Integrand size = 22, antiderivative size = 180 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c d^3}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c d^3} \]
1/12*b/c/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+1/4*x*(a+b*arccosh(c*x))/d^3/(-c^ 2*x^2+1)^2+3/8*x*(a+b*arccosh(c*x))/d^3/(-c^2*x^2+1)+3/4*(a+b*arccosh(c*x) )*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^3+3/8*b*polylog(2,-c*x-(c*x -1)^(1/2)*(c*x+1)^(1/2))/c/d^3-3/8*b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^( 1/2))/c/d^3-3/8*b/c/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.80 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.76 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {4 a x}{\left (-1+c^2 x^2\right )^2}-\frac {6 a x}{-1+c^2 x^2}+\frac {b \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{3 c (1+c x)^2}+\frac {b \left ((2-c x) \sqrt {-1+c x} \sqrt {1+c x}+3 \text {arccosh}(c x)\right )}{3 c (-1+c x)^2}+\frac {3 b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )}{c}+\frac {3 b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )}{c}-\frac {3 a \log (1-c x)}{c}+\frac {3 a \log (1+c x)}{c}-\frac {3 b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {3 b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}}{16 d^3} \]
((4*a*x)/(-1 + c^2*x^2)^2 - (6*a*x)/(-1 + c^2*x^2) + (b*(Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))/(3*c*(1 + c*x)^2) + (b*((2 - c*x) *Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 3*ArcCosh[c*x]))/(3*c*(-1 + c*x)^2) + (3*b *(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)))/c + (3*b*(Sqr t[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)))/c - (3*a*Log[1 - c*x])/ c + (3*a*Log[1 + c*x])/c - (3*b*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ ArcCosh[c*x]]) - 4*PolyLog[2, -E^ArcCosh[c*x]]))/(2*c) + (3*b*(ArcCosh[c*x ]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x] ]))/(2*c))/(16*d^3)
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6316, 27, 83, 6316, 83, 6318, 3042, 26, 4670, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \int \frac {a+b \text {arccosh}(c x)}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}-\frac {b c \int \frac {x}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}-\frac {b c \int \frac {x}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {3 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{4 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\) |
b/(12*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (x*(a + b*ArcCosh[c*x]))/( 4*d^3*(1 - c^2*x^2)^2) + (3*(-1/2*b/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x* (a + b*ArcCosh[c*x]))/(2*(1 - c^2*x^2)) - ((I/2)*((2*I)*(a + b*ArcCosh[c*x ])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog [2, E^ArcCosh[c*x]]))/c))/(4*d^3)
3.1.50.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> 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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[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]
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] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(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]
Time = 0.69 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c}\) | \(256\) |
| default | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c}\) | \(256\) |
| parts | \(-\frac {a \left (\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3} c}\) | \(273\) |
1/c*(-a/d^3*(1/16/(c*x+1)^2+3/16/(c*x+1)-3/16*ln(c*x+1)-1/16/(c*x-1)^2+3/1 6/(c*x-1)+3/16*ln(c*x-1))-b/d^3*(1/24*(9*c^3*x^3*arccosh(c*x)+9*(c*x-1)^(1 /2)*(c*x+1)^(1/2)*c^2*x^2-15*c*x*arccosh(c*x)-11*(c*x-1)^(1/2)*(c*x+1)^(1/ 2))/(c^4*x^4-2*c^2*x^2+1)+3/8*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^ (1/2))+3/8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/8*arccosh(c*x)*ln( 1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/8*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1 )^(1/2))))
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
-(Integral(a/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral(b* acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
1/2048*(18432*c^5*integrate(1/32*x^5*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4*d^3 *x^4 + 3*c^2*d^3*x^2 - d^3), x) - 48*c^4*(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)) - 6144*c^4*integrate(1/32*x^4*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4*d ^3*x^4 + 3*c^2*d^3*x^2 - d^3), x) + 18*(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^3 + 80*c^2*(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) + log(c*x - 1)/(c^3*d^3)) + 12288*c^2*integrate(1/32*x^2*log(c*x - 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x) + 9*(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*log(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 - 32*(3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1)^2 + 6*(c^4*x^4 - 2*c^2* x^2 + 1)*log(c*x + 1)*log(c*x - 1) + 4*(6*c^3*x^3 - 10*c*x - 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x - 1))*lo g(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3 ) + 2048*integrate(-1/16*(6*c^3*x^3 - 10*c*x - 3*(c^4*x^4 - 2*c^2*x^2 + 1) *log(c*x + 1) + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x - 1))/(c^7*d^3*x^7 - 3 *c^5*d^3*x^5 + 3*c^3*d^3*x^3 - c*d^3*x + (c^6*d^3*x^6 - 3*c^4*d^3*x^4 +...
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]