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

Optimal. Leaf size=64 \[ \frac{x^2}{15 a^3}-\frac{x^3 e^{\text{sech}^{-1}(a x)}}{15 a^2}-\frac{2 x e^{\text{sech}^{-1}(a x)}}{15 a^4}+\frac{x^4}{20 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}(a x)} \]

[Out]

(-2*E^ArcSech[a*x]*x)/(15*a^4) + x^2/(15*a^3) - (E^ArcSech[a*x]*x^3)/(15*a^2) + x^4/(20*a) + (E^ArcSech[a*x]*x
^5)/5

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Rubi [A]  time = 0.0384958, antiderivative size = 83, normalized size of antiderivative = 1.3, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 30, 100, 12, 74} \[ -\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{a x+1}}}-\frac{2 \sqrt{1-a x}}{15 a^5 \sqrt{\frac{1}{a x+1}}}+\frac{x^4}{20 a}+\frac{1}{5} x^5 e^{\text{sech}^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

x^4/(20*a) + (E^ArcSech[a*x]*x^5)/5 - (2*Sqrt[1 - a*x])/(15*a^5*Sqrt[(1 + a*x)^(-1)]) - (x^2*Sqrt[1 - a*x])/(1
5*a^3*Sqrt[(1 + a*x)^(-1)])

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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^4 \, dx &=\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5+\frac{\int x^3 \, dx}{5 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x^3}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{5 a}\\ &=\frac{x^4}{20 a}+\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5-\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{1+a x}}}-\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int -\frac{2 x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{15 a^3}\\ &=\frac{x^4}{20 a}+\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5-\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (2 \sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{15 a^3}\\ &=\frac{x^4}{20 a}+\frac{1}{5} e^{\text{sech}^{-1}(a x)} x^5-\frac{2 \sqrt{1-a x}}{15 a^5 \sqrt{\frac{1}{1+a x}}}-\frac{x^2 \sqrt{1-a x}}{15 a^3 \sqrt{\frac{1}{1+a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0899093, size = 65, normalized size = 1.02 \[ \frac{15 a^4 x^4+4 \sqrt{\frac{1-a x}{a x+1}} (a x+1)^2 \left (3 a^3 x^3-3 a^2 x^2+2 a x-2\right )}{60 a^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(15*a^4*x^4 + 4*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(-2 + 2*a*x - 3*a^2*x^2 + 3*a^3*x^3))/(60*a^5)

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Maple [A]  time = 0.189, size = 64, normalized size = 1. \begin{align*}{\frac{x \left ({a}^{2}{x}^{2}-1 \right ) \left ( 3\,{a}^{2}{x}^{2}+2 \right ) }{15\,{a}^{4}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}}+{\frac{{x}^{4}}{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(a^2*x^2-1)*(3*a^2*x^2+2)/a^4+1/4*x^4/a

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Maxima [A]  time = 1.07833, size = 63, normalized size = 0.98 \begin{align*} \frac{x^{4}}{4 \, a} + \frac{{\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{15 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4/a + 1/15*(3*a^4*x^4 - a^2*x^2 - 2)*sqrt(a*x + 1)*sqrt(-a*x + 1)/a^5

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Fricas [A]  time = 1.71408, size = 135, normalized size = 2.11 \begin{align*} \frac{15 \, a^{3} x^{4} + 4 \,{\left (3 \, a^{4} x^{5} - a^{2} x^{3} - 2 \, x\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{60 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(15*a^3*x^4 + 4*(3*a^4*x^5 - a^2*x^3 - 2*x)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)))/a^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x^{3}\, dx + \int a x^{4} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(x**3, x) + Integral(a*x**4*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), 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} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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