3.138 \(\int \frac{x \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{a^2}-\frac{x \sqrt{a x-1}}{a \sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.176885, antiderivative size = 73, normalized size of antiderivative = 1.49, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5798, 5718, 8} \[ -\frac{x \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

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

Rule 5798

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

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{x \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int 1 \, dx}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x}}{a \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0842637, size = 55, normalized size = 1.12 \[ \frac{\left (a^2 x^2-1\right ) \cosh ^{-1}(a x)-a x \sqrt{a x-1} \sqrt{a x+1}}{a^2 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]

[Out]

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

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Maple [B]  time = 0.12, size = 123, normalized size = 2.5 \begin{align*} -{\frac{-1+{\rm arccosh} \left (ax\right )}{2\,{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \sqrt{ax+1}\sqrt{ax-1}ax+{a}^{2}{x}^{2}-1 \right ) }-{\frac{1+{\rm arccosh} \left (ax\right )}{2\,{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.16619, size = 38, normalized size = 0.78 \begin{align*} \frac{i \, x}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x/a - sqrt(-a^2*x^2 + 1)*arccosh(a*x)/a^2

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Fricas [A]  time = 2.06582, size = 151, normalized size = 3.08 \begin{align*} \frac{\sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} a x +{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{acosh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.186, size = 54, normalized size = 1.1 \begin{align*} -\frac{i \, x}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*x/a - sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))/a^2