3.2.41 \(\int \frac {\cosh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [141]

Optimal. Leaf size=48 \[ -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{x}-\frac {a \sqrt {-1+a x} \log (x)}{\sqrt {1-a x}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5917, 29} \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{x}-\frac {a \sqrt {a x-1} \log (x)}{\sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 5917

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Dist[b*c*(n/(f*(m + 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*A
rcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m
+ 2*p + 3, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 57, normalized size = 1.19 \begin {gather*} \frac {\left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)-a x \sqrt {-1+a x} \sqrt {1+a x} \log (x)}{x \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + a^2*x^2)*ArcCosh[a*x] - a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[x])/(x*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. \(167\) vs. \(2(42)=84\).
time = 4.10, size = 168, normalized size = 3.50

method	result	size
default	\(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) a}{a^{2} x^{2}-1}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \mathrm {arccosh}\left (a x \right )}{\left (a^{2} x^{2}-1\right ) x}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.47, size = 73, normalized size = 1.52 \begin {gather*} -\frac {1}{2} \, {\left (a^{2} \sqrt {-\frac {1}{a^{4}}} \log \left (x^{2} - \frac {1}{a^{2}}\right ) + i \, \left (-1\right )^{-2 \, a^{2} x^{2} + 2} \log \left (-2 \, a^{2} + \frac {2}{x^{2}}\right )\right )} a - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (42) = 84\).
time = 0.37, size = 89, normalized size = 1.85 \begin {gather*} \frac {a x \arctan \left (\frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} {\left (x^{2} + 1\right )}}{a^{2} x^{4} - {\left (a^{2} + 1\right )} x^{2} + 1}\right ) - \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x*arctan(sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*(x^2 + 1)/(a^2*x^4 - (a^2 + 1)*x^2 + 1)) - sqrt(-a^2*x^2 + 1)
*log(a*x + sqrt(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 {\operatorname {acosh}{\left (a x \right )}}{x^{2} \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(acosh(a*x)/x**2/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 80, normalized size = 1.67 \begin {gather*} \frac {1}{2} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - i \, a \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*log(a*x + sq
rt(a^2*x^2 - 1)) - I*a*log(abs(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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