### 3.29 $$\int e^{\text{csch}^{-1}(a x)} x^2 \, dx$$

Optimal. Leaf size=31 $\frac{1}{3} x^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}+\frac{x^2}{2 a}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0219103, antiderivative size = 31, normalized size of antiderivative = 1., 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 = {6336, 30, 264} $\frac{1}{3} x^3 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}+\frac{x^2}{2 a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6336

Int[E^ArcCsch[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Dist[1/a, Int[x^(m - p), x], x] + Int[x^m*Sqrt[1 + 1/
(a^2*x^(2*p))], x] /; FreeQ[{a, m, p}, x]

Rule 30

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

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.0379749, size = 38, normalized size = 1.23 $\frac{2 \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^3+x\right )+3 a x^2}{6 a^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.171, size = 43, normalized size = 1.4 \begin{align*}{\frac{x \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}+{\frac{{x}^{2}}{2\,a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02121, size = 34, normalized size = 1.1 \begin{align*} \frac{1}{3} \, x^{3}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + \frac{x^{2}}{2 \, a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.53478, size = 90, normalized size = 2.9 \begin{align*} \frac{3 \, a x^{2} + 2 \,{\left (a^{2} x^{3} + x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}}}{6 \, a^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.45725, size = 41, normalized size = 1.32 \begin{align*} \frac{x^{2} \sqrt{a^{2} x^{2} + 1}}{3 a} + \frac{x^{2}}{2 a} + \frac{\sqrt{a^{2} x^{2} + 1}}{3 a^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1483, size = 61, normalized size = 1.97 \begin{align*} -\frac{{\left | a \right |} \mathrm{sgn}\left (x\right )}{3 \, a^{4}} + \frac{2 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 3 \,{\left (a^{2} x^{2} + 1\right )} a}{6 \, a^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*abs(a)*sgn(x)/a^4 + 1/6*(2*(a^2*x^2 + 1)^(3/2)*abs(a)*sgn(x) + 3*(a^2*x^2 + 1)*a)/a^4