3.47 \(\int e^{\text{sech}^{-1}(a x^2)} x^5 \, dx\)

Optimal. Leaf size=58 \[ -\frac{\sqrt{1-a x^2}}{6 a^3 \sqrt{\frac{1}{a x^2+1}}}+\frac{x^4}{12 a}+\frac{1}{6} x^6 e^{\text{sech}^{-1}\left (a x^2\right )} \]

[Out]

x^4/(12*a) + (E^ArcSech[a*x^2]*x^6)/6 - Sqrt[1 - a*x^2]/(6*a^3*Sqrt[(1 + a*x^2)^(-1)])

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Rubi [A]  time = 0.0412168, antiderivative size = 71, normalized size of antiderivative = 1.22, 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, 30, 259, 261} \[ -\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{6 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 e^{\text{sech}^{-1}\left (a x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x^2]*x^5,x]

[Out]

x^4/(12*a) + (E^ArcSech[a*x^2]*x^6)/6 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(6*a^3)

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^5 \, dx &=\frac{1}{6} e^{\text{sech}^{-1}\left (a x^2\right )} x^6+\frac{\int x^3 \, dx}{3 a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^3}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{3 a}\\ &=\frac{x^4}{12 a}+\frac{1}{6} e^{\text{sech}^{-1}\left (a x^2\right )} x^6+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^3}{\sqrt{1-a^2 x^4}} \, dx}{3 a}\\ &=\frac{x^4}{12 a}+\frac{1}{6} e^{\text{sech}^{-1}\left (a x^2\right )} x^6-\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{6 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0941505, size = 56, normalized size = 0.97 \[ \frac{\left (a x^2-1\right ) \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )^2}{6 a^3}+\frac{x^4}{4 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^2]*x^5,x]

[Out]

x^4/(4*a) + ((-1 + a*x^2)*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2)^2)/(6*a^3)

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Maple [A]  time = 0.178, size = 60, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ({a}^{2}{x}^{4}-1 \right ) }{6\,{a}^{2}}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}}}+{\frac{{x}^{4}}{4\,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^5,x)

[Out]

1/6*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(a^2*x^4-1)/a^2+1/4*x^4/a

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Maxima [A]  time = 1.06829, size = 57, normalized size = 0.98 \begin{align*} \frac{x^{4}}{4 \, a} + \frac{{\left (a^{2} x^{4} - 1\right )} \sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1}}{6 \, a^{3}} \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^5,x, algorithm="maxima")

[Out]

1/4*x^4/a + 1/6*(a^2*x^4 - 1)*sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/a^3

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Fricas [A]  time = 2.06129, size = 126, normalized size = 2.17 \begin{align*} \frac{3 \, a x^{4} + 2 \,{\left (a^{2} x^{6} - x^{2}\right )} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}}}{12 \, a^{2}} \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^5,x, algorithm="fricas")

[Out]

1/12*(3*a*x^4 + 2*(a^2*x^6 - x^2)*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)))/a^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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**5,x)

[Out]

Timed out

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Giac [B]  time = 1.18225, size = 240, normalized size = 4.14 \begin{align*} \frac{3 \,{\left (a^{2} x^{2} + a\right )}^{2} - 6 \,{\left (a^{2} x^{2} + a\right )} a - \frac{6 \, a^{3} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right ) - \sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a}{\left ({\left (2 \, a^{2} x^{2} + 5 \, a\right )}{\left (a^{2} x^{2} - a\right )} + 3 \, a^{2}\right )} + 3 \,{\left (\sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a} a^{2} x^{2} - 2 \, a^{2} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right )\right )} a}{a}}{12 \, a^{5}} \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^5,x, algorithm="giac")

[Out]

1/12*(3*(a^2*x^2 + a)^2 - 6*(a^2*x^2 + a)*a - (6*a^3*arcsin(1/2*sqrt(2)*sqrt(-a^2*x^2 + a)/sqrt(a)) - sqrt(a^2
*x^2 + a)*sqrt(-a^2*x^2 + a)*((2*a^2*x^2 + 5*a)*(a^2*x^2 - a) + 3*a^2) + 3*(sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2 +
a)*a^2*x^2 - 2*a^2*arcsin(1/2*sqrt(2)*sqrt(-a^2*x^2 + a)/sqrt(a)))*a)/a)/a^5