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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.254105, antiderivative size = 72, normalized size of antiderivative = 1.5, 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 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]

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 29

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

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*}

Mathematica [A]  time = 0.0302518, 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])

________________________________________________________________________________________

Maple [B]  time = 0.136, size = 168, normalized size = 3.5 \begin{align*} -2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\rm arccosh} \left (ax\right )a}{{a}^{2}{x}^{2}-1}}-{\frac{{\rm arccosh} \left (ax\right )}{ \left ({a}^{2}{x}^{2}-1 \right ) x}\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }+{\frac{a}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1+ \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) } \end{align*}

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

________________________________________________________________________________________

Maxima [C]  time = 1.78803, size = 99, normalized size = 2.06 \begin{align*} -\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{align*}

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [C]  time = 1.20838, size = 116, normalized size = 2.42 \begin{align*} -\frac{1}{2} i \, a \log \left (-i \, a^{2} x^{2}\right ) + \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 ) \end{align*}

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