3.33 \(\int e^{\text{sech}^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=84 \[ -\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{8 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 e^{\text{sech}^{-1}(a x)} \]

[Out]

x^3/(12*a) + (E^ArcSech[a*x]*x^4)/4 - (x*Sqrt[1 - a*x])/(8*a^3*Sqrt[(1 + a*x)^(-1)]) + (Sqrt[(1 + a*x)^(-1)]*S
qrt[1 + a*x]*ArcSin[a*x])/(8*a^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0319382, antiderivative size = 84, normalized size of antiderivative = 1., 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, 90, 41, 216} \[ -\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{a x+1}}}+\frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{8 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 e^{\text{sech}^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[a*x]*x^3,x]

[Out]

x^3/(12*a) + (E^ArcSech[a*x]*x^4)/4 - (x*Sqrt[1 - a*x])/(8*a^3*Sqrt[(1 + a*x)^(-1)]) + (Sqrt[(1 + a*x)^(-1)]*S
qrt[1 + a*x]*ArcSin[a*x])/(8*a^4)

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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}(a x)} x^3 \, dx &=\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4+\frac{\int x^2 \, dx}{4 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x^2}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{4 a}\\ &=\frac{x^3}{12 a}+\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4-\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{8 a^3}\\ &=\frac{x^3}{12 a}+\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4-\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^3}\\ &=\frac{x^3}{12 a}+\frac{1}{4} e^{\text{sech}^{-1}(a x)} x^4-\frac{x \sqrt{1-a x}}{8 a^3 \sqrt{\frac{1}{1+a x}}}+\frac{\sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \sin ^{-1}(a x)}{8 a^4}\\ \end{align*}

Mathematica [C]  time = 0.126645, size = 97, normalized size = 1.15 \[ \frac{8 a^3 x^3-3 a \sqrt{\frac{1-a x}{a x+1}} \left (-2 a^3 x^4-2 a^2 x^3+a x^2+x\right )+3 i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{24 a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x]*x^3,x]

[Out]

(8*a^3*x^3 - 3*a*Sqrt[(1 - a*x)/(1 + a*x)]*(x + a*x^2 - 2*a^2*x^3 - 2*a^3*x^4) + (3*I)*Log[(-2*I)*a*x + 2*Sqrt
[(1 - a*x)/(1 + a*x)]*(1 + a*x)])/(24*a^4)

________________________________________________________________________________________

Maple [C]  time = 0.179, size = 118, normalized size = 1.4 \begin{align*}{\frac{x{\it csgn} \left ( a \right ) }{8\,{a}^{3}}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( 2\,{\it csgn} \left ( a \right ){x}^{3}{a}^{3}\sqrt{-{a}^{2}{x}^{2}+1}-x\sqrt{-{a}^{2}{x}^{2}+1}{\it csgn} \left ( a \right ) a+\arctan \left ({x{\it csgn} \left ( a \right ) a{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{x}^{3}}{3\,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^3,x)

[Out]

1/8*(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(2*csgn(a)*x^3*a^3*(-a^2*x^2+1)^(1/2)-x*(-a^2*x^2+1)^(1/2)*csgn
(a)*a+arctan(csgn(a)*a*x/(-a^2*x^2+1)^(1/2)))*csgn(a)/(-a^2*x^2+1)^(1/2)/a^3+1/3*x^3/a

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{3}}{3 \, a} + \frac{-\frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, a^{2}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{8 \, a^{2}} + \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}} a^{2}}}{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^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.77774, size = 203, normalized size = 2.42 \begin{align*} \frac{8 \, a^{3} x^{3} + 3 \,{\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - 3 \, \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right )}{24 \, 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^3,x, algorithm="fricas")

[Out]

1/24*(8*a^3*x^3 + 3*(2*a^4*x^4 - a^2*x^2)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 3*arctan(sqrt((a*x +
1)/(a*x))*sqrt(-(a*x - 1)/(a*x))))/a^4

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int x^{2}\, dx + \int a x^{3} \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**3,x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\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^3,x, algorithm="giac")

[Out]

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