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

Optimal. Leaf size=98 \[ -\frac{3}{2} a^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \cosh ^{-1}(a x)^2-3 a^2 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac{\cosh ^{-1}(a x)^3}{2 x^2}+\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.31681, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5662, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac{3}{2} a^2 \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \cosh ^{-1}(a x)^2-3 a^2 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac{\cosh ^{-1}(a x)^3}{2 x^2}+\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{x^3} \, dx &=-\frac{\cosh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} (3 a) \int \frac{\cosh ^{-1}(a x)^2}{x^2 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \int \frac{\cosh ^{-1}(a x)}{x} \, dx\\ &=\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{3}{2} a^2 \cosh ^{-1}(a x)^2+\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{2 x^2}-\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{3}{2} a^2 \cosh ^{-1}(a x)^2+\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac{3}{2} a^2 \cosh ^{-1}(a x)^2+\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=\frac{3}{2} a^2 \cosh ^{-1}(a x)^2+\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac{\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \text{Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.793931, size = 92, normalized size = 0.94 \[ \frac{3}{2} a^2 \left (\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )-\frac{\cosh ^{-1}(a x)^3}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.082, size = 113, normalized size = 1.2 \begin{align*}{\frac{3\,{a}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{2\,{x}^{2}}}-3\,{a}^{2}{\rm arccosh} \left (ax\right )\ln \left ( 1+ \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) -{\frac{3\,{a}^{2}}{2}{\it polylog} \left ( 2,- \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) }+{\frac{3\,a \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,x}\sqrt{ax-1}\sqrt{ax+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3/x^3,x)

[Out]

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

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

[Out]

-1/2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^2 + integrate(3/2*(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^5 - a*x^3 + (a^2*x^4 - x^2)*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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccosh(a*x)^3/x^3, 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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)**3/x**3, 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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(a*x)^3/x^3, x)