3.54 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^4 (d-c^2 d x^2)^3} \, dx\)

Optimal. Leaf size=310 \[ \frac{35 b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{35 b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac{29 b c^3}{24 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{19 b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d^3}+\frac{b c}{6 d^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2}} \]

[Out]

-(b*c^3)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (b*c)/(6*d^3*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (29*
b*c^3)/(24*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a +
b*ArcCosh[c*x]))/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcCosh[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*c^
4*x*(a + b*ArcCosh[c*x]))/(8*d^3*(1 - c^2*x^2)) + (19*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^3) + (3
5*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*d^3) + (35*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(8*d^3) -
(35*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(8*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.387042, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {5746, 103, 12, 104, 21, 92, 205, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac{35 b c^3 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{35 b c^3 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac{29 b c^3}{24 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{19 b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d^3}+\frac{b c}{6 d^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]

[Out]

-(b*c^3)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (b*c)/(6*d^3*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (29*
b*c^3)/(24*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a +
b*ArcCosh[c*x]))/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcCosh[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*c^
4*x*(a + b*ArcCosh[c*x]))/(8*d^3*(1 - c^2*x^2)) + (19*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^3) + (3
5*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*d^3) + (35*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(8*d^3) -
(35*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(8*d^3)

Rule 5746

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

Rule 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

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

Rule 104

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] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 92

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 5689

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

Rule 74

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 5694

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac{1}{3} \left (7 c^2\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{1}{x^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{3 d^3}\\ &=\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{1}{3} \left (35 c^4\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{5 c^2}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{6 d^3}+\frac{\left (7 b c^3\right ) \int \frac{1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{3 d^3}\\ &=-\frac{7 b c^3}{9 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac{\left (7 b c^2\right ) \int \frac{3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{9 d^3}+\frac{\left (5 b c^3\right ) \int \frac{1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{6 d^3}-\frac{\left (35 b c^5\right ) \int \frac{x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{12 d^3}+\frac{\left (35 c^4\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac{b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac{\left (5 b c^2\right ) \int \frac{3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{18 d^3}-\frac{\left (7 b c^3\right ) \int \frac{1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^3}+\frac{\left (35 b c^5\right ) \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3}+\frac{\left (35 c^4\right ) \int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac{b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{49 b c^3}{24 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{\left (7 b c^2\right ) \int \frac{c+c^2 x}{x \sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{3 d^3}-\frac{\left (35 c^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}-\frac{\left (5 b c^3\right ) \int \frac{1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^3}\\ &=-\frac{b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{29 b c^3}{24 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{\left (5 b c^2\right ) \int \frac{c+c^2 x}{x \sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{6 d^3}+\frac{\left (7 b c^3\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 d^3}+\frac{\left (35 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}-\frac{\left (35 b c^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}\\ &=-\frac{b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{29 b c^3}{24 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{\left (5 b c^3\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{6 d^3}+\frac{\left (35 b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{\left (35 b c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{\left (7 b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{3 d^3}\\ &=-\frac{b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{29 b c^3}{24 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{7 b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{3 d^3}+\frac{35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 b c^3 \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{35 b c^3 \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac{\left (5 b c^4\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d^3}\\ &=-\frac{b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac{b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac{29 b c^3}{24 d^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac{7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac{35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac{19 b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d^3}+\frac{35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac{35 b c^3 \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac{35 b c^3 \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}\\ \end{align*}

Mathematica [A]  time = 1.96871, size = 471, normalized size = 1.52 \[ \frac{-\frac{105}{2} b c^3 \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )-4 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )\right )+\frac{105}{2} b c^3 \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (1-e^{\cosh ^{-1}(c x)}\right )\right )-4 \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )\right )-\frac{66 a c^4 x}{c^2 x^2-1}+\frac{12 a c^4 x}{\left (c^2 x^2-1\right )^2}-\frac{144 a c^2}{x}-105 a c^3 \log (1-c x)+105 a c^3 \log (c x+1)-\frac{16 a}{x^3}+144 b c^2 \left (\frac{c \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\cosh ^{-1}(c x)}{x}\right )+\frac{8 b \left (\frac{c x \left (c^2 x^2+c^2 x^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )-1\right )}{\sqrt{c x-1} \sqrt{c x+1}}-2 \cosh ^{-1}(c x)\right )}{x^3}-\frac{b c^3 \left ((c x-2) \sqrt{c x-1} \sqrt{c x+1}-3 \cosh ^{-1}(c x)\right )}{(c x-1)^2}+\frac{b c^3 \left (\sqrt{c x-1} \sqrt{c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right )}{(c x+1)^2}+33 b c^3 \left (\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right )+33 b c^3 \left (\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right )}{48 d^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]

