Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4} \]
1/3*b^2/d/e^4/(d*x+c)-1/3*(a+b*arccosh(d*x+c))^2/d/e^4/(d*x+c)^3+2/3*b*(a+ b*arccosh(d*x+c))*arctan(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/d/e^4-1/3* I*b^2*polylog(2,-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+1/3*I*b^ 2*polylog(2,I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))/d/e^4+1/3*b*(a+b*ar ccosh(d*x+c))*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^4/(d*x+c)^2
Time = 0.95 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {-\frac {a^2}{(c+d x)^3}+a b \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{(c+d x)^2}-\frac {2 \text {arccosh}(c+d x)}{(c+d x)^3}+2 \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(c+d x)\right )\right )\right )+b^2 \left (\frac {1}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{(c+d x)^2}-\frac {\text {arccosh}(c+d x)^2}{(c+d x)^3}-i \text {arccosh}(c+d x) \log \left (1-i e^{-\text {arccosh}(c+d x)}\right )+i \text {arccosh}(c+d x) \log \left (1+i e^{-\text {arccosh}(c+d x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c+d x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c+d x)}\right )\right )}{3 d e^4} \]
(-(a^2/(c + d*x)^3) + a*b*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d* x))/(c + d*x)^2 - (2*ArcCosh[c + d*x])/(c + d*x)^3 + 2*ArcTan[Tanh[ArcCosh [c + d*x]/2]]) + b^2*((c + d*x)^(-1) + (Sqrt[(-1 + c + d*x)/(1 + c + d*x)] *(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x)^2 - ArcCosh[c + d*x]^2/(c + d*x )^3 - I*ArcCosh[c + d*x]*Log[1 - I/E^ArcCosh[c + d*x]] + I*ArcCosh[c + d*x ]*Log[1 + I/E^ArcCosh[c + d*x]] - I*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] + I*PolyLog[2, I/E^ArcCosh[c + d*x]]))/(3*d*e^4)
Time = 0.94 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.83, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6411, 27, 6298, 6348, 15, 6362, 3042, 4668, 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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^2}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^2}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {\frac {2}{3} b \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1}}d(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6348 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)-\frac {1}{2} b \int \frac {1}{(c+d x)^2}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}\right )-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}d(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}+\frac {b}{2 (c+d x)}\right )-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{c+d x}d\text {arccosh}(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}+\frac {b}{2 (c+d x)}\right )-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (\frac {1}{2} \int (a+b \text {arccosh}(c+d x)) \csc \left (i \text {arccosh}(c+d x)+\frac {\pi }{2}\right )d\text {arccosh}(c+d x)+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}+\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (\frac {1}{2} \left (-i b \int \log \left (1-i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+i b \int \log \left (1+i e^{\text {arccosh}(c+d x)}\right )d\text {arccosh}(c+d x)+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}+\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (\frac {1}{2} \left (-i b \int e^{-\text {arccosh}(c+d x)} \log \left (1-i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}+i b \int e^{-\text {arccosh}(c+d x)} \log \left (1+i e^{\text {arccosh}(c+d x)}\right )de^{\text {arccosh}(c+d x)}+2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}+\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (\frac {1}{2} \left (2 \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )\right )+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{2 (c+d x)^2}+\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
(-1/3*(a + b*ArcCosh[c + d*x])^2/(c + d*x)^3 + (2*b*(b/(2*(c + d*x)) + (Sq rt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(2*(c + d*x)^ 2) + (2*(a + b*ArcCosh[c + d*x])*ArcTan[E^ArcCosh[c + d*x]] - I*b*PolyLog[ 2, (-I)*E^ArcCosh[c + d*x]] + I*b*PolyLog[2, I*E^ArcCosh[c + d*x]])/2))/3) /(d*e^4)
3.2.12.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) *(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d2*f*( m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)* (d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp[b*c*(n/(f *(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2, f, p}, x] && Eq Q[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[m]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.94 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.69
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\) | \(314\) |
| default | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\) | \(314\) |
| parts | \(-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{2} \left (-\frac {-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4} d}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4} d}\) | \(319\) |
1/d*(-1/3*a^2/e^4/(d*x+c)^3+b^2/e^4*(-1/3*(-(d*x+c)*arccosh(d*x+c)*(d*x+c- 1)^(1/2)*(d*x+c+1)^(1/2)+arccosh(d*x+c)^2-(d*x+c)^2)/(d*x+c)^3-1/3*I*arcco sh(d*x+c)*ln(1+I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))+1/3*I*arccosh(d* x+c)*ln(1-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))-1/3*I*dilog(1+I*(d*x+ c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))+1/3*I*dilog(1-I*(d*x+c+(d*x+c-1)^(1/2) *(d*x+c+1)^(1/2))))+2*a*b/e^4*(-1/3/(d*x+c)^3*arccosh(d*x+c)-1/6*(d*x+c-1) ^(1/2)*(d*x+c+1)^(1/2)*(arctan(1/((d*x+c)^2-1)^(1/2))*(d*x+c)^2-((d*x+c)^2 -1)^(1/2))/(d*x+c)^2/((d*x+c)^2-1)^(1/2)))
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^2*arccosh(d*x + c)^2 + 2*a*b*arccosh(d*x + c) + a^2)/(d^4*e^4* x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(Integral(a**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 4*x**4), x) + Integral(b**2*acosh(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2* d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(2*a*b*acosh(c + d*x) /(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/e **4
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*b^2*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1/3*a^2/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) + integrate(2/3*((3*a*b*d^ 3 + b^2*d^3)*x^3 + 3*(c^3 - c)*a*b + (c^3 - c)*b^2 + 3*(3*a*b*c*d^2 + b^2* c*d^2)*x^2 + (b^2*c^2 + 3*(c^2 - 1)*a*b + (3*a*b*d^2 + b^2*d^2)*x^2 + 2*(3 *a*b*c*d + b^2*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (3*(3*c^2*d - d)*a*b + (3*c^2*d - d)*b^2)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4 - c^5*e^4 + (21*c^2*d^5* e^4 - d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 - c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 - 2*c^2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 - 10*c^3*d^2*e^4)*x^2 + (d^6*e^4*x^ 6 + 6*c*d^5*e^4*x^5 + c^6*e^4 - c^4*e^4 + (15*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 - c*d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^4 - 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 - 2*c^3*d*e^4)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (7*c^6*d*e^4 - 5*c^4*d*e^4)*x), x)
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]