3.31 \(\int \frac{\cosh ^{-1}(a x)^3}{x^5} \, dx\)

Optimal. Leaf size=174 \[ -\frac{1}{2} a^4 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}-\frac{a^3 \sqrt{a x-1} \sqrt{a x+1}}{4 x}+\frac{1}{2} a^4 \cosh ^{-1}(a x)^2+\frac{a^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x}-a^4 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{4 x^3}-\frac{\cosh ^{-1}(a x)^3}{4 x^4} \]

[Out]

-(a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*x) + (a^2*ArcCosh[a*x])/(4*x^2) + (a^4*ArcCosh[a*x]^2)/2 + (a*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(4*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x) - ArcCos
h[a*x]^3/(4*x^4) - a^4*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] - (a^4*PolyLog[2, -E^(2*ArcCosh[a*x])])/2

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Rubi [A]  time = 0.577703, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5662, 5748, 5724, 5660, 3718, 2190, 2279, 2391, 95} \[ -\frac{1}{2} a^4 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}-\frac{a^3 \sqrt{a x-1} \sqrt{a x+1}}{4 x}+\frac{1}{2} a^4 \cosh ^{-1}(a x)^2+\frac{a^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x}-a^4 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{4 x^3}-\frac{\cosh ^{-1}(a x)^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/x^5,x]

[Out]

-(a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*x) + (a^2*ArcCosh[a*x])/(4*x^2) + (a^4*ArcCosh[a*x]^2)/2 + (a*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(4*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x) - ArcCos
h[a*x]^3/(4*x^4) - a^4*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] - (a^4*PolyLog[2, -E^(2*ArcCosh[a*x])])/2

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 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 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

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

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && 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 95

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

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{x^5} \, dx &=-\frac{\cosh ^{-1}(a x)^3}{4 x^4}+\frac{1}{4} (3 a) \int \frac{\cosh ^{-1}(a x)^2}{x^4 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-\frac{1}{2} a^2 \int \frac{\cosh ^{-1}(a x)}{x^3} \, dx+\frac{1}{2} a^3 \int \frac{\cosh ^{-1}(a x)^2}{x^2 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-\frac{1}{4} a^3 \int \frac{1}{x^2 \sqrt{-1+a x} \sqrt{1+a x}} \, dx-a^4 \int \frac{\cosh ^{-1}(a x)}{x} \, dx\\ &=-\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x}}{4 x}+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x}}{4 x}+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \cosh ^{-1}(a x)^2+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x}}{4 x}+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \cosh ^{-1}(a x)^2+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+a^4 \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x}}{4 x}+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \cosh ^{-1}(a x)^2+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=-\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x}}{4 x}+\frac{a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \cosh ^{-1}(a x)^2+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac{1}{2} a^4 \text{Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.598095, size = 220, normalized size = 1.26 \[ \frac{2 a^4 x^4 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )-a^5 x^5+a^3 x^3-a x (a x+1) \left (2 a^3 x^3 \left (\sqrt{\frac{a x-1}{a x+1}}-1\right )+2 a^2 x^2-a x+1\right ) \cosh ^{-1}(a x)^2-a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x) \left (4 a^2 x^2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )-1\right )-\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{4 x^4 \sqrt{a x-1} \sqrt{a x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/x^5,x]

[Out]

(a^3*x^3 - a^5*x^5 - a*x*(1 + a*x)*(1 - a*x + 2*a^2*x^2 + 2*a^3*x^3*(-1 + Sqrt[(-1 + a*x)/(1 + a*x)]))*ArcCosh
[a*x]^2 - Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3 - a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a
*x]*(-1 + 4*a^2*x^2*Log[1 + E^(-2*ArcCosh[a*x])]) + 2*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*PolyLog[2,
-E^(-2*ArcCosh[a*x])])/(4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

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Maple [A]  time = 0.122, size = 180, normalized size = 1. \begin{align*}{\frac{{a}^{3} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,x}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{a}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2}}-{\frac{{a}^{3}}{4\,x}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{a}^{4}}{4}}+{\frac{a \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{4\,{x}^{3}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{a}^{2}{\rm arccosh} \left (ax\right )}{4\,{x}^{2}}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{4\,{x}^{4}}}-{a}^{4}{\rm arccosh} \left (ax\right )\ln \left ( 1+ \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) -{\frac{{a}^{4}}{2}{\it polylog} \left ( 2,- \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)^3/x^5,x)

[Out]

1/2*a^3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x+1/2*a^4*arccosh(a*x)^2-1/4*a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2
)/x+1/4*a^4+1/4*a*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^3+1/4*a^2*arccosh(a*x)/x^2-1/4*arccosh(a*x)^3/x
^4-a^4*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-1/2*a^4*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1
/2))^2)

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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 )^{3}}{4 \, x^{4}} + \int \frac{3 \,{\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 )^{2}}{4 \,{\left (a^{3} x^{7} - a x^{5} +{\left (a^{2} x^{6} - x^{4}\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)^3/x^5,x, algorithm="maxima")

[Out]

-1/4*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^4 + integrate(3/4*(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))^2/(a^3*x^7 - a*x^5 + (a^2*x^6 - x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))
, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcosh}\left (a x\right )^{3}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^5,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^3/x^5, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{3}{\left (a x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/x**5,x)

[Out]

Integral(acosh(a*x)**3/x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^5,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^3/x^5, x)