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

Optimal. Leaf size=53 \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]

[Out]

(2*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c) + PolyLog[2, -E^ArcCosh[a*x]]/(a*c) - PolyLog[2, E^ArcCosh[a*x]
]/(a*c)

________________________________________________________________________________________

Rubi [A]  time = 0.0519342, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5694, 4182, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c - a^2*c*x^2),x]

[Out]

(2*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c) + PolyLog[2, -E^ArcCosh[a*x]]/(a*c) - PolyLog[2, E^ArcCosh[a*x]
]/(a*c)

Rule 5694

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(c*d)^(-1), Subst[Int[
(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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)}{c-a^2 c x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{\operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}-\frac{\operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{\text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{\text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}\\ \end{align*}

Mathematica [A]  time = 0.0466481, size = 77, normalized size = 1.45 \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{\cosh ^{-1}(a x) \log \left (1-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{\cosh ^{-1}(a x) \log \left (e^{\cosh ^{-1}(a x)}+1\right )}{a c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.017, size = 321, normalized size = 6.1 \begin{align*}{\frac{{\it Artanh} \left ( ax \right ){\rm arccosh} \left (ax\right )}{ac}}+{\frac{2\,i{\it Artanh} \left ( ax \right ) }{ac \left ( ax-1 \right ) \left ( ax+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{ax}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{ax}{2}}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{2\,i{\it Artanh} \left ( ax \right ) }{ac \left ( ax-1 \right ) \left ( ax+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{ax}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{ax}{2}}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{2\,i}{ac \left ( ax-1 \right ) \left ( ax+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{ax}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{ax}{2}}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{2\,i}{ac \left ( ax-1 \right ) \left ( ax+1 \right ) }\sqrt{{\frac{1}{2}}+{\frac{ax}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-{\frac{1}{2}}+{\frac{ax}{2}}}{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/c*arctanh(a*x)*arccosh(a*x)+2*I/a/c*(1/2+1/2*a*x)^(1/2)*(-a^2*x^2+1)^(1/2)*(-1/2+1/2*a*x)^(1/2)/(a*x-1)/(a
*x+1)*arctanh(a*x)*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))-2*I/a/c*(1/2+1/2*a*x)^(1/2)*(-a^2*x^2+1)^(1/2)*(-1/2+1/2
*a*x)^(1/2)/(a*x-1)/(a*x+1)*arctanh(a*x)*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))+2*I/a/c*(1/2+1/2*a*x)^(1/2)*(-a^2*
x^2+1)^(1/2)*(-1/2+1/2*a*x)^(1/2)/(a*x-1)/(a*x+1)*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))-2*I/a/c*(1/2+1/2*a*x)^
(1/2)*(-a^2*x^2+1)^(1/2)*(-1/2+1/2*a*x)^(1/2)/(a*x-1)/(a*x+1)*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \,{\left (\log \left (a x + 1\right ) - \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right ) - \log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2}}{8 \, a c} + \frac{\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )}{2 \, a c} + \int \frac{\log \left (a x + 1\right ) - \log \left (a x - 1\right )}{2 \,{\left (a^{3} c x^{3} - a c x +{\left (a^{2} c x^{2} - c\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)/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/8*(4*(log(a*x + 1) - log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)) - log(a*x + 1)^2 - 2*log(a*x + 1)*
log(a*x - 1) + log(a*x - 1)^2)/(a*c) + 1/2*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/(a*c) + i
ntegrate(1/2*(log(a*x + 1) - log(a*x - 1))/(a^3*c*x^3 - a*c*x + (a^2*c*x^2 - c)*sqrt(a*x + 1)*sqrt(a*x - 1)),
x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-arccosh(a*x)/(a^2*c*x^2 - c), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)/(-a**2*c*x**2+c),x)

[Out]

-Integral(acosh(a*x)/(a**2*x**2 - 1), x)/c

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-arccosh(a*x)/(a^2*c*x^2 - c), x)