3.19 \(\int \frac{\cosh ^{-1}(a x)^2}{x^3} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.192071, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5662, 5724, 29} \[ a^2 (-\log (x))-\frac{\cosh ^{-1}(a x)^2}{2 x^2}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^2/x^3,x]

[Out]

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

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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

Mathematica [A]  time = 0.0191179, size = 48, normalized size = 1. \[ a^2 (-\log (x))-\frac{\cosh ^{-1}(a x)^2}{2 x^2}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]^2/x^3,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.063, size = 73, normalized size = 1.5 \begin{align*}{a}^{2}{\rm arccosh} \left (ax\right )+{\frac{a{\rm arccosh} \left (ax\right )}{x}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,{x}^{2}}}-{a}^{2}\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)^2/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.73286, size = 53, normalized size = 1.1 \begin{align*} -a^{2} \log \left (x\right ) + \frac{\sqrt{a^{2} x^{2} - 1} a \operatorname{arcosh}\left (a x\right )}{x} - \frac{\operatorname{arcosh}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.51547, size = 158, normalized size = 3.29 \begin{align*} -\frac{2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt{a^{2} x^{2} - 1} a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{2}{\left (a x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**2/x**3,x)

[Out]

Integral(acosh(a*x)**2/x**3, x)

________________________________________________________________________________________

Giac [B]  time = 1.42507, size = 149, normalized size = 3.1 \begin{align*}{\left (a{\left (\frac{\log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |}} - \frac{{\left | a \right |} \log \left ({\left | x \right |}\right )}{a^{2}}\right )}{\left | a \right |} + \frac{2 \,{\left | a \right |} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1}\right )} a - \frac{\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^3,x, algorithm="giac")

[Out]

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