\(\int \frac {(a+b \text {arccosh}(c x))^2}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\) [223]

Optimal result

Integrand size = 29, antiderivative size = 562 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

-1/3*(a+b*arccosh(c*x))^2/d/x^3/(-c^2*d*x^2+d)^(3/2)-2*c^2*(a+b*arccosh(c*x))^2/d/x/(-c^2*d*x^2+d)^(3/2)+8/3*c
^4*x*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)+1/3*b^2*c^2/d^2/x/(-c^2*d*x^2+d)^(1/2)-2/3*b^2*c^4*x/d^2/(-c^
2*d*x^2+d)^(1/2)+16/3*c^4*x*(a+b*arccosh(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*b*c*(a+b*arccosh(c*x))*(c*x-1)^(
1/2)*(c*x+1)^(1/2)/d^2/x^2/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)+16/3*c^3*(a+b*arccosh(c*x))^2*(c*x-1)^(1/2)*(c*x+
1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-32/3*b*c^3*(a+b*arccosh(c*x))*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(
c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-32/3*b*c^3*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x
+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*b^2*c^3*polylog(2,-(c*x+(c*x-1)^(1/2)*(
c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*b^2*c^3*polylog(2,(c*x+(c*x-1)^(1/2)
*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {5932, 5901, 5899, 5913, 3797, 2221, 2317, 2438, 5912, 5914, 39, 5936, 5916, 5569, 4267, 105, 12} \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {32 b c^3 \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

Int[(a + b*ArcCosh[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(b^2*c^2)/(3*d^2*x*Sqrt[d - c^2*d*x^2]) - (2*b^2*c^4*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*x^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(3*d*x^
3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*ArcCosh[c*x])^2)/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcCos
h[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (16*c
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*b^2
*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*b^2*c^3*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5899

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Dist[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Int[x
*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ
[n, 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 5912

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

Rule 5913

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 5916

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Dist[-d^(-1), Subst[I
nt[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] &
& IGtQ[n, 0]

Rule 5932

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5936

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*f*(p + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCo
sh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &
&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\left (2 c^2\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx-\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x^3 (-1+c x)^2 (1+c x)^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\left (8 c^4\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx-\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x (-1+c x)^2 (1+c x)^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (16 c^4\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (-1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{(-1+c x)^2 (1+c x)^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {8 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 c^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {8 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \coth (x) \, dx,x,\text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {csch}(2 x) \, dx,x,\text {arccosh}(c x))}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (64 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text {arccosh}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (32 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {arccosh}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {arccosh}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 2.80 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )}{x^3 \left (-1+c^2 x^2\right )}+a b c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {1}{c^2 x^2}+\frac {1}{1-c^2 x^2}+\frac {2 \left (\frac {-1+c x}{1+c x}\right )^{3/2} \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \text {arccosh}(c x)}{c^3 x^3 (-1+c x)^3}-16 \log (c x)-16 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )+b^2 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\frac {c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\frac {\text {arccosh}(c x)}{c^2 x^2}+\frac {\text {arccosh}(c x)}{1-c^2 x^2}-16 \text {arccosh}(c x)^2-\frac {c x \text {arccosh}(c x)^2}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}+\frac {8 c x \text {arccosh}(c x)^2}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2}{c^3 x^3}+\frac {8 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2}{c x}-16 \text {arccosh}(c x) \log \left (1-e^{-2 \text {arccosh}(c x)}\right )-16 \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+8 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+8 \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/(x^3*(-1 + c^2*x^2)) + a*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1
+ c*x)*(1/(c^2*x^2) + (1 - c^2*x^2)^(-1) + (2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^
6*x^6)*ArcCosh[c*x])/(c^3*x^3*(-1 + c*x)^3) - 16*Log[c*x] - 16*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]) + b^
2*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (Sqrt[(-1 + c*x)/(1 +
c*x)]*(1 + c*x))/(c*x) + ArcCosh[c*x]/(c^2*x^2) + ArcCosh[c*x]/(1 - c^2*x^2) - 16*ArcCosh[c*x]^2 - (c*x*ArcCo
sh[c*x]^2)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) + (8*c*x*ArcCosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1
+ c*x)) + (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c^3*x^3) + (8*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 +
c*x)*ArcCosh[c*x]^2)/(c*x) - 16*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 16*ArcCosh[c*x]*Log[1 + E^(-2*Arc
Cosh[c*x])] + 8*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 8*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*d^2*Sqrt[d - c^2*d*x
^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3744\) vs. \(2(544)=1088\).

