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

Optimal. Leaf size=67 \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{3 a^{3/2}}+\frac{1}{3} x^3 e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{2 x}{3 a} \]

[Out]

(2*x)/(3*a) + (E^ArcSech[a*x^2]*x^3)/3 + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x]
, -1])/(3*a^(3/2))

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Rubi [A]  time = 0.0279784, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6335, 8, 248, 221} \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{3 a^{3/2}}+\frac{1}{3} x^3 e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{2 x}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/(3*a) + (E^ArcSech[a*x^2]*x^3)/3 + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x]
, -1])/(3*a^(3/2))

Rule 6335

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

Rule 8

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

Rule 248

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

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]

Rubi steps

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

Mathematica [C]  time = 0.189033, size = 116, normalized size = 1.73 \[ -\frac{2 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )}{3 (-a)^{3/2} \left (a x^2-1\right )}+\frac{\sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^3+x\right )}{3 a}+\frac{x}{a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

x/a + (Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(x + a*x^3))/(3*a) - (((2*I)/3)*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*Sqrt[1 - a^
2*x^4]*EllipticF[I*ArcSinh[Sqrt[-a]*x], -1])/((-a)^(3/2)*(-1 + a*x^2))

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Maple [A]  time = 0.204, size = 102, normalized size = 1.5 \begin{align*}{\frac{{x}^{2}}{3\,{a}^{2}{x}^{4}-3}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({x}^{5}{a}^{{\frac{5}{2}}}-2\,{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}-x\sqrt{a} \right ){\frac{1}{\sqrt{a}}}}+{\frac{x}{a}} \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^2,x)

[Out]

1/3*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(x^5*a^(5/2)-2*EllipticF(x*a^(1/2),I)*(-a*x^2+1)^(1/2
)*(a*x^2+1)^(1/2)-x*a^(1/2))/(a^2*x^4-1)/a^(1/2)+x/a

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

[Out]

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

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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\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^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)

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

[Out]

(Integral(1, x) + Integral(a*x**2*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 x^{2}{\left (\sqrt{\frac{1}{a x^{2}} + 1} \sqrt{\frac{1}{a x^{2}} - 1} + \frac{1}{a x^{2}}\right )}\,{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^2,x, algorithm="giac")

[Out]

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