3.20 \(\int \frac{\cosh ^{-1}(a x)^2}{x^4} \, dx\)

Optimal. Leaf size=114 \[ -\frac{1}{3} i a^3 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+\frac{1}{3} i a^3 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+\frac{a^2}{3 x}+\frac{2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3} \]

[Out]

a^2/(3*x) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*x^2) - ArcCosh[a*x]^2/(3*x^3) + (2*a^3*ArcCosh[a*
x]*ArcTan[E^ArcCosh[a*x]])/3 - (I/3)*a^3*PolyLog[2, (-I)*E^ArcCosh[a*x]] + (I/3)*a^3*PolyLog[2, I*E^ArcCosh[a*
x]]

________________________________________________________________________________________

Rubi [A]  time = 0.392414, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5662, 5748, 5761, 4180, 2279, 2391, 30} \[ -\frac{1}{3} i a^3 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+\frac{1}{3} i a^3 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+\frac{a^2}{3 x}+\frac{2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^2/x^4,x]

[Out]

a^2/(3*x) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*x^2) - ArcCosh[a*x]^2/(3*x^3) + (2*a^3*ArcCosh[a*
x]*ArcTan[E^ArcCosh[a*x]])/3 - (I/3)*a^3*PolyLog[2, (-I)*E^ArcCosh[a*x]] + (I/3)*a^3*PolyLog[2, I*E^ArcCosh[a*
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 5748

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)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], 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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)^2}{x^4} \, dx &=-\frac{\cosh ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} (2 a) \int \frac{\cosh ^{-1}(a x)}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} a^2 \int \frac{1}{x^2} \, dx+\frac{1}{3} a^3 \int \frac{\cosh ^{-1}(a x)}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a^2}{3 x}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} a^3 \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{a^2}{3 x}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac{1}{3} \left (i a^3\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\frac{1}{3} \left (i a^3\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{a^2}{3 x}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac{1}{3} \left (i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+\frac{1}{3} \left (i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=\frac{a^2}{3 x}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x^2}-\frac{\cosh ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\frac{1}{3} i a^3 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+\frac{1}{3} i a^3 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.242953, size = 144, normalized size = 1.26 \[ \frac{1}{3} a^3 \left (-i \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+i \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )-\frac{\cosh ^{-1}(a x)^2}{a^3 x^3}+\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a^2 x^2}+\frac{1}{a x}-i \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+i \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^2/x^4,x]

[Out]

(a^3*(1/(a*x) + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x])/(a^2*x^2) - ArcCosh[a*x]^2/(a^3*x^3) - I*A
rcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x]] + I*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] - I*PolyLog[2, (-I)/E^ArcCosh
[a*x]] + I*PolyLog[2, I/E^ArcCosh[a*x]]))/3

________________________________________________________________________________________

Maple [A]  time = 0.106, size = 177, normalized size = 1.6 \begin{align*}{\frac{a{\rm arccosh} \left (ax\right )}{3\,{x}^{2}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{a}^{2}}{3\,x}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{i}{3}}{a}^{3}{\rm arccosh} \left (ax\right )\ln \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) +{\frac{i}{3}}{a}^{3}{\rm arccosh} \left (ax\right )\ln \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) -{\frac{i}{3}}{a}^{3}{\it dilog} \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) +{\frac{i}{3}}{a}^{3}{\it dilog} \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^2/x^4,x)

[Out]

1/3*a*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^2+1/3*a^2/x-1/3*arccosh(a*x)^2/x^3-1/3*I*a^3*arccosh(a*x)*ln(
1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*I*a^3*arccosh(a*x)*ln(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-1/3*I*
a^3*dilog(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+1/3*I*a^3*dilog(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2}}{3 \, x^{3}} + \int \frac{2 \,{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}{3 \,{\left (a^{3} x^{6} - a x^{4} +{\left (a^{2} x^{5} - x^{3}\right )} \sqrt{a x + 1} \sqrt{a x - 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^4,x, algorithm="maxima")

[Out]

-1/3*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/x^3 + integrate(2/3*(a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x
- a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/(a^3*x^6 - a*x^4 + (a^2*x^5 - x^3)*sqrt(a*x + 1)*sqrt(a*x - 1)),
x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcosh}\left (a x\right )^{2}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^2/x^4, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{2}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**2/x**4,x)

[Out]

Integral(acosh(a*x)**2/x**4, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^2/x^4, x)