\(\int \frac {\text {arccosh}(a x)^3}{(c-a^2 c x^2)^2} \, dx\) [244]

Optimal result

Integrand size = 20, antiderivative size = 260 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c^2} \]

[Out]

1/2*x*arccosh(a*x)^3/c^2/(-a^2*x^2+1)-6*arccosh(a*x)*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+arccosh(a*
x)^3*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+3/2*ar
ccosh(a*x)^2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+3*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/
c^2-3/2*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3*arccosh(a*x)*polylog(3,-a*x-(a*x-1)^
(1/2)*(a*x+1)^(1/2))/a/c^2+3*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2+3*polylog(4,-a*x-(a
*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3*polylog(4,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^2-3/2*arccosh(a*x)^2/a/c^2/(
a*x-1)^(1/2)/(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5901, 5903, 4267, 2611, 6744, 2320, 6724, 5915, 5889, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {6 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}} \]

[In]

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

[Out]

(-3*ArcCosh[a*x]^2)/(2*a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^3)/(2*c^2*(1 - a^2*x^2)) - (6*Arc
Cosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) + (ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - (3*PolyLog[2,
-E^ArcCosh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^2) + (3*PolyLog[2, E^ArcCosh
[a*x]])/(a*c^2) - (3*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^2) - (3*ArcCosh[a*x]*PolyLog[3, -E^ArcC
osh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(a*c^2) + (3*PolyLog[4, -E^ArcCosh[a*x]])/(a*
c^2) - (3*PolyLog[4, E^ArcCosh[a*x]])/(a*c^2)

Rule 2317

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 2320

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 2438

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]

Rule 2611

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 4267

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 5889

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2
*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5901

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[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(
1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 5903

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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 6724

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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {(3 a) \int \frac {x \text {arccosh}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{2 c^2}+\frac {\int \frac {\text {arccosh}(a x)^3}{c-a^2 c x^2} \, dx}{2 c} \\ & = -\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {3 \int \frac {\text {arccosh}(a x)}{(-1+a x) (1+a x)} \, dx}{c^2}-\frac {\text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\text {arccosh}(a x)\right )}{2 a c^2} \\ & = -\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \int \frac {\text {arccosh}(a x)}{-1+a^2 x^2} \, dx}{c^2}+\frac {3 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 a c^2}-\frac {3 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{2 a c^2} \\ & = -\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2}+\frac {3 \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arccosh}(a x))}{a c^2}-\frac {3 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2}+\frac {3 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2} \\ & = -\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2} \\ & = -\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c^2} \\ & = -\frac {3 \text {arccosh}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {\text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^2}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c^2}+\frac {3 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{a c^2}-\frac {3 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.06 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {-\pi ^4+2 \text {arccosh}(a x)^4-12 \text {arccosh}(a x)^2 \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )-2 \text {arccosh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+48 \text {arccosh}(a x) \log \left (1-e^{-\text {arccosh}(a x)}\right )-48 \text {arccosh}(a x) \log \left (1+e^{-\text {arccosh}(a x)}\right )+8 \text {arccosh}(a x)^3 \log \left (1+e^{-\text {arccosh}(a x)}\right )-8 \text {arccosh}(a x)^3 \log \left (1-e^{\text {arccosh}(a x)}\right )-24 \left (-2+\text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(a x)}\right )-48 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(a x)}\right )-24 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )-48 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(a x)}\right )+48 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-48 \operatorname {PolyLog}\left (4,-e^{-\text {arccosh}(a x)}\right )-48 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x)^3 \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+12 \text {arccosh}(a x)^2 \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )}{16 a c^2} \]

[In]

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

[Out]

(-Pi^4 + 2*ArcCosh[a*x]^4 - 12*ArcCosh[a*x]^2*Coth[ArcCosh[a*x]/2] - 2*ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 +
48*ArcCosh[a*x]*Log[1 - E^(-ArcCosh[a*x])] - 48*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] + 8*ArcCosh[a*x]^3*Lo
g[1 + E^(-ArcCosh[a*x])] - 8*ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] - 24*(-2 + ArcCosh[a*x]^2)*PolyLog[2, -E^(
-ArcCosh[a*x])] - 48*PolyLog[2, E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] - 48*ArcCosh
[a*x]*PolyLog[3, -E^(-ArcCosh[a*x])] + 48*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] - 48*PolyLog[4, -E^(-ArcCosh
[a*x])] - 48*PolyLog[4, E^ArcCosh[a*x]] - 2*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + 12*ArcCosh[a*x]^2*Tanh[Arc
Cosh[a*x]/2])/(16*a*c^2)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccosh}\left (a x \right )^{2} \left (a x \,\operatorname {arccosh}\left (a x \right )+3 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}}{a}\)	\(416\)
default	\(\frac {-\frac {\operatorname {arccosh}\left (a x \right )^{2} \left (a x \,\operatorname {arccosh}\left (a x \right )+3 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 \left (a^{2} x^{2}-1\right ) c^{2}}+\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{2}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}-\frac {3 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}+\frac {3 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2}}}{a}\)	\(416\)

[In]

int(arccosh(a*x)^3/(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]

[In]

integrate(arccosh(a*x)^3/(-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))^
3/(a^3*c^2*x^2 - a*c^2) - integrate(-3/4*(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))^2/(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)

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2}} \,d x } \]

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]

[In]

int(acosh(a*x)^3/(c - a^2*c*x^2)^2,x)

[Out]

int(acosh(a*x)^3/(c - a^2*c*x^2)^2, x)