[Out]

((-16*a)/x^3 - (144*a*c^2)/x + (12*a*c^4*x)/(-1 + c^2*x^2)^2 - (66*a*c^4*x)/(-1 + c^2*x^2) - (b*c^3*((-2 + c*x
)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x]))/(-1 + c*x)^2 + (b*c^3*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x
) - 3*ArcCosh[c*x]))/(1 + c*x)^2 + 33*b*c^3*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) + 33*b*
c^3*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) + 144*b*c^2*(-(ArcCosh[c*x]/x) + (c*Sqrt[-1 + c^2*x^
2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (8*b*(-2*ArcCosh[c*x] + (c*x*(-1 + c^2*x^2 +
c^2*x^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/x^3 - 105*a*c^3*Log[1
- c*x] + 105*a*c^3*Log[1 + c*x] - (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4*Pol
yLog[2, -E^ArcCosh[c*x]]))/2 + (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog
[2, E^ArcCosh[c*x]]))/2)/(48*d^3)

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Maple [A]  time = 0.237, size = 504, normalized size = 1.6 \begin{align*}{\frac{{c}^{3}a}{16\,{d}^{3} \left ( cx-1 \right ) ^{2}}}-{\frac{11\,{c}^{3}a}{16\,{d}^{3} \left ( cx-1 \right ) }}-{\frac{35\,{c}^{3}a\ln \left ( cx-1 \right ) }{16\,{d}^{3}}}-{\frac{a}{3\,{d}^{3}{x}^{3}}}-3\,{\frac{{c}^{2}a}{{d}^{3}x}}-{\frac{{c}^{3}a}{16\,{d}^{3} \left ( cx+1 \right ) ^{2}}}-{\frac{11\,{c}^{3}a}{16\,{d}^{3} \left ( cx+1 \right ) }}+{\frac{35\,{c}^{3}a\ln \left ( cx+1 \right ) }{16\,{d}^{3}}}-{\frac{35\,{c}^{6}b{\rm arccosh} \left (cx\right ){x}^{3}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}-{\frac{29\,{c}^{5}b{x}^{2}}{24\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}+{\frac{175\,{c}^{4}b{\rm arccosh} \left (cx\right )x}{24\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{9\,b{c}^{3}}{8\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{7\,{c}^{2}b{\rm arccosh} \left (cx\right )}{3\,{d}^{3}x \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ) }}+{\frac{bc}{6\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ){x}^{2}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )}{3\,{d}^{3} \left ({c}^{4}{x}^{4}-2\,{c}^{2}{x}^{2}+1 \right ){x}^{3}}}+{\frac{19\,b{c}^{3}}{3\,{d}^{3}}\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{35\,b{c}^{3}}{8\,{d}^{3}}{\it dilog} \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{35\,b{c}^{3}}{8\,{d}^{3}}{\it dilog} \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{35\,b{c}^{3}{\rm arccosh} \left (cx\right )}{8\,{d}^{3}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x)

[Out]

