3.2.3 \(\int \sqrt {1-x^2} \cosh ^{-1}(x) \, dx\) [103]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5895, 5893, 30} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]*ArcCosh[x],x]

[Out]

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

Rule 30

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

Rule 5893

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

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*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] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

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*}

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Mathematica [A]
time = 0.08, size = 54, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {-((-1+x) (1+x))} \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 {-1+x}{1+x}} (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]*ArcCosh[x],x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(151\) vs. \(2(50)=100\).
time = 3.58, size = 152, normalized size = 2.30

method	result	size
default	\(-\frac {\sqrt {-x^{2}+1}\, \mathrm {arccosh}\left (x \right )^{2}}{4 \sqrt {x -1}\, \sqrt {1+x}}+\frac {\sqrt {-x^{2}+1}\, \left (2 x^{3}-2 x +2 \sqrt {1+x}\, \sqrt {x -1}\, x^{2}-\sqrt {x -1}\, \sqrt {1+x}\right ) \left (-1+2 \,\mathrm {arccosh}\left (x \right )\right )}{16 \left (1+x \right ) \left (x -1\right )}+\frac {\sqrt {-x^{2}+1}\, \left (-2 \sqrt {1+x}\, \sqrt {x -1}\, x^{2}+2 x^{3}+\sqrt {x -1}\, \sqrt {1+x}-2 x \right ) \left (1+2 \,\mathrm {arccosh}\left (x \right )\right )}{16 \left (1+x \right ) \left (x -1\right )}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \left (x - 1\right ) \left (x + 1\right )} \operatorname {acosh}{\left (x \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {acosh}\left (x\right )\,\sqrt {1-x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(x)*(1 - x^2)^(1/2),x)

[Out]

int(acosh(x)*(1 - x^2)^(1/2), x)

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