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

Optimal. Leaf size=79 \[ -\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{a^2}-\frac{2 \sqrt{1-a x} \sqrt{a x+1}}{a^2}-\frac{2 x \sqrt{a x-1} \cosh ^{-1}(a x)}{a \sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.271709, antiderivative size = 109, normalized size of antiderivative = 1.38, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {5798, 5718, 5654, 74} \[ -\frac{2 (1-a x) (a x+1)}{a^2 \sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - a*x)*(1 + a*x))/(a^2*Sqrt[1 - a^2*x^2]) - (2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(a*Sqrt[1 -
a^2*x^2]) - ((1 - a*x)*(1 + a*x)*ArcCosh[a*x]^2)/(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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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

Mathematica [A]  time = 0.0943196, size = 54, normalized size = 0.68 \[ \frac{\sqrt{1-a^2 x^2} \left (-\cosh ^{-1}(a x)^2+\frac{2 a x \cosh ^{-1}(a x)}{\sqrt{a x-1} \sqrt{a x+1}}-2\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.13, size = 139, normalized size = 1.8 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}-2\,{\rm arccosh} \left (ax\right )+2}{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{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+2\,{\rm arccosh} \left (ax\right )+2}{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)^2/(-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)*(arccosh(a*x)^2-2*arccosh(a*x)+2)/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)*(arccosh(a*x)^2+2*arccosh(a*x)+2)/a
^2/(a^2*x^2-1)

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Maxima [C]  time = 1.10081, size = 68, normalized size = 0.86 \begin{align*} \frac{2 i \, x \operatorname{arcosh}\left (a x\right )}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )^{2}}{a^{2}} - \frac{2 i \, \sqrt{a^{2} x^{2} - 1}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.14564, size = 246, normalized size = 3.11 \begin{align*} \frac{2 \, \sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) +{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 2 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-a^{2} x^{2} + 1}}{a^{4} x^{2} - a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 - 1)) + (-a^2*x^2 + 1)^(3/2)*log(a*x + sqrt
(a^2*x^2 - 1))^2 - 2*(a^2*x^2 - 1)*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}^{2}{\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)**2/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [C]  time = 1.1736, size = 103, normalized size = 1.3 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{a^{2}} - \frac{2 i \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{\sqrt{a^{2} x^{2} - 1}}{a}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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