\(\int \frac {\text {arccosh}(a x)^2}{(c-a^2 c x^2)^3} \, dx\) [169]

Optimal result

Integrand size = 20, antiderivative size = 258 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \text {arctanh}(a x)}{6 a c^3}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \]

[Out]

-1/12*x/c^3/(-a^2*x^2+1)+1/6*arccosh(a*x)/a/c^3/(a*x-1)^(3/2)/(a*x+1)^(3/2)+1/4*x*arccosh(a*x)^2/c^3/(-a^2*x^2
+1)^2+3/8*x*arccosh(a*x)^2/c^3/(-a^2*x^2+1)+3/4*arccosh(a*x)^2*arctanh(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-
5/6*arctanh(a*x)/a/c^3+3/4*arccosh(a*x)*polylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/4*arccosh(a*x)*pol
ylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/4*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3+3/4*polylo
g(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c^3-3/4*arccosh(a*x)/a/c^3/(a*x-1)^(1/2)/(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5901, 5903, 4267, 2611, 2320, 6724, 5915, 35, 213, 41, 205} \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\text {arccosh}(a x)}{6 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}-\frac {5 \text {arctanh}(a x)}{6 a c^3} \]

[In]

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

[Out]

-1/12*x/(c^3*(1 - a^2*x^2)) + ArcCosh[a*x]/(6*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - (3*ArcCosh[a*x])/(4*a*
c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^2)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x]^2)/(8*c^3*(
1 - a^2*x^2)) + (3*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) - (5*ArcTanh[a*x])/(6*a*c^3) + (3*ArcCosh
[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(4*a*c^3) - (3*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(4*a*c^3) - (3*Poly
Log[3, -E^ArcCosh[a*x]])/(4*a*c^3) + (3*PolyLog[3, E^ArcCosh[a*x]])/(4*a*c^3)

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {a \int \frac {x \text {arccosh}(a x)}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{2 c^3}+\frac {3 \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c} \\ & = \frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{(-1+a x)^2 (1+a x)^2} \, dx}{6 c^3}+\frac {(3 a) \int \frac {x \text {arccosh}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\text {arccosh}(a x)^2}{c-a^2 c x^2} \, dx}{8 c^2} \\ & = \frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (-1+a^2 x^2\right )^2} \, dx}{6 c^3}+\frac {3 \int \frac {1}{(-1+a x) (1+a x)} \, dx}{4 c^3}-\frac {3 \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\text {arccosh}(a x)\right )}{8 a c^3} \\ & = -\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{12 c^3}+\frac {3 \int \frac {1}{-1+a^2 x^2} \, dx}{4 c^3}+\frac {3 \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3} \\ & = -\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \text {arctanh}(a x)}{6 a c^3}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{4 a c^3} \\ & = -\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \text {arctanh}(a x)}{6 a c^3}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \\ & = -\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\text {arccosh}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \text {arccosh}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \text {arccosh}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {arccosh}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \text {arccosh}(a x)^2 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {5 \text {arctanh}(a x)}{6 a c^3}+\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{4 a c^3}-\frac {3 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{4 a c^3}+\frac {3 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{4 a c^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.30 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.24 \[ \int \frac {\text {arccosh}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx=-\frac {80 \text {arccosh}(a x) \coth \left (\frac {1}{2} \text {arccosh}(a x)\right )+2 \left (-2+9 \text {arccosh}(a x)^2\right ) \text {csch}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x) \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-3 \text {arccosh}(a x)^2 \text {csch}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-160 \log \left (\tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )\right )+72 \left (\text {arccosh}(a x)^2 \log \left (1-e^{-\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \log \left (1+e^{-\text {arccosh}(a x)}\right )+2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(a x)}\right )-2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arccosh}(a x)}\right )\right )+2 \left (-2+9 \text {arccosh}(a x)^2\right ) \text {sech}^2\left (\frac {1}{2} \text {arccosh}(a x)\right )+3 \text {arccosh}(a x)^2 \text {sech}^4\left (\frac {1}{2} \text {arccosh}(a x)\right )-\frac {32 \text {arccosh}(a x) \sinh ^4\left (\frac {1}{2} \text {arccosh}(a x)\right )}{\left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3}-80 \text {arccosh}(a x) \tanh \left (\frac {1}{2} \text {arccosh}(a x)\right )}{192 a c^3} \]

[In]

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

[Out]

-1/192*(80*ArcCosh[a*x]*Coth[ArcCosh[a*x]/2] + 2*(-2 + 9*ArcCosh[a*x]^2)*Csch[ArcCosh[a*x]/2]^2 - 2*Sqrt[(-1 +
a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]*Csch[ArcCosh[a*x]/2]^4 - 3*ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - 160
*Log[Tanh[ArcCosh[a*x]/2]] + 72*(ArcCosh[a*x]^2*Log[1 - E^(-ArcCosh[a*x])] - ArcCosh[a*x]^2*Log[1 + E^(-ArcCos
h[a*x])] + 2*ArcCosh[a*x]*PolyLog[2, -E^(-ArcCosh[a*x])] - 2*ArcCosh[a*x]*PolyLog[2, E^(-ArcCosh[a*x])] + 2*Po
lyLog[3, -E^(-ArcCosh[a*x])] - 2*PolyLog[3, E^(-ArcCosh[a*x])]) + 2*(-2 + 9*ArcCosh[a*x]^2)*Sech[ArcCosh[a*x]/
2]^2 + 3*ArcCosh[a*x]^2*Sech[ArcCosh[a*x]/2]^4 - (32*ArcCosh[a*x]*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*
x))^(3/2)*(1 + a*x)^3) - 80*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2])/(a*c^3)

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}+18 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \operatorname {arccosh}\left (a x \right )^{2}-22 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )-2 a^{3} x^{3}+2 a x}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arctanh}\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{3 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\)	\(320\)
default	\(\frac {-\frac {9 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{2}+18 a^{2} x^{2} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-15 a x \operatorname {arccosh}\left (a x \right )^{2}-22 \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )-2 a^{3} x^{3}+2 a x}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \,\operatorname {arctanh}\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{3 c^{3}}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\)	\(320\)

[In]

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

[Out]

1/a*(-1/24*(9*a^3*x^3*arccosh(a*x)^2+18*a^2*x^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-15*a*x*arccosh(a*x)^2
-22*(a*x-1)^(1/2)*(a*x+1)^(1/2)*arccosh(a*x)-2*a^3*x^3+2*a*x)/(a^4*x^4-2*a^2*x^2+1)/c^3-5/3/c^3*arctanh(a*x+(a
*x-1)^(1/2)*(a*x+1)^(1/2))+3/8/c^3*arccosh(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+3/4/c^3*arccosh(a*x)*p
olylog(2,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/4/c^3*polylog(3,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/8/c^3*arccosh
(a*x)^2*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))-3/4/c^3*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))+
3/4/c^3*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral(-arccosh(a*x)^2/(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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(arccosh(a*x)^2/(-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))^2/(a^5*c^3*x^4 - 2*a^3*c^3*x^2 + a*c^3) - integrate(-1/8*(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))/(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(acosh(a*x)^2/(c - a^2*c*x^2)^3,x)

[Out]

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