1/16*c^3*a/d^3/(c*x-1)^2-11/16*c^3*a/d^3/(c*x-1)-35/16*c^3*a/d^3*ln(c*x-1)-1/3*a/d^3/x^3-3*c^2*a/d^3/x-1/16*c^
3*a/d^3/(c*x+1)^2-11/16*c^3*a/d^3/(c*x+1)+35/16*c^3*a/d^3*ln(c*x+1)-35/8*c^6*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcco
sh(c*x)*x^3-29/24*c^5*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+175/24*c^4*b/d^3/(c^4*x^4-2*
c^2*x^2+1)*arccosh(c*x)*x+9/8*c^3*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)-7/3*c^2*b/d^3/x/(c^4
*x^4-2*c^2*x^2+1)*arccosh(c*x)+1/6*c*b/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)-1/3*b/d^3/(c^
4*x^4-2*c^2*x^2+1)/x^3*arccosh(c*x)+19/3*c^3*b/d^3*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+35/8*c^3*b/d^3*dilo
g(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+35/8*c^3*b/d^3*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+35/8*c^3*b/d^3*arcc
osh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")

[Out]

1/6144*(1935360*c^9*integrate(1/96*x^7*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x
) - 1680*c^8*(2*(5*c^2*x^3 - 3*x)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3*log(c*x + 1)/(c^5*d^3) - 3*log(c
*x - 1)/(c^5*d^3)) - 645120*c^8*integrate(1/96*x^6*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 -
d^3*x^2), x) + 630*(c*(2*(5*c^2*x^2 + 3*c*x - 6)/(c^8*d^3*x^3 - c^7*d^3*x^2 - c^6*d^3*x + c^5*d^3) - 5*log(c*
x + 1)/(c^5*d^3) + 5*log(c*x - 1)/(c^5*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1)/(c^8*d^3*x^4 - 2*c^6*d^3*x^2 +
c^4*d^3))*c^7 + 2800*c^6*(2*(c^2*x^3 + x)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x + 1)/(c^3*d^3) + l
og(c*x - 1)/(c^3*d^3)) + 1290240*c^6*integrate(1/96*x^4*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*
x^4 - d^3*x^2), x) + 315*(c*(2*(3*c^2*x^2 - 3*c*x - 2)/(c^6*d^3*x^3 - c^5*d^3*x^2 - c^4*d^3*x + c^3*d^3) - 3*l
og(c*x + 1)/(c^3*d^3) + 3*log(c*x - 1)/(c^3*d^3)) - 16*log(c*x - 1)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3))*c
^5 + 896*c^4*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log(c*x + 1)/(c*d^3) + 3*log(c*x - 1
)/(c*d^3)) - 645120*c^4*integrate(1/96*x^2*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2
), x) + 128*c^2*(2*(15*c^4*x^4 - 25*c^2*x^2 + 8)/(c^4*d^3*x^5 - 2*c^2*d^3*x^3 + d^3*x) - 15*c*log(c*x + 1)/d^3
+ 15*c*log(c*x - 1)/d^3) - 32*(105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*log(c*x + 1)^2 + 210*(c^7*x^7 - 2*c^5*x^5
+ c^3*x^3)*log(c*x + 1)*log(c*x - 1) + 4*(210*c^6*x^6 - 350*c^4*x^4 + 112*c^2*x^2 - 105*(c^7*x^7 - 2*c^5*x^5 +
c^3*x^3)*log(c*x + 1) + 105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*log(c*x - 1) + 16)*log(c*x + sqrt(c*x + 1)*sqrt(c
*x - 1)))/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3) + 6144*integrate(-1/48*(210*c^7*x^6 - 350*c^5*x^4 + 112*c^3*
x^2 - 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(c*x + 1) + 105*(c^8*x^7 - 2*c^6*x^5 + c^4*x^3)*log(c*x - 1) + 16
*c)/(c^7*d^3*x^10 - 3*c^5*d^3*x^8 + 3*c^3*d^3*x^6 - c*d^3*x^4 + (c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 -
d^3*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x))*b + 1/48*a*(105*c^3*log(c*x + 1)/d^3 - 105*c^3*log(c*x - 1)/d^3 -
2*(105*c^6*x^6 - 175*c^4*x^4 + 56*c^2*x^2 + 8)/(c^4*d^3*x^7 - 2*c^2*d^3*x^5 + d^3*x^3))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{arcosh}\left (c x\right ) + a}{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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")

[Out]

integral(-(b*arccosh(c*x) + a)/(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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="giac")

[Out]

integrate(-(b*arccosh(c*x) + a)/((c^2*d*x^2 - d)^3*x^4), x)