3.2.37 \(\int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [137]

Optimal. Leaf size=88 \[ -\frac {x^2 \sqrt {-1+a x}}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{2 a^2}+\frac {\sqrt {-1+a x} \cosh ^{-1}(a x)^2}{4 a^3 \sqrt {1-a x}} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5938, 5892, 30} \begin {gather*} \frac {\sqrt {a x-1} \cosh ^{-1}(a x)^2}{4 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{2 a^2}-\frac {x^2 \sqrt {a x-1}}{4 a \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

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

Rule 5892

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

Rule 5938

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

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 75, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {-((-1+a x) (1+a x))} \left (-\cosh \left (2 \cosh ^{-1}(a x)\right )+2 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)+\sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{8 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(-Cosh[2*ArcCosh[a*x]] + 2*ArcCosh[a*x]*(ArcCosh[a*x] + Sinh[2*ArcCosh[a*x
]])))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(72)=144\).
time = 5.95, size = 223, normalized size = 2.53

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2}}{4 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-1+2 \,\mathrm {arccosh}\left (a x \right )\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (1+2 \,\mathrm {arccosh}\left (a x \right )\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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