3.2.38 \(\int \frac {x \cosh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\) [138]

Optimal. Leaf size=49 \[ -\frac {x \sqrt {-1+a x}}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{a^2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5914, 8} \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{a^2}-\frac {x \sqrt {a x-1}}{a \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5914

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 55, normalized size = 1.12 \begin {gather*} \frac {-a x \sqrt {-1+a x} \sqrt {1+a x}+\left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(43)=86\).
time = 3.02, size = 123, normalized size = 2.51

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (1+\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.26, size = 28, normalized size = 0.57 \begin {gather*} \frac {i \, x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 72, normalized size = 1.47 \begin {gather*} \frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \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*acosh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 0.41, size = 40, normalized size = 0.82 \begin {gather*} -\frac {i \, x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,\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*acosh(a*x))/(1 - a^2*x^2)^(1/2),x)

[Out]

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

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