3.43 \(\int e^{-2 \tanh ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=43 \[ -\frac{x^2}{a^2}+\frac{2 x}{a^3}-\frac{2 \log (a x+1)}{a^4}+\frac{2 x^3}{3 a}-\frac{x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.0392676, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 77} \[ -\frac{x^2}{a^2}+\frac{2 x}{a^3}-\frac{2 \log (a x+1)}{a^4}+\frac{2 x^3}{3 a}-\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^(2*ArcTanh[a*x]),x]

[Out]

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

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

Mathematica [A]  time = 0.0187701, size = 43, normalized size = 1. \[ -\frac{x^2}{a^2}+\frac{2 x}{a^3}-\frac{2 \log (a x+1)}{a^4}+\frac{2 x^3}{3 a}-\frac{x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/E^(2*ArcTanh[a*x]),x]

[Out]

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

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Maple [A]  time = 0.033, size = 40, normalized size = 0.9 \begin{align*} 2\,{\frac{x}{{a}^{3}}}-{\frac{{x}^{2}}{{a}^{2}}}+{\frac{2\,{x}^{3}}{3\,a}}-{\frac{{x}^{4}}{4}}-2\,{\frac{\ln \left ( ax+1 \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.945287, size = 58, normalized size = 1.35 \begin{align*} -\frac{3 \, a^{3} x^{4} - 8 \, a^{2} x^{3} + 12 \, a x^{2} - 24 \, x}{12 \, a^{3}} - \frac{2 \, \log \left (a x + 1\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-1/12*(3*a^3*x^4 - 8*a^2*x^3 + 12*a*x^2 - 24*x)/a^3 - 2*log(a*x + 1)/a^4

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Fricas [A]  time = 1.76151, size = 101, normalized size = 2.35 \begin{align*} -\frac{3 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 24 \, a x + 24 \, \log \left (a x + 1\right )}{12 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fricas")

[Out]

-1/12*(3*a^4*x^4 - 8*a^3*x^3 + 12*a^2*x^2 - 24*a*x + 24*log(a*x + 1))/a^4

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Sympy [A]  time = 0.338888, size = 37, normalized size = 0.86 \begin{align*} - \frac{x^{4}}{4} + \frac{2 x^{3}}{3 a} - \frac{x^{2}}{a^{2}} + \frac{2 x}{a^{3}} - \frac{2 \log{\left (a x + 1 \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a*x+1)**2*(-a**2*x**2+1),x)

[Out]

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

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Giac [A]  time = 1.23158, size = 89, normalized size = 2.07 \begin{align*} \frac{{\left (a x + 1\right )}^{4}{\left (\frac{20}{a x + 1} - \frac{54}{{\left (a x + 1\right )}^{2}} + \frac{84}{{\left (a x + 1\right )}^{3}} - 3\right )}}{12 \, a^{4}} + \frac{2 \, \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="giac")

[Out]

1/12*(a*x + 1)^4*(20/(a*x + 1) - 54/(a*x + 1)^2 + 84/(a*x + 1)^3 - 3)/a^4 + 2*log(abs(a*x + 1)/((a*x + 1)^2*ab
s(a)))/a^4