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

Optimal. Leaf size=163 \[ \frac{\cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{a x-1} \sqrt{a x+1}}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.287199, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5689, 5718, 207, 5694, 4182, 2531, 2282, 6589} \[ \frac{\cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{a x-1} \sqrt{a x+1}}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcCosh[a*x]/(a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (x*ArcCosh[a*x]^2)/(2*c^2*(1 - a^2*x^2)) + (ArcCosh[a*x
]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - ArcTanh[a*x]/(a*c^2) + (ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c^
2) - (ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c^2) - PolyLog[3, -E^ArcCosh[a*x]]/(a*c^2) + PolyLog[3, E^Ar
cCosh[a*x]]/(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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 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}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac{a \int \frac{x \cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c^2}+\frac{\int \frac{\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{c^2}-\frac{\operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac{\operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac{\operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ &=-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac{\text{Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\text{Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ \end{align*}

Mathematica [A]  time = 0.918435, size = 191, normalized size = 1.17 \[ \frac{-8 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )+8 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )-8 \text{PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )+8 \text{PolyLog}\left (3,e^{-\cosh ^{-1}(a x)}\right )-4 \cosh ^{-1}(a x)^2 \log \left (1-e^{-\cosh ^{-1}(a x)}\right )+4 \cosh ^{-1}(a x)^2 \log \left (e^{-\cosh ^{-1}(a x)}+1\right )+4 \cosh ^{-1}(a x) \tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )-4 \cosh ^{-1}(a x) \coth \left (\frac{1}{2} \cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^2 \left (-\text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )\right )-\cosh ^{-1}(a x)^2 \text{sech}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+8 \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )\right )}{8 a c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*ArcCosh[a*x]*Coth[ArcCosh[a*x]/2] - ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^2 - 4*ArcCosh[a*x]^2*Log[1 - E^(-A
rcCosh[a*x])] + 4*ArcCosh[a*x]^2*Log[1 + E^(-ArcCosh[a*x])] + 8*Log[Tanh[ArcCosh[a*x]/2]] - 8*ArcCosh[a*x]*Pol
yLog[2, -E^(-ArcCosh[a*x])] + 8*ArcCosh[a*x]*PolyLog[2, E^(-ArcCosh[a*x])] - 8*PolyLog[3, -E^(-ArcCosh[a*x])]
+ 8*PolyLog[3, E^(-ArcCosh[a*x])] - ArcCosh[a*x]^2*Sech[ArcCosh[a*x]/2]^2 + 4*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2
])/(8*a*c^2)

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Maple [A]  time = 0.084, size = 288, normalized size = 1.8 \begin{align*} -{\frac{x \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){c}^{2}}}-{\frac{{\rm arccosh} \left (ax\right )}{a \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,a{c}^{2}}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{{\rm arccosh} \left (ax\right )}{a{c}^{2}}{\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{1}{a{c}^{2}}{\it polylog} \left ( 3,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{2\,a{c}^{2}}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{{\rm arccosh} \left (ax\right )}{a{c}^{2}}{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{1}{a{c}^{2}}{\it polylog} \left ( 3,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-2\,{\frac{{\it Artanh} \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

integral(arccosh(a*x)^2/(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}^{2}{\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)**2/(-a**2*c*x**2+c)**2,x)

[Out]

Integral(acosh(a*x)**2/(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 )^{2}}{{\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)^2/(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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