3.48 \(\int e^{\text{sech}^{-1}(a x^2)} x^4 \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{5 a^{5/2}}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}\left (a x^2\right )} \]

[Out]

(2*x^3)/(15*a) + (E^ArcSech[a*x^2]*x^5)/5 + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticE[ArcSin[Sqrt[a]
*x], -1])/(5*a^(5/2)) - (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/(5*a^(5/2)
)

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Rubi [A]  time = 0.0665785, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6335, 30, 259, 307, 221, 1199, 424} \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{5 a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}\left (a x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[a*x^2]*x^4,x]

[Out]

(2*x^3)/(15*a) + (E^ArcSech[a*x^2]*x^5)/5 + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticE[ArcSin[Sqrt[a]
*x], -1])/(5*a^(5/2)) - (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/(5*a^(5/2)
)

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 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 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 )} x^4 \, dx &=\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5+\frac{2 \int x^2 \, dx}{5 a}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^2}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{5 a}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^2}{\sqrt{1-a^2 x^4}} \, dx}{5 a}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5-\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}{5 a^2}+\frac{\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}{5 a^2}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5-\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 )}{5 a^{5/2}}+\frac{\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}{5 a^2}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} e^{\text{sech}^{-1}\left (a x^2\right )} x^5+\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 )}{5 a^{5/2}}-\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 )}{5 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.435428, size = 140, normalized size = 1.25 \[ \frac{1}{15} \left (\frac{6 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-a} x\right )\right |-1\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )\right )}{(-a)^{5/2} \left (a x^2-1\right )}+\frac{5 x^3}{a}+\frac{3 \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^5+x^3\right )}{a}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^2]*x^4,x]

[Out]

((5*x^3)/a + (3*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(x^3 + a*x^5))/a + ((6*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]))/((-a)^(5/2)*(-1 + a*x
^2)))/15

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Maple [A]  time = 0.181, size = 136, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{5\,{a}^{2}{x}^{4}-5}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({a}^{{\frac{7}{2}}}{x}^{7}-{x}^{3}{a}^{{\frac{3}{2}}}+2\,{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}-2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}{\it EllipticE} \left ( x\sqrt{a},i \right ) \right ){a}^{-{\frac{3}{2}}}}+{\frac{{x}^{3}}{3\,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^4,x)

[Out]

1/5*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(a^(7/2)*x^7-x^3*a^(3/2)+2*EllipticF(x*a^(1/2),I)*(-a
*x^2+1)^(1/2)*(a*x^2+1)^(1/2)-2*(-a*x^2+1)^(1/2)*(a*x^2+1)^(1/2)*EllipticE(x*a^(1/2),I))/(a^2*x^4-1)/a^(3/2)+1
/3*x^3/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{3}}{3 \, a} + \frac{\int \sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1} x^{2}\,{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^4,x, algorithm="maxima")

[Out]

1/3*x^3/a + integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)*x^2, x)/a

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x^{4} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + x^{2}}{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^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x^{2}\, dx + \int a x^{4} \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**4,x)

[Out]

(Integral(x**2, x) + Integral(a*x**4*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^{4}{\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^4,x, algorithm="giac")

[Out]

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