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

Optimal. Leaf size=109 \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac{1}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.0804365, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5689, 74, 5694, 4182, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac{1}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c - a^2*c*x^2)^2,x]

[Out]

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

Rule 5689

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcCosh[c*x])^n)/(2*d*(p + 1)), x] + (-Dist[(b*c*n*(-d)^p)/(2*(p + 1)), Int[x*(1 + c*x)^(p + 1/2)
*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p
+ 1)*(a + b*ArcCosh[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p,
-1] && IntegerQ[p]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

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

Mathematica [A]  time = 0.83427, size = 120, normalized size = 1.1 \[ \frac{2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-\frac{2 \left (\cosh ^{-1}(a x) \left (\left (a^2 x^2-1\right ) \log \left (1-e^{\cosh ^{-1}(a x)}\right )+\left (1-a^2 x^2\right ) \log \left (e^{\cosh ^{-1}(a x)}+1\right )+a x\right )+\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right )}{a^2 x^2-1}}{4 a c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x]*(a*x + (-1 + a^2*x^2)*Log[1 - E^ArcCosh[a*x]] + (1 -
a^2*x^2)*Log[1 + E^ArcCosh[a*x]])))/(-1 + a^2*x^2) + 2*PolyLog[2, -E^ArcCosh[a*x]] - 2*PolyLog[2, E^ArcCosh[a
*x]])/(4*a*c^2)

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Maple [A]  time = 0.078, size = 184, normalized size = 1.7 \begin{align*} -{\frac{x{\rm arccosh} \left (ax\right )}{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){c}^{2}}}-{\frac{1}{2\,a \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\rm arccosh} \left (ax\right )}{2\,a{c}^{2}}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{1}{2\,a{c}^{2}}{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{{\rm arccosh} \left (ax\right )}{2\,a{c}^{2}}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{1}{2\,a{c}^{2}}{\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+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)^2,x)

[Out]

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

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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 + 1\right )^{2} + 2 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 4 \, a x + 4 \,{\left (2 \, a x -{\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 ) - 2 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )}{16 \,{\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} + \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 )}{4 \, a c^{2}} - \frac{\log \left (a x + 1\right )}{8 \, a c^{2}} + \int -\frac{2 \, a x -{\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 )}{4 \,{\left (a^{5} c^{2} x^{5} - 2 \, a^{3} c^{2} x^{3} + a c^{2} x +{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\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)^2,x, algorithm="maxima")

[Out]

-1/16*((a^2*x^2 - 1)*log(a*x + 1)^2 + 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a*x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^2
+ 4*a*x + 4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*
x - 1)) - 2*(a^2*x^2 - 1)*log(a*x - 1))/(a^3*c^2*x^2 - a*c^2) + 1/4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-
1/2*a*x + 1/2))/(a*c^2) - 1/8*log(a*x + 1)/(a*c^2) + integrate(-1/4*(2*a*x - (a^2*x^2 - 1)*log(a*x + 1) + (a^2
*x^2 - 1)*log(a*x - 1))/(a^5*c^2*x^5 - 2*a^3*c^2*x^3 + a*c^2*x + (a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*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 )}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{acosh}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)/(a**4*x**4 - 2*a**2*x**2 + 1), x)/c**2

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

[Out]

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