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

Optimal. Leaf size=166 \[ -\frac{3 a \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{3 a \sqrt{a x-1} \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{x}+\frac{a \sqrt{a x-1} \cosh ^{-1}(a x)^3}{\sqrt{1-a x}}-\frac{3 a \sqrt{a x-1} \cosh ^{-1}(a x)^2 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.492011, antiderivative size = 229, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5798, 5724, 5660, 3718, 2190, 2531, 2282, 6589} \[ -\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2] - ((1 - a*x)*(1 + a*x)*ArcCosh[a*x]^3)/(x*Sq
rt[1 - a^2*x^2]) - (3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 + E^(2*ArcCosh[a*x])])/Sqrt[1 - a^2*
x^2] - (3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])])/Sqrt[1 - a^2*x^2] + (3*
a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, -E^(2*ArcCosh[a*x])])/(2*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 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{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)^3}{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)^3}{x \sqrt{1-a^2 x^2}}-\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}-\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}-\frac{\left (6 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (6 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{3 a \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.512007, size = 137, normalized size = 0.83 \[ \frac{a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (6 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )+3 \text{PolyLog}\left (3,-e^{-2 \cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^2 \left (\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-3 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )}{2 \sqrt{-(a x-1) (a x+1)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(2*ArcCosh[a*x]^2*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*
x)*ArcCosh[a*x])/(a*x) - 3*Log[1 + E^(-2*ArcCosh[a*x])]) + 6*ArcCosh[a*x]*PolyLog[2, -E^(-2*ArcCosh[a*x])] + 3
*PolyLog[3, -E^(-2*ArcCosh[a*x])]))/(2*Sqrt[-((-1 + a*x)*(1 + a*x))])

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Maple [A]  time = 0.165, size = 313, normalized size = 1.9 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{x \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }-2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}a}{{a}^{2}{x}^{2}-1}}+3\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\ln \left ( 1+ \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) a}{{a}^{2}{x}^{2}-1}}+3\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,- \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) a}{{a}^{2}{x}^{2}-1}}-{\frac{3\,a}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it polylog} \left ( 3,- \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^2/(-a^2*x^2+1)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )^{3}}{a^{2} x^{4} - x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3/(a^2*x^4 - x^2), 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^{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)**3/x**2/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(acosh(a*x)**3/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), 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}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^2), x)