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

Optimal. Leaf size=183 \[ -\frac{2 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{2 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{2 i \sqrt{a x-1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{2 i \sqrt{a x-1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.420395, antiderivative size = 248, normalized size of antiderivative = 1.36, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5798, 5761, 4180, 2531, 2282, 6589} \[ -\frac{2 i \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 i \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 i \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{2 i \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] - ((2*I)*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]*ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] + ((2*I)*Sqrt[-1 + a*x]*Sqrt[
1 + a*x]*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] + ((2*I)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Po
lyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a^2*x^2] - ((2*I)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, I*E^ArcCosh[
a*x]])/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 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 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)^2}{x \sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (2 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (2 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (2 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (2 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (2 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (2 i \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{2 i \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.18457, size = 151, normalized size = 0.83 \[ \frac{i \sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (-2 \cosh ^{-1}(a x) \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right )-2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^2 \left (-\left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )\right )\right )}{\sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{2}{\left (a x \right )}}{x \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)**2/x/(-a**2*x**2+1)**(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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