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

Optimal. Leaf size=151 \[ -\frac{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 a^2}-\frac{x \sqrt{1-a x} \sqrt{a x+1}}{4 a^2}+\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \cosh ^{-1}(a x)}{4 a^3 \sqrt{1-a x}}-\frac{x^2 \sqrt{a x-1} \cosh ^{-1}(a x)}{2 a \sqrt{1-a x}} \]

[Out]

-(x*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(4*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x])/(4*a^3*Sqrt[1 - a*x]) - (x^2*Sqrt[-1
+ a*x]*ArcCosh[a*x])/(2*a*Sqrt[1 - a*x]) - (x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(2*a^2) + (Sqrt[-1 + a*x]*ArcC
osh[a*x]^3)/(6*a^3*Sqrt[1 - a*x])

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Rubi [A]  time = 0.510342, antiderivative size = 207, normalized size of antiderivative = 1.37, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5798, 5759, 5676, 5662, 90, 52} \[ -\frac{x (1-a x) (a x+1)}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt{1-a^2 x^2}}-\frac{x (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt{1-a^2 x^2}}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{2 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{4 a^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(1 - a*x)*(1 + a*x))/(4*a^2*Sqrt[1 - a^2*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(4*a^3*Sqrt[1
- a^2*x^2]) - (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(2*a*Sqrt[1 - a^2*x^2]) - (x*(1 - a*x)*(1 + a*x
)*ArcCosh[a*x]^2)/(2*a^2*Sqrt[1 - a^2*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(6*a^3*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 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5676

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

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && 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 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{x^2 \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^2 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 a^2 \sqrt{1-a^2 x^2}}-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int x \cosh ^{-1}(a x) \, dx}{a \sqrt{1-a^2 x^2}}\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{2 a \sqrt{1-a^2 x^2}}-\frac{x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{1-a^2 x^2}}\\ &=-\frac{x (1-a x) (1+a x)}{4 a^2 \sqrt{1-a^2 x^2}}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{2 a \sqrt{1-a^2 x^2}}-\frac{x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{4 a^2 \sqrt{1-a^2 x^2}}\\ &=-\frac{x (1-a x) (1+a x)}{4 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{4 a^3 \sqrt{1-a^2 x^2}}-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{2 a \sqrt{1-a^2 x^2}}-\frac{x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.170177, size = 87, normalized size = 0.58 \[ -\frac{\sqrt{-(a x-1) (a x+1)} \left (4 \cosh ^{-1}(a x)^3-6 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)+\left (6 \cosh ^{-1}(a x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{24 a^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[-((-1 + a*x)*(1 + a*x))]*(4*ArcCosh[a*x]^3 - 6*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] + (3 + 6*ArcCosh[a*x]^
2)*Sinh[2*ArcCosh[a*x]]))/(24*a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

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Maple [A]  time = 0.162, size = 239, normalized size = 1.6 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{6\,{a}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}-2\,{\rm arccosh} \left (ax\right )+1}{16\,{a}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax+2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+2\,{\rm arccosh} \left (ax\right )+1}{16\,{a}^{3} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( 2\,{x}^{3}{a}^{3}-2\,ax-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/(a^2*x^2-1)*arccosh(a*x)^3-1/16*(-a^2*x^2+1)^(1/2)*(2*
x^3*a^3-2*a*x+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2-2*arccosh(a
*x)+1)/a^3/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(2*x^3*a^3-2*a*x-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2+(a*x-1)^
(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2+2*arccosh(a*x)+1)/a^3/(a^2*x^2-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{2} \operatorname{arcosh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \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**2*acosh(a*x)**2/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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