\(\int \frac {\text {arccosh}(a x)^3}{(c-a^2 c x^2)^3} \, dx\) [245]

Optimal result

Integrand size = 20, antiderivative size = 387 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^3}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}+\frac {5 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \]

[Out]

-1/4*x*arccosh(a*x)/c^3/(-a^2*x^2+1)+1/4*arccosh(a*x)^2/a/c^3/(a*x-1)^(3/2)/(a*x+1)^(3/2)+1/4*x*arccosh(a*x)^3
/c^3/(-a^2*x^2+1)^2+3/8*x*arccosh(a*x)^3/c^3/(-a^2*x^2+1)-5*arccosh(a*x)*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/
2))/a/c^3+3/4*arccosh(a*x)^3*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-5/2*polylog(2,-a*x-(a*x-1)^(1/2)*(
a*x+1)^(1/2))/a/c^3+9/8*arccosh(a*x)^2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+5/2*polylog(2,a*x+(a*
x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-9/8*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-9/4*arccos
h(a*x)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+9/4*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^
(1/2))/a/c^3+9/4*polylog(4,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-9/4*polylog(4,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/
2))/a/c^3+1/4/a/c^3/(a*x-1)^(1/2)/(a*x+1)^(1/2)-9/8*arccosh(a*x)^2/a/c^3/(a*x-1)^(1/2)/(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5901, 5903, 4267, 2611, 6744, 2320, 6724, 5915, 5889, 2317, 2438, 75} \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^3}+\frac {5 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^3}+\frac {9 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac {1}{4 a c^3 \sqrt {a x-1} \sqrt {a x+1}} \]

[In]

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

[Out]

1/(4*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (x*ArcCosh[a*x])/(4*c^3*(1 - a^2*x^2)) + ArcCosh[a*x]^2/(4*a*c^3*(-
1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - (9*ArcCosh[a*x]^2)/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^
3)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x]^3)/(8*c^3*(1 - a^2*x^2)) - (5*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*
x]])/(a*c^3) + (3*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) - (5*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^3
) + (9*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) + (5*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^3) - (9*A
rcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3) - (9*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(4*a*c^3) +
(9*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(4*a*c^3) + (9*PolyLog[4, -E^ArcCosh[a*x]])/(4*a*c^3) - (9*PolyLo
g[4, E^ArcCosh[a*x]])/(4*a*c^3)

Rule 75

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 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}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {(3 a) \int \frac {x \text {arccosh}(a x)^2}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c} \\ & = \frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arccosh}(a x)}{(-1+a x)^2 (1+a x)^2} \, dx}{2 c^3}+\frac {(9 a) \int \frac {x \text {arccosh}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{8 c^3}+\frac {3 \int \frac {\text {arccosh}(a x)^3}{c-a^2 c x^2} \, dx}{8 c^2} \\ & = \frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {\int \frac {\text {arccosh}(a x)}{\left (-1+a^2 x^2\right )^2} \, dx}{2 c^3}+\frac {9 \int \frac {\text {arccosh}(a x)}{(-1+a x) (1+a x)} \, dx}{4 c^3}-\frac {3 \text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\text {arccosh}(a x)\right )}{8 a c^3} \\ & = -\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {\int \frac {\text {arccosh}(a x)}{-1+a^2 x^2} \, dx}{4 c^3}+\frac {9 \int \frac {\text {arccosh}(a x)}{-1+a^2 x^2} \, dx}{4 c^3}+\frac {9 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{8 a c^3}-\frac {9 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{8 a c^3}-\frac {a \int \frac {x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{4 c^3} \\ & = \frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}+\frac {\text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arccosh}(a x))}{4 a c^3}+\frac {9 \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arccosh}(a x))}{4 a c^3}-\frac {9 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3} \\ & = \frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^3}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {\text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}+\frac {\text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3} \\ & = \frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^3}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \\ & = \frac {1}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {x \text {arccosh}(a x)}{4 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)^2}{4 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {9 \text {arccosh}(a x)^2}{8 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 \text {arccosh}(a x) \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c^3}+\frac {3 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{2 a c^3}+\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{8 a c^3}+\frac {5 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{2 a c^3}-\frac {9 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{8 a c^3}-\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {9 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {9 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 6.83 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.18 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {3 \pi ^4-6 \text {arccosh}(a x)^4-8 \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )+40 \text {arccosh}(a x)^2 \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )-4 \text {arccosh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+6 \text {arccosh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )-\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)^2 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-\text {arccosh}(a x)^3 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-160 \text {arccosh}(a x) \log \left (1-e^{-\text {arccosh}(a x)}\right )+160 \text {arccosh}(a x) \log \left (1+e^{-\text {arccosh}(a x)}\right )-24 \text {arccosh}(a x)^3 \log \left (1+e^{-\text {arccosh}(a x)}\right )+24 \text {arccosh}(a x)^3 \log \left (1-e^{\text {arccosh}(a x)}\right )+8 \left (-20+9 \text {arccosh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(a x)}\right )+160 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(a x)}\right )+72 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )+144 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(a x)}\right )-144 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )+144 \operatorname {PolyLog}\left (4,-e^{-\text {arccosh}(a x)}\right )+144 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )-4 \text {arccosh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+6 \text {arccosh}(a x)^3 \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+\text {arccosh}(a x)^3 \text {sech}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-\frac {16 \text {arccosh}(a x)^2 \sinh ^4\left (\frac {1}{2} \text {arccosh}(a x)\right )}{\left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3}+8 \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )-40 \text {arccosh}(a x)^2 \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )}{64 a c^3} \]

