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

Optimal. Leaf size=103 \[ -\frac{i \sqrt{a x-1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{i \sqrt{a x-1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{2 \sqrt{a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.276047, antiderivative size = 142, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {5798, 5761, 4180, 2279, 2391} \[ -\frac{i \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.141795, size = 113, normalized size = 1.1 \[ \frac{i \sqrt{-(a x-1) (a x+1)} \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.142, size = 270, normalized size = 2.6 \begin{align*}{\frac{i{\rm arccosh} \left (ax\right )}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }-{\frac{i{\rm arccosh} \left (ax\right )}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }+{\frac{i}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it dilog} \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }-{\frac{i}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\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)/x/(-a^2*x^2+1)^(1/2),x)

[Out]

I*(-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)*ln(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1
/2)))-I*(-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)*ln(1-I*(a*x+(a*x-1)^(1/2)*(a*x
+1)^(1/2)))+I*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*dilog(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^
(1/2)))-I*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*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*} \int \frac{\operatorname{arcosh}\left (a x\right )}{\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)/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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 )}{a^{2} x^{3} - x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Integral(acosh(a*x)/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)

________________________________________________________________________________________

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

[Out]

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