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

Optimal. Leaf size=147 \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{\sqrt{a}}-\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{a x}+x e^{\text{sech}^{-1}\left (a x^2\right )}-\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\frac{2}{a x} \]

[Out]

-2/(a*x) + E^ArcSech[a*x^2]*x - (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(a*x) - (2*Sqrt[(
1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticE[ArcSin[Sqrt[a]*x], -1])/Sqrt[a] + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1
+ a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/Sqrt[a]

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Rubi [A]  time = 0.0680163, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6330, 30, 259, 325, 307, 221, 1199, 424} \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{a x}+x e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\frac{2}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a*x) + E^ArcSech[a*x^2]*x - (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(a*x) - (2*Sqrt[(
1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticE[ArcSin[Sqrt[a]*x], -1])/Sqrt[a] + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1
+ a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/Sqrt[a]

Rule 6330

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

Rule 30

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

Rule 259

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 325

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] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} \, dx &=e^{\text{sech}^{-1}\left (a x^2\right )} x+\frac{2 \int \frac{1}{x^2} \, dx}{a}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^2 \sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{a}\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^4}} \, dx}{a}\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}-\left (2 a \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^2}{\sqrt{1-a^2 x^4}} \, dx\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}+\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{\sqrt{1-a^2 x^4}} \, dx-\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1+a x^2}{\sqrt{1-a^2 x^4}} \, dx\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}+\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{\sqrt{1+a x^2}}{\sqrt{1-a x^2}} \, dx\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}+\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [C]  time = 0.281666, size = 135, normalized size = 0.92 \[ -\frac{2 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-a} x\right )\right |-1\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )\right )}{\sqrt{-a} \left (a x^2-1\right )}+\sqrt{\frac{1-a x^2}{a x^2+1}} \left (-\frac{1}{a x}-x\right )-\frac{1}{a x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(1/(a*x)) + (-(1/(a*x)) - x)*Sqrt[(1 - a*x^2)/(1 + a*x^2)] - ((2*I)*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*Sqrt[1 - a^
2*x^4]*(EllipticE[I*ArcSinh[Sqrt[-a]*x], -1] - EllipticF[I*ArcSinh[Sqrt[-a]*x], -1]))/(Sqrt[-a]*(-1 + a*x^2))

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Maple [A]  time = 0.187, size = 132, normalized size = 0.9 \begin{align*} -{\frac{1}{ax}}-{\frac{x}{{a}^{2}{x}^{4}-1}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({a}^{2}{x}^{4}+2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}x{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{a}-2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}x{\it EllipticE} \left ( x\sqrt{a},i \right ) \sqrt{a}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/x-(-(a*x^2-1)/a/x^2)^(1/2)*x*((a*x^2+1)/a/x^2)^(1/2)*(a^2*x^4+2*(-a*x^2+1)^(1/2)*(a*x^2+1)^(1/2)*x*Ellipt
icF(x*a^(1/2),I)*a^(1/2)-2*(-a*x^2+1)^(1/2)*(a*x^2+1)^(1/2)*x*EllipticE(x*a^(1/2),I)*a^(1/2)-1)/(a^2*x^4-1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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