### 3.51 $$\int e^{2 \text{csch}^{-1}(a x)} x^2 \, dx$$

Optimal. Leaf size=52 $\frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{a^3}+\frac{2 x}{a^2}+\frac{x^3}{3}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.239738, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {6338, 6742, 266, 47, 63, 208} $\frac{x^2 \sqrt{\frac{1}{a^2 x^2}+1}}{a}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{a^3}+\frac{2 x}{a^2}+\frac{x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 6338

Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && Int
egerQ[n]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0378865, size = 57, normalized size = 1.1 $\frac{a x \left (a^2 x^2+3 a x \sqrt{\frac{1}{a^2 x^2}+1}+6\right )+3 \log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{3 a^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.171, size = 90, normalized size = 1.7 \begin{align*}{\frac{{x}^{3}}{3}}+2\,{\frac{x}{{a}^{2}}}+{\frac{x}{{a}^{3}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( x\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+\ln \left ( x+\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}} \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))^2*x^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.988807, size = 120, normalized size = 2.31 \begin{align*} \frac{1}{3} \, x^{3} + \frac{\frac{2 \, \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{a^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - a^{2}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{a^{2}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{a^{2}}}{2 \, a} + \frac{2 \, x}{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))^2*x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.5246, size = 159, normalized size = 3.06 \begin{align*} \frac{a^{3} x^{3} + 3 \, a^{2} x^{2} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 6 \, a x - 3 \, \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{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))^2*x^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.37071, size = 36, normalized size = 0.69 \begin{align*} \frac{x^{3}}{3} + \frac{x \sqrt{a^{2} x^{2} + 1}}{a^{2}} + \frac{2 x}{a^{2}} + \frac{\operatorname{asinh}{\left (a x \right )}}{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))**2*x**2,x)

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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))^2*x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError