3.103 \(\int \sqrt{1-x^2} \cosh ^{-1}(x) \, dx\)

Optimal. Leaf size=66 \[ -\frac{\sqrt{1-x} x^2}{4 \sqrt{x-1}}+\frac{1}{2} \sqrt{1-x^2} x \cosh ^{-1}(x)-\frac{\sqrt{1-x} \cosh ^{-1}(x)^2}{4 \sqrt{x-1}} \]

[Out]

-(Sqrt[1 - x]*x^2)/(4*Sqrt[-1 + x]) + (x*Sqrt[1 - x^2]*ArcCosh[x])/2 - (Sqrt[1 - x]*ArcCosh[x]^2)/(4*Sqrt[-1 +
x])

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Rubi [A]  time = 0.104011, antiderivative size = 84, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5713, 5683, 5676, 30} \[ -\frac{\sqrt{1-x^2} x^2}{4 \sqrt{x-1} \sqrt{x+1}}+\frac{1}{2} \sqrt{1-x^2} x \cosh ^{-1}(x)-\frac{\sqrt{1-x^2} \cosh ^{-1}(x)^2}{4 \sqrt{x-1} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]*ArcCosh[x],x]

[Out]

-(x^2*Sqrt[1 - x^2])/(4*Sqrt[-1 + x]*Sqrt[1 + x]) + (x*Sqrt[1 - x^2]*ArcCosh[x])/2 - (Sqrt[1 - x^2]*ArcCosh[x]
^2)/(4*Sqrt[-1 + x]*Sqrt[1 + x])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5683

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

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.110636, size = 54, normalized size = 0.82 \[ -\frac{\sqrt{-(x-1) (x+1)} \left (\cosh \left (2 \cosh ^{-1}(x)\right )+2 \cosh ^{-1}(x) \left (\cosh ^{-1}(x)-\sinh \left (2 \cosh ^{-1}(x)\right )\right )\right )}{8 \sqrt{\frac{x-1}{x+1}} (x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[1 - x^2]*ArcCosh[x],x]

[Out]

-(Sqrt[-((-1 + x)*(1 + x))]*(Cosh[2*ArcCosh[x]] + 2*ArcCosh[x]*(ArcCosh[x] - Sinh[2*ArcCosh[x]])))/(8*Sqrt[(-1
+ x)/(1 + x)]*(1 + x))

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Maple [B]  time = 0.147, size = 152, normalized size = 2.3 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (x\right ) \right ) ^{2}}{4}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{-1+x}}}{\frac{1}{\sqrt{1+x}}}}+{\frac{-1+2\,{\rm arccosh} \left (x\right )}{ \left ( -16+16\,x \right ) \left ( 1+x \right ) }\sqrt{-{x}^{2}+1} \left ( 2\,{x}^{3}-2\,x+2\,\sqrt{1+x}\sqrt{-1+x}{x}^{2}-\sqrt{-1+x}\sqrt{1+x} \right ) }+{\frac{1+2\,{\rm arccosh} \left (x\right )}{ \left ( -16+16\,x \right ) \left ( 1+x \right ) }\sqrt{-{x}^{2}+1} \left ( -2\,\sqrt{1+x}\sqrt{-1+x}{x}^{2}+2\,{x}^{3}+\sqrt{-1+x}\sqrt{1+x}-2\,x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-x^2+1)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2)*arccosh(x)^2+1/16*(-x^2+1)^(1/2)*(2*x^3-2*x+2*(1+x)^(1/2)*(-1+x)^
(1/2)*x^2-(-1+x)^(1/2)*(1+x)^(1/2))*(-1+2*arccosh(x))/(-1+x)/(1+x)+1/16*(-x^2+1)^(1/2)*(-2*(1+x)^(1/2)*(-1+x)^
(1/2)*x^2+2*x^3+(-1+x)^(1/2)*(1+x)^(1/2)-2*x)*(1+2*arccosh(x))/(-1+x)/(1+x)

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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(arccosh(x)*(-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 (\sqrt{-x^{2} + 1} \operatorname{arcosh}\left (x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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