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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0493616, 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} \[ \frac{2 x}{a^2}+\frac{2 \log (1-a x)}{a^3}+\frac{x^2}{a}+\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCoth[a*x])*x^2,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 = 0.0140906, size = 33, normalized size = 1. \[ \frac{2 x}{a^2}+\frac{2 \log (1-a x)}{a^3}+\frac{x^2}{a}+\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 31, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{{x}^{2}}{a}}+2\,{\frac{x}{{a}^{2}}}+2\,{\frac{\ln \left ( ax-1 \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00945, 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(1/(a*x-1)*(a*x+1)*x^2,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

________________________________________________________________________________________

Fricas [A]  time = 1.80224, 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(1/(a*x-1)*(a*x+1)*x^2,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

________________________________________________________________________________________

Sympy [A]  time = 0.720187, 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(1/(a*x-1)*(a*x+1)*x**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13857, 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(1/(a*x-1)*(a*x+1)*x^2,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