[In]

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

[Out]

-1/64*(3*Pi^4 - 6*ArcCosh[a*x]^4 - 8*Coth[ArcCosh[a*x]/2] + 40*ArcCosh[a*x]^2*Coth[ArcCosh[a*x]/2] - 4*ArcCosh
[a*x]*Csch[ArcCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 - Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*
ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^4 - 160*ArcCosh[a*x]*Log[1 - E^(-A
rcCosh[a*x])] + 160*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^3*Log[1 + E^(-ArcCosh[a*x])] + 2
4*ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] + 8*(-20 + 9*ArcCosh[a*x]^2)*PolyLog[2, -E^(-ArcCosh[a*x])] + 160*Pol
yLog[2, E^(-ArcCosh[a*x])] + 72*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] + 144*ArcCosh[a*x]*PolyLog[3, -E^(-A
rcCosh[a*x])] - 144*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] + 144*PolyLog[4, -E^(-ArcCosh[a*x])] + 144*PolyLog
[4, E^ArcCosh[a*x]] - 4*ArcCosh[a*x]*Sech[ArcCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + ArcCos
h[a*x]^3*Sech[ArcCosh[a*x]/2]^4 - (16*ArcCosh[a*x]^2*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*x))^(3/2)*(1
+ a*x)^3) + 8*Tanh[ArcCosh[a*x]/2] - 40*ArcCosh[a*x]^2*Tanh[ArcCosh[a*x]/2])/(a*c^3)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}+9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5 a x \operatorname {arccosh}\left (a x \right )^{3}-11 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a x \,\operatorname {arccosh}\left (a x \right )+2 \sqrt {a x -1}\, \sqrt {a x +1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}-\frac {5 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {9 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {9 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\)	\(527\)
default	\(\frac {-\frac {3 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}+9 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5 a x \operatorname {arccosh}\left (a x \right )^{3}-11 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-2 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 a x \,\operatorname {arccosh}\left (a x \right )+2 \sqrt {a x -1}\, \sqrt {a x +1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}-\frac {5 \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {5 \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {9 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {9 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {9 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {9 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\)	\(527\)

[In]

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

[Out]

1/a*(-1/8*(3*a^3*x^3*arccosh(a*x)^3+9*a^2*x^2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-5*a*x*arccosh(a*x)^3-
11*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-2*a^3*x^3*arccosh(a*x)-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*a
*x*arccosh(a*x)+2*(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a^4*x^4-2*a^2*x^2+1)/c^3-5/2/c^3*arccosh(a*x)*ln(1+a*x+(a*x-1)
^(1/2)*(a*x+1)^(1/2))-5/2/c^3*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+5/2/c^3*arccosh(a*x)*ln(1-a*x-(a*x-1
)^(1/2)*(a*x+1)^(1/2))+5/2/c^3*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*arccosh(a*x)^3*ln(1+a*x+(a*x
-1)^(1/2)*(a*x+1)^(1/2))+9/8/c^3*arccosh(a*x)^2*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-9/4/c^3*arccosh(a*
x)*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+9/4/c^3*polylog(4,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/8/c^3*arc
cosh(a*x)^3*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-9/8/c^3*arccosh(a*x)^2*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(
1/2))+9/4/c^3*arccosh(a*x)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-9/4/c^3*polylog(4,a*x+(a*x-1)^(1/2)*(a*x
+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral(-arccosh(a*x)^3/(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]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx=- \frac {\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-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
))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) - integrate(-3/16*(6*a^5*x^5
- 16*a^3*x^3 + (6*a^4*x^4 - 10*a^2*x^2 - 3*(a^5*x^5 - 2*a^3*x^3 + a*x)*log(a*x + 1) + 3*(a^5*x^5 - 2*a^3*x^3
+ a*x)*log(a*x - 1))*sqrt(a*x + 1)*sqrt(a*x - 1) + 10*a*x - 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x +
1) + 3*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(acosh(a*x)^3/(c - a^2*c*x^2)^3,x)

[Out]

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