3.30 $$\int e^{\text{csch}^{-1}(a x)} x \, dx$$

Optimal. Leaf size=47 $\frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^2}+1}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{2 a^2}+\frac{x}{a}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0264616, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.75, Rules used = {6336, 8, 266, 47, 63, 208} $\frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^2}+1}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^2}+1}\right )}{2 a^2}+\frac{x}{a}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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^{\text{csch}^{-1}(a x)} x \, dx &=\frac{\int 1 \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^2}} x \, dx\\ &=\frac{x}{a}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{x}{a}+\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^2}} x^2-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^2}\\ &=\frac{x}{a}+\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^2}} x^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^2}}\right )\\ &=\frac{x}{a}+\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^2}} x^2+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^2}}\right )}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0303789, size = 47, normalized size = 1. $\frac{a x \left (a x \sqrt{\frac{1}{a^2 x^2}+1}+2\right )+\log \left (x \left (\sqrt{\frac{1}{a^2 x^2}+1}+1\right )\right )}{2 a^2}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.194, size = 85, normalized size = 1.8 \begin{align*}{\frac{x}{2\,{a}^{2}}\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}}}}}}}+{\frac{x}{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,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.967232, size = 105, normalized size = 2.23 \begin{align*} \frac{x}{a} + \frac{\sqrt{\frac{1}{a^{2} x^{2}} + 1}}{2 \,{\left (a^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - a^{2}\right )}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{4 \, a^{2}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{4 \, 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,x, algorithm="maxima")

[Out]

x/a + 1/2*sqrt(1/(a^2*x^2) + 1)/(a^2*(1/(a^2*x^2) + 1) - a^2) + 1/4*log(sqrt(1/(a^2*x^2) + 1) + 1)/a^2 - 1/4*l
og(sqrt(1/(a^2*x^2) + 1) - 1)/a^2

________________________________________________________________________________________

Fricas [A]  time = 2.57032, size = 140, normalized size = 2.98 \begin{align*} \frac{a^{2} x^{2} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 2 \, a x - \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right )}{2 \, 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,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 3.4527, size = 29, normalized size = 0.62 \begin{align*} \frac{x \sqrt{a^{2} x^{2} + 1}}{2 a} + \frac{x}{a} + \frac{\operatorname{asinh}{\left (a x \right )}}{2 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,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17104, size = 70, normalized size = 1.49 \begin{align*} \frac{\sqrt{a^{2} x^{2} + 1} x{\left | a \right |} \mathrm{sgn}\left (x\right )}{2 \, a^{2}} + \frac{x}{a} - \frac{\log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right ) \mathrm{sgn}\left (x\right )}{2 \, 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,x, algorithm="giac")

[Out]

1/2*sqrt(a^2*x^2 + 1)*x*abs(a)*sgn(x)/a^2 + x/a - 1/2*log(-x*abs(a) + sqrt(a^2*x^2 + 1))*sgn(x)/a^2