3.57 \(\int \frac{\cosh ^{-1}(a x)}{(c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=164 \[ \frac{3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{3}{8 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac{3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \]

[Out]

1/(12*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - 3/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x])/(4
*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x])/(8*c^3*(1 - a^2*x^2)) + (3*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(4
*a*c^3) + (3*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) - (3*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.112888, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{3}{8 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac{3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]/(c - a^2*c*x^2)^3,x]

[Out]

1/(12*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - 3/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x])/(4
*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x])/(8*c^3*(1 - a^2*x^2)) + (3*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(4
*a*c^3) + (3*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) - (3*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3)

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 )^3} \, dx &=\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{a \int \frac{x}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{4 c^3}+\frac{3 \int \frac{\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac{(3 a) \int \frac{x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{8 c^3}+\frac{3 \int \frac{\cosh ^{-1}(a x)}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=\frac{1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}-\frac{3 \operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}\\ &=\frac{1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}-\frac{3 \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}\\ &=\frac{1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}\\ &=\frac{1}{12 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3}{8 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}-\frac{3 \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{8 a c^3}\\ \end{align*}

Mathematica [A]  time = 2.34331, size = 223, normalized size = 1.36 \[ \frac{36 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-36 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-\frac{2 \sqrt{a x+1} (a x-2)}{(a x-1)^{3/2}}+\frac{2 \sqrt{a x-1} (a x+2)}{(a x+1)^{3/2}}+\frac{6 \cosh ^{-1}(a x)}{(a x-1)^2}-\frac{6 \cosh ^{-1}(a x)}{(a x+1)^2}+18 \left (\frac{\cosh ^{-1}(a x)}{1-a x}-\frac{1}{\sqrt{\frac{a x-1}{a x+1}}}\right )+18 \left (\sqrt{\frac{a x-1}{a x+1}}-\frac{\cosh ^{-1}(a x)}{a x+1}\right )+9 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)-4 \log \left (1-e^{\cosh ^{-1}(a x)}\right )\right )-9 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)-4 \log \left (e^{\cosh ^{-1}(a x)}+1\right )\right )}{96 a c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]/(c - a^2*c*x^2)^3,x]

[Out]

((-2*(-2 + a*x)*Sqrt[1 + a*x])/(-1 + a*x)^(3/2) + (2*Sqrt[-1 + a*x]*(2 + a*x))/(1 + a*x)^(3/2) + (6*ArcCosh[a*
x])/(-1 + a*x)^2 - (6*ArcCosh[a*x])/(1 + a*x)^2 + 18*(-(1/Sqrt[(-1 + a*x)/(1 + a*x)]) + ArcCosh[a*x]/(1 - a*x)
) + 18*(Sqrt[(-1 + a*x)/(1 + a*x)] - ArcCosh[a*x]/(1 + a*x)) + 9*ArcCosh[a*x]*(ArcCosh[a*x] - 4*Log[1 - E^ArcC
osh[a*x]]) - 9*ArcCosh[a*x]*(ArcCosh[a*x] - 4*Log[1 + E^ArcCosh[a*x]]) + 36*PolyLog[2, -E^ArcCosh[a*x]] - 36*P
olyLog[2, E^ArcCosh[a*x]])/(96*a*c^3)

________________________________________________________________________________________

Maple [A]  time = 0.088, size = 276, normalized size = 1.7 \begin{align*} -{\frac{3\,{a}^{2}{x}^{3}{\rm arccosh} \left (ax\right )}{ \left ( 8\,{x}^{4}{a}^{4}-16\,{a}^{2}{x}^{2}+8 \right ){c}^{3}}}-{\frac{3\,a{x}^{2}}{ \left ( 8\,{x}^{4}{a}^{4}-16\,{a}^{2}{x}^{2}+8 \right ){c}^{3}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{5\,x{\rm arccosh} \left (ax\right )}{ \left ( 8\,{x}^{4}{a}^{4}-16\,{a}^{2}{x}^{2}+8 \right ){c}^{3}}}+{\frac{11}{24\,a \left ({x}^{4}{a}^{4}-2\,{a}^{2}{x}^{2}+1 \right ){c}^{3}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{3\,{\rm arccosh} \left (ax\right )}{8\,a{c}^{3}}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{3}{8\,a{c}^{3}}{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{3\,{\rm arccosh} \left (ax\right )}{8\,a{c}^{3}}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{3}{8\,a{c}^{3}}{\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)^3,x)

[Out]

-3/8*a^2/(a^4*x^4-2*a^2*x^2+1)/c^3*x^3*arccosh(a*x)-3/8*a/(a^4*x^4-2*a^2*x^2+1)/c^3*(a*x+1)^(1/2)*(a*x-1)^(1/2
)*x^2+5/8/(a^4*x^4-2*a^2*x^2+1)/c^3*x*arccosh(a*x)+11/24/a/(a^4*x^4-2*a^2*x^2+1)/c^3*(a*x-1)^(1/2)*(a*x+1)^(1/
2)+3/8/a/c^3*arccosh(a*x)*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/8*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2)
)/a/c^3-3/8/a/c^3*arccosh(a*x)*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/8*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(
1/2))/a/c^3

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{10 \, a^{3} x^{3} + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 14 \, a x + 4 \,{\left (6 \, a^{3} x^{3} - 10 \, a x - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 2 \, 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 ) - 7 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )}{64 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} + \frac{3 \,{\left (\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 )\right )}}{16 \, a c^{3}} - \frac{7 \, \log \left (a x + 1\right )}{64 \, a c^{3}} + \int -\frac{6 \, a^{3} x^{3} - 10 \, a x - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )}{16 \,{\left (a^{7} c^{3} x^{7} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{3} c^{3} x^{3} - a c^{3} x +{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\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)^3,x, algorithm="maxima")

[Out]

-1/64*(10*a^3*x^3 + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)^2 + 6*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1)*log(
a*x - 1) - 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1)^2 - 14*a*x + 4*(6*a^3*x^3 - 10*a*x - 3*(a^4*x^4 - 2*a^2*x^
2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)) - 7*(a^
4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) + 3/16*(log(a*x - 1)*log(1/2*a*x +
1/2) + dilog(-1/2*a*x + 1/2))/(a*c^3) - 7/64*log(a*x + 1)/(a*c^3) + integrate(-1/16*(6*a^3*x^3 - 10*a*x - 3*(a
^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) + 3*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x - 1))/(a^7*c^3*x^7 - 3*a^5*c^3*x^5
+ 3*a^3*c^3*x^3 - a*c^3*x + (a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)*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^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral(-arccosh(a*x)/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\operatorname{acosh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)/(-a**2*c*x**2+c)**3,x)

[Out]

-Integral(acosh(a*x)/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)/c**3

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

integrate(-arccosh(a*x)/(a^2*c*x^2 - c)^3, x)