Time = 1.39 (sec) , antiderivative size = 3745, normalized size of antiderivative = 6.66

method	result	size
default	\(\text {Expression too large to display}\)	\(3745\)
parts	\(\text {Expression too large to display}\)	\(3745\)

[In]

int((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(-1/3/d/x^3/(-c^2*d*x^2+d)^(3/2)+2*c^2*(-1/d/x/(-c^2*d*x^2+d)^(3/2)+4*c^2*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/
3/d^2*x/(-c^2*d*x^2+d)^(1/2))))+128/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*
x^2-1)*x^5*(c*x-1)*(c*x+1)*c^8-8*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1
)*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^9+256/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-1
0*c^2*x^2-1)*x^9*(c*x-1)*(c*x+1)*arccosh(c*x)*c^12-640/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6
+35*c^4*x^4-10*c^2*x^2-1)*x^7*(c*x-1)*(c*x+1)*arccosh(c*x)*c^10+64*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-
36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*c^9+160*b^2*(-d*(c^2*x^2-1)
)^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*(c*x-1)*(c*x+1)*arccosh(c*x)*c^8-128*b^2*(-d*(
c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(
c*x)^2*c^7-80/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*(c*x-1)*(c*
x+1)*arccosh(c*x)*c^6+176/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^2
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2*c^5+8/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3
/(c^2*x^2-1)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^3+16/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)
*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^3+64/3*b^2*(-d*(c^2*x^2-1))^(1/2)/
d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^9*(c*x-1)*(c*x+1)*c^12-160/3*b^2*(-d*(c^2*x^2-1))^(1/2)/
d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*(c*x-1)*(c*x+1)*c^10+16*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3
/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^7-22/3*b^2*(-d*(c^2*x^2-1))
^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5-32/3*b^2*(-d*(c
^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)^2*c^3+16/3*b^2*(-d*(c^2*x^2-1))^(1/2
)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^3+16/3*b^2*(-d*(c^
2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^2
*c^3-4*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*(c*x-1)^(1/2)*(c*x+1)^(1
/2)*arccosh(c*x)*c^3-64/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^11*
c^14+224/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^9*c^12-88*b^2*(-d*
(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*c^10+100/3*b^2*(-d*(c^2*x^2-1))^(1/
2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*c^8+14/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8
-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*c^6-3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^
4-10*c^2*x^2-1)*x*c^4-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/x*c^2
+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/x^3*arccosh(c*x)^2+16/3*b^
2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x
+1)^(1/2))^2)*c^3-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/x^2*(c*x-
1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c+16/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2
-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^3+16/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+
1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^3-16/3*b^2*(-d*(c^2*x^2-1))^(1/2
)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x*(c*x-1)*(c*x+1)*arccosh(c*x)*c^4+4*b^2*(-d*(c^2*x^2-1)
)^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^5-3
2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*(c*x-1)*(c*x+1)*c^6+1/3
*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(16*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^7*c^7-
32*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^6*x^6-32*arccosh(c*x)*c^7*x^7-32*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(
1/2))^4-1)*x^5*c^5+48*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^4*x^4+64*arccosh(c*x)*c^5*x^5+16*ln((c*x+(c*x
-1)^(1/2)*(c*x+1)^(1/2))^4-1)*x^3*c^3-12*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2-32*c^3*x^3*arccosh(c
*x)+c^3*x^3-2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-c*x)/d^3/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/x^3-256/3*b^2*
(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^11*arccosh(c*x)*c^14+896/3*b^2*(-
d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^9*arccosh(c*x)*c^12-64*b^2*(-d*(c^2
*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*arccosh(c*x)^2*c^10-1120/3*b^2*(-d*(c^2
*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^7*arccosh(c*x)*c^10+160*b^2*(-d*(c^2*x^2-
1))^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*arccosh(c*x)^2*c^8+560/3*b^2*(-d*(c^2*x^2-1)
)^(1/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^5*arccosh(c*x)*c^8-344/3*b^2*(-d*(c^2*x^2-1))^(1
/2)/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*arccosh(c*x)^2*c^6-64/3*b^2*(-d*(c^2*x^2-1))^(1/2)
/d^3/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x^3*arccosh(c*x)*c^6+12*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(1
2*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x*arccosh(c*x)^2*c^4-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8
*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*x*arccosh(c*x)*c^4-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-36*
c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^3+6*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(12*c^8*x^8-
36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/x*arccosh(c*x)^2*c^2

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 +
3*c^2*d^3*x^6 - d^3*x^4), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*acosh(c*x))**2/x**4/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*a*b*c*(8*c^2*sqrt(-d)*log(c*x + 1)/d^3 + 8*c^2*sqrt(-d)*log(c*x - 1)/d^3 + 16*c^2*sqrt(-d)*log(x)/d^3 + sq
rt(-d)/(c^2*d^3*x^4 - d^3*x^2)) + 2/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d
) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*a*b*arccosh(c*x) + 1/3*(16*c^4*x/(s
qrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*
d*x^2 + d)^(3/2)*d*x^3))*a^2 + b^2*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/((-c^2*d*x^2 + d)^(5/2)*
x^4), x)

Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*x^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[In]

int((a + b*acosh(c*x))^2/(x^4*(d - c^2*d*x^2)^(5/2)),x)

[Out]

int((a + b*acosh(c*x))^2/(x^4*(d - c^2*d*x^2)^(5/2)), x)