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

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

[Out]

(2*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c) + (2*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c) - (2*Arc
Cosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c) - (2*PolyLog[3, -E^ArcCosh[a*x]])/(a*c) + (2*PolyLog[3, E^ArcCosh[
a*x]])/(a*c)

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Rubi [A]  time = 0.0978093, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5694, 4182, 2531, 2282, 6589} \[ \frac{2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^2/(c - a^2*c*x^2),x]

[Out]

(2*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c) + (2*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c) - (2*Arc
Cosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c) - (2*PolyLog[3, -E^ArcCosh[a*x]])/(a*c) + (2*PolyLog[3, 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 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}{c-a^2 c x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}-\frac{2 \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}+\frac{2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}\\ &=\frac{2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac{2 \text{Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac{2 \text{Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}\\ \end{align*}

Mathematica [A]  time = 0.0794439, size = 95, normalized size = 0.97 \[ \frac{2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^2 \left (-\log \left (1-e^{\cosh ^{-1}(a x)}\right )\right )+\cosh ^{-1}(a x)^2 \log \left (e^{\cosh ^{-1}(a x)}+1\right )}{a c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 201, normalized size = 2.1 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{ac}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-2\,{\frac{{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{ac}}+2\,{\frac{{\it polylog} \left ( 3,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{ac}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{ac}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+2\,{\frac{{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{ac}}-2\,{\frac{{\it polylog} \left ( 3,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{ac}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(a*x + 1) - log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a*c) - integrate(((a*x*log(a*x + 1
) - a*x*log(a*x - 1))*sqrt(a*x + 1)*sqrt(a*x - 1) + (a^2*x^2 - 1)*log(a*x + 1) - (a^2*x^2 - 1)*log(a*x - 1))*l
og(a*x + sqrt(a*x + 1)*sqrt(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)

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

[Out]

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

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

[Out]

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

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

[Out]

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