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

Optimal. Leaf size=12 \[ -\frac{x e^{\text{sech}^{-1}(a x)}}{a} \]

[Out]

-((E^ArcSech[a*x]*x)/a)

________________________________________________________________________________________

Rubi [B]  time = 1.05158, antiderivative size = 26, normalized size of antiderivative = 2.17, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6725, 260, 6341, 1956, 74} \[ -\frac{\sqrt{1-a x}}{a^2 \sqrt{\frac{1}{a x+1}}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(x*(-1 + a*E^ArcSech[a*x]*x))/(1 - a^2*x^2),x]

[Out]

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 260

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

Rule 6341

Int[(E^ArcSech[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/(a*c), Int[((d*x)^(m
- 1)*Sqrt[1/(1 + c*x)])/Sqrt[1 - c*x], x], x] + Dist[d/c, Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a,
b, c, d, m}, x] && EqQ[b + a*c^2, 0]

Rule 1956

Int[(x_)^(m_.)*((e_.)*((a_) + (b_.)*(x_)^(n_.))^(r_.))^(p_)*((f_.)*((c_) + (d_.)*(x_)^(n_.))^(s_))^(q_), x_Sym
bol] :> Dist[((e*(a + b*x^n)^r)^p*(f*(c + d*x^n)^s)^q)/((a + b*x^n)^(p*r)*(c + d*x^n)^(q*s)), Int[x^m*(a + b*x
^n)^(p*r)*(c + d*x^n)^(q*s), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r, s}, 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 \frac{x \left (-1+a e^{\text{sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx &=\int \left (\frac{x}{-1+a^2 x^2}-\frac{a e^{\text{sech}^{-1}(a x)} x^2}{-1+a^2 x^2}\right ) \, dx\\ &=-\left (a \int \frac{e^{\text{sech}^{-1}(a x)} x^2}{-1+a^2 x^2} \, dx\right )+\int \frac{x}{-1+a^2 x^2} \, dx\\ &=\frac{\log \left (1-a^2 x^2\right )}{2 a^2}+\int \frac{x \sqrt{\frac{1}{1+a x}}}{\sqrt{1-a x}} \, dx-\int \frac{x}{-1+a^2 x^2} \, dx\\ &=\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=-\frac{\sqrt{1-a x}}{a^2 \sqrt{\frac{1}{1+a x}}}\\ \end{align*}

Mathematica [B]  time = 0.243834, size = 28, normalized size = 2.33 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*(-1 + a*E^ArcSech[a*x]*x))/(1 - a^2*x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.15, size = 36, normalized size = 3. \begin{align*} -{\frac{x}{a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.04182, size = 69, normalized size = 5.75 \begin{align*} -\frac{x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))/a

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (a x{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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