Integrand size = 25, antiderivative size = 171 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^3} \]
1/12*b*c*x/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+1/4*(a+b*arccosh(c*x))/d^3/(-c^ 2*x^2+1)^2+1/2*(a+b*arccosh(c*x))/d^3/(-c^2*x^2+1)+2*(a+b*arccosh(c*x))*ar ctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^3+1/2*b*polylog(2,-(c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2))^2)/d^3-1/2*b*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^ (1/2))^2)/d^3-2/3*b*c*x/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.97 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.92 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {a}{\left (-1+c^2 x^2\right )^2}-\frac {2 a}{-1+c^2 x^2}-\frac {b \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{12 (1+c x)^2}+\frac {b \left ((2-c x) \sqrt {-1+c x} \sqrt {1+c x}+3 \text {arccosh}(c x)\right )}{12 (-1+c x)^2}+\frac {5}{4} b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-\frac {5}{4} b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )+2 b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )+b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{4 d^3} \]
(a/(-1 + c^2*x^2)^2 - (2*a)/(-1 + c^2*x^2) - (b*(Sqrt[-1 + c*x]*Sqrt[1 + c *x]*(2 + c*x) - 3*ArcCosh[c*x]))/(12*(1 + c*x)^2) + (b*((2 - c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 3*ArcCosh[c*x]))/(12*(-1 + c*x)^2) + (5*b*(-(1/Sqrt [(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)))/4 - (5*b*(Sqrt[(-1 + c* x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)))/4 + 4*a*Log[x] - 2*a*Log[1 - c^2* x^2] + 2*b*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]) + b*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log [1 + E^ArcCosh[c*x]]) - 4*PolyLog[2, -E^ArcCosh[c*x]]) + b*(ArcCosh[c*x]*( ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x]])) /(4*d^3)
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {6351, 27, 42, 41, 6351, 41, 6331, 5984, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{d^2 x \left (1-c^2 x^2\right )^2}dx}{d}-\frac {b c \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^3}-\frac {b c \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 42 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^3}-\frac {b c \left (-\frac {2}{3} \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 6351 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 6331 |
| \(\displaystyle \frac {-\int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {-2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-2 i \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-2 i \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-2 i \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )+\frac {a+b \text {arccosh}(c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {c x-1} \sqrt {c x+1}}}{d^3}+\frac {a+b \text {arccosh}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{4 d^3}\) |
-1/4*(b*c*(-1/3*x/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (2*x)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/d^3 + (a + b*ArcCosh[c*x])/(4*d^3*(1 - c^2*x^2)^2) + (-1/2*(b*c*x)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(2*(1 - c^2*x^2)) - (2*I)*(I*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])] + (I/4)*b*PolyLog[2, -E^(2*ArcCosh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcCosh [c*x])]))/d^3
3.1.51.3.1 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[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 3)/(2*a*c*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 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[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[-d^(-1) Subst[Int[(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] && IG tQ[n, 0]
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] + (Simp[(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] - Simp[ 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*ArcCosh[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])
Time = 0.79 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.89
| method | result | size |
| parts | \(-\frac {a \left (-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\ln \left (x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-8 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-9 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +16 c^{2} x^{2}-9 \,\operatorname {arccosh}\left (c x \right )-8}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(324\) |
| derivativedivides | \(-\frac {a \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-8 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-9 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +16 c^{2} x^{2}-9 \,\operatorname {arccosh}\left (c x \right )-8}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(326\) |
| default | \(-\frac {a \left (-\ln \left (c x \right )-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-8 c^{4} x^{4}+6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-9 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +16 c^{2} x^{2}-9 \,\operatorname {arccosh}\left (c x \right )-8}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3}}\) | \(326\) |
-a/d^3*(-1/16/(c*x+1)^2-5/16/(c*x+1)+1/2*ln(c*x+1)-ln(x)-1/16/(c*x-1)^2+5/ 16/(c*x-1)+1/2*ln(c*x-1))-b/d^3*(1/12*(8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x ^3-8*c^4*x^4+6*c^2*x^2*arccosh(c*x)-9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+16*c ^2*x^2-9*arccosh(c*x)-8)/(c^4*x^4-2*c^2*x^2+1)-arccosh(c*x)*ln(1+(c*x+(c*x -1)^(1/2)*(c*x+1)^(1/2))^2)-1/2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 ))^2)+arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+polylog(2,c*x+(c* x-1)^(1/2)*(c*x+1)^(1/2))+arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 ))+polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2)))
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \]
-(Integral(a/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x) + Integral(b* acosh(c*x)/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x), x))/d**3
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
-1/4*a*((2*c^2*x^2 - 3)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) + 2*log(c*x + 1)/d^3 + 2*log(c*x - 1)/d^3 - 4*log(x)/d^3) - b*integrate(log(c*x + sqrt(c *x + 1)*sqrt(c*x - 1))/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3* x), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]