### 3.43 $$\int \frac{e^{\text{csch}^{-1}(a x^2)}}{x} \, dx$$

Optimal. Leaf size=46 $-\frac{1}{2} \sqrt{\frac{1}{a^2 x^4}+1}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^4}+1}\right )-\frac{1}{2 a x^2}$

[Out]

-Sqrt[1 + 1/(a^2*x^4)]/2 - 1/(2*a*x^2) + ArcTanh[Sqrt[1 + 1/(a^2*x^4)]]/2

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Rubi [A]  time = 0.0327176, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {6336, 30, 266, 50, 63, 208} $-\frac{1}{2} \sqrt{\frac{1}{a^2 x^4}+1}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^4}+1}\right )-\frac{1}{2 a x^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sqrt[1 + 1/(a^2*x^4)]/2 - 1/(2*a*x^2) + ArcTanh[Sqrt[1 + 1/(a^2*x^4)]]/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 30

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

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x} \, dx &=\frac{\int \frac{1}{x^3} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x} \, dx\\ &=-\frac{1}{2 a x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}}-\frac{1}{2 a x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}}-\frac{1}{2 a x^2}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^4}}\right )\\ &=-\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}}-\frac{1}{2 a x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^4}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0453229, size = 22, normalized size = 0.48 $\tanh ^{-1}\left (e^{\text{csch}^{-1}\left (a x^2\right )}\right )-\frac{1}{2} e^{\text{csch}^{-1}\left (a x^2\right )}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-E^ArcCsch[a*x^2]/2 + ArcTanh[E^ArcCsch[a*x^2]]

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Maple [B]  time = 0.272, size = 86, normalized size = 1.9 \begin{align*} -{\frac{1}{2}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -\ln \left ({x}^{2}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}} \right ){x}^{2}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}}}}-{\frac{1}{2\,a{x}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x,x)

[Out]

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

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Maxima [A]  time = 1.00797, size = 73, normalized size = 1.59 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{a^{2} x^{4}} + 1} - \frac{1}{2 \, a x^{2}} + \frac{1}{4} \, \log \left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right ) - \frac{1}{4} \, \log \left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.55955, size = 163, normalized size = 3.54 \begin{align*} -\frac{a x^{2} \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - a x^{2}\right ) + a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + a x^{2} + 1}{2 \, a x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.13401, size = 54, normalized size = 1.17 \begin{align*} - \frac{a x^{2}}{2 \sqrt{a^{2} x^{4} + 1}} + \frac{\operatorname{asinh}{\left (a x^{2} \right )}}{2} - \frac{1}{2 a x^{2}} - \frac{1}{2 a x^{2} \sqrt{a^{2} x^{4} + 1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x**2+(1+1/a**2/x**4)**(1/2))/x,x)

[Out]

-a*x**2/(2*sqrt(a**2*x**4 + 1)) + asinh(a*x**2)/2 - 1/(2*a*x**2) - 1/(2*a*x**2*sqrt(a**2*x**4 + 1))

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Giac [A]  time = 1.22897, size = 80, normalized size = 1.74 \begin{align*} \frac{{\left (a \log \left (a + \sqrt{a^{2} + \frac{1}{x^{4}}}\right ) - a \log \left (-a + \sqrt{a^{2} + \frac{1}{x^{4}}}\right ) - 2 \, \sqrt{a^{2} + \frac{1}{x^{4}}}\right )}{\left | a \right |}}{4 \, a^{2}} - \frac{1}{2 \, a x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1+1/a^2/x^4)^(1/2))/x,x, algorithm="giac")

[Out]

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