3.141 \(\int \frac {\cosh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\)

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

[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 = 0.25, antiderivative size = 72, normalized size of antiderivative = 1.50, 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 = {5798, 5724, 29} \[ -\frac {a \sqrt {a x-1} \sqrt {a x+1} \log (x)}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 29

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

Rule 5724

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh
[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5798

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

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 = 0.03, size = 57, normalized size = 1.19 \[ \frac {\left (a^2 x^2-1\right ) \cosh ^{-1}(a x)-a x \sqrt {a x-1} \sqrt {a x+1} \log (x)}{x \sqrt {1-a^2 x^2}} \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> the translation of the FriCAS object sage2 to sage is not yet impleme
nted

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giac [C]  time = 0.52, size = 80, normalized size = 1.67 \[ \frac {1}{2} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - i \, a \log \left ({\left | x \right |}\right ) \]

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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maple [B]  time = 0.36, size = 168, normalized size = 3.50 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) a}{a^{2} x^{2}-1}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \mathrm {arccosh}\left (a x \right )}{x \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1} \]

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)

[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)/x/(a^2*x^2-1)+(-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]  time = 0.86, size = 73, normalized size = 1.52 \[ -\frac {1}{2} \, {\left (a^{2} \sqrt {-\frac {1}{a^{4}}} \log \left (x^{2} - \frac {1}{a^{2}}\right ) + i \, \left (-1\right )^{-2 \, a^{2} x^{2} + 2} \log \left (-2 \, a^{2} + \frac {2}{x^{2}}\right )\right )} a - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}{x} \]

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

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

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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