3.34 \(\int e^{\text{sech}^{-1}(a x)} x^2 \, dx\)

Optimal. Leaf size=38 \[ -\frac{x e^{\text{sech}^{-1}(a x)}}{3 a^2}+\frac{x^2}{6 a}+\frac{1}{3} x^3 e^{\text{sech}^{-1}(a x)} \]

[Out]

-(E^ArcSech[a*x]*x)/(3*a^2) + x^2/(6*a) + (E^ArcSech[a*x]*x^3)/3

________________________________________________________________________________________

Rubi [A]  time = 0.024187, antiderivative size = 52, normalized size of antiderivative = 1.37, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6335, 30, 74} \[ -\frac{\sqrt{1-a x}}{3 a^3 \sqrt{\frac{1}{a x+1}}}+\frac{x^2}{6 a}+\frac{1}{3} x^3 e^{\text{sech}^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x]*x^2,x]

[Out]

x^2/(6*a) + (E^ArcSech[a*x]*x^3)/3 - Sqrt[1 - a*x]/(3*a^3*Sqrt[(1 + a*x)^(-1)])

Rule 6335

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*E^ArcSech[a*x^p])/(m + 1), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/(a*(m + 1)), Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^2 \, dx &=\frac{1}{3} e^{\text{sech}^{-1}(a x)} x^3+\frac{\int x \, dx}{3 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{3 a}\\ &=\frac{x^2}{6 a}+\frac{1}{3} e^{\text{sech}^{-1}(a x)} x^3-\frac{\sqrt{1-a x}}{3 a^3 \sqrt{\frac{1}{1+a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0587718, size = 48, normalized size = 1.26 \[ \frac{3 a^2 x^2+2 (a x-1) \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2}{6 a^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x]*x^2,x]

[Out]

(3*a^2*x^2 + 2*(-1 + a*x)*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2)/(6*a^3)

________________________________________________________________________________________

Maple [A]  time = 0.173, size = 54, normalized size = 1.4 \begin{align*}{\frac{x \left ({a}^{2}{x}^{2}-1 \right ) }{3\,{a}^{2}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{{x}^{2}}{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.06955, size = 51, normalized size = 1.34 \begin{align*} \frac{x^{2}}{2 \, a} + \frac{{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{3 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.82112, size = 111, normalized size = 2.92 \begin{align*} \frac{3 \, a x^{2} + 2 \,{\left (a^{2} x^{3} - x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{6 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x\, dx + \int a x^{2} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x, x) + Integral(a*x**2*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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