abs(a*x + 1))/a^3