Integrand size = 23, antiderivative size = 195 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=-\frac {2 b (a+b \text {arccosh}(c+d x))^3}{d e^3}+\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arccosh}(c+d x))^4}{2 d e^3 (c+d x)^2}-\frac {6 b^2 (a+b \text {arccosh}(c+d x))^2 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )}{d e^3}+\frac {6 b^3 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )}{d e^3}+\frac {3 b^4 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(c+d x)}\right )}{d e^3} \]
-2*b*(a+b*arccosh(d*x+c))^3/d/e^3-1/2*(a+b*arccosh(d*x+c))^4/d/e^3/(d*x+c) ^2-6*b^2*(a+b*arccosh(d*x+c))^2*ln(1+1/(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1 /2))^2)/d/e^3+6*b^3*(a+b*arccosh(d*x+c))*polylog(2,-1/(d*x+c+(d*x+c-1)^(1/ 2)*(d*x+c+1)^(1/2))^2)/d/e^3+3*b^4*polylog(3,-1/(d*x+c+(d*x+c-1)^(1/2)*(d* x+c+1)^(1/2))^2)/d/e^3+2*b*(a+b*arccosh(d*x+c))^3*(d*x+c-1)^(1/2)*(d*x+c+1 )^(1/2)/d/e^3/(d*x+c)
Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(195)=390\).
Time = 1.85 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=\frac {-\frac {a^4}{(c+d x)^2}+\frac {4 a^3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{c+d x}-\frac {4 a^3 b \text {arccosh}(c+d x)}{(c+d x)^2}-\frac {b^4 \text {arccosh}(c+d x)^4}{(c+d x)^2}+12 a^2 b^2 \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{c+d x}-\frac {\text {arccosh}(c+d x)^2}{2 (c+d x)^2}-\log (c+d x)\right )+4 a b^3 \left (-\text {arccosh}(c+d x) \left (3 \text {arccosh}(c+d x)-\frac {3 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{c+d x}+\frac {\text {arccosh}(c+d x)^2}{(c+d x)^2}+6 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )\right )+3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )\right )+2 b^4 \left (2 \text {arccosh}(c+d x)^2 \left (-\text {arccosh}(c+d x)+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{c+d x}-3 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )\right )+6 \text {arccosh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(c+d x)}\right )\right )}{2 d e^3} \]
(-(a^4/(c + d*x)^2) + (4*a^3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(c + d*x) - (4*a^3*b*ArcCosh[c + d*x])/(c + d*x)^2 - (b^4*ArcCosh[c + d*x]^4)/( c + d*x)^2 + 12*a^2*b^2*((Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) *ArcCosh[c + d*x])/(c + d*x) - ArcCosh[c + d*x]^2/(2*(c + d*x)^2) - Log[c + d*x]) + 4*a*b^3*(-(ArcCosh[c + d*x]*(3*ArcCosh[c + d*x] - (3*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x])/(c + d*x) + ArcCos h[c + d*x]^2/(c + d*x)^2 + 6*Log[1 + E^(-2*ArcCosh[c + d*x])])) + 3*PolyLo g[2, -E^(-2*ArcCosh[c + d*x])]) + 2*b^4*(2*ArcCosh[c + d*x]^2*(-ArcCosh[c + d*x] + (Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x ])/(c + d*x) - 3*Log[1 + E^(-2*ArcCosh[c + d*x])]) + 6*ArcCosh[c + d*x]*Po lyLog[2, -E^(-2*ArcCosh[c + d*x])] + 3*PolyLog[3, -E^(-2*ArcCosh[c + d*x]) ]))/(2*d*e^3)
Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6411, 27, 6298, 6333, 6297, 25, 3042, 26, 4201, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{(c+d x)^3}d(c+d x)}{d e^3}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {2 b \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}d(c+d x)-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 6333 |
| \(\displaystyle \frac {2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 b \int \frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}d(c+d x)\right )-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 \int -(a+b \text {arccosh}(c+d x))^2 \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )d(a+b \text {arccosh}(c+d x))\right )-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \left (3 \int (a+b \text {arccosh}(c+d x))^2 \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )d(a+b \text {arccosh}(c+d x))+\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}\right )-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}+3 \int -i (a+b \text {arccosh}(c+d x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )d(a+b \text {arccosh}(c+d x))\right )}{d e^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 i \int (a+b \text {arccosh}(c+d x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )d(a+b \text {arccosh}(c+d x))\right )}{d e^3}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 i \left (2 i \int \frac {e^{\frac {2 (a-c-d x)}{b}} (a+b \text {arccosh}(c+d x))^2}{1+e^{\frac {2 (a-c-d x)}{b}}}d(a+b \text {arccosh}(c+d x))-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )\right )}{d e^3}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 i \left (2 i \left (b \int (a+b \text {arccosh}(c+d x)) \log \left (1+e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )\right )}{d e^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )\right )}{d e^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 (a-c-d x)}{b}} \operatorname {PolyLog}(2,-c-d x)de^{\frac {2 (a-c-d x)}{b}}+\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )\right )}{d e^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \text {arccosh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (\frac {\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}{c+d x}-3 i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )+\frac {1}{4} b^2 \operatorname {PolyLog}(3,-c-d x)\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )\right )}{d e^3}\) |
(-1/2*(a + b*ArcCosh[c + d*x])^4/(c + d*x)^2 + 2*b*((Sqrt[-1 + c + d*x]*Sq rt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(c + d*x) - (3*I)*((-1/3*I)*(a + b*ArcCosh[c + d*x])^3 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c + d*x])^2*Log[1 + E^((2*(a - c - d*x))/b)]) + b*((b*(a + b*ArcCosh[c + d*x])*PolyLog[2, - E^((2*(a - c - d*x))/b)])/2 + (b^2*PolyLog[3, -c - d*x])/4)))))/(d*e^3)
3.2.28.3.1 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[(((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] - Si mp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 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[b*c*(n/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Sim p[(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*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]
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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(491\) vs. \(2(231)=462\).
Time = 0.86 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.52
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{4} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3} \left (4 \left (d x +c \right )^{2}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+4 \operatorname {arccosh}\left (d x +c \right )^{3}-6 \operatorname {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )\right )}{e^{3}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2} \left (-3 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+3 \left (d x +c \right )^{2}+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+3 \operatorname {arccosh}\left (d x +c \right )^{2}-3 \,\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e^{3}}+\frac {6 a^{2} b^{2} \left (2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {\operatorname {arccosh}\left (d x +c \right ) \left (2 \left (d x +c \right )^{2}-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-\ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )\right )}{e^{3}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\) | \(492\) |
| default | \(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{4} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3} \left (4 \left (d x +c \right )^{2}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+4 \operatorname {arccosh}\left (d x +c \right )^{3}-6 \operatorname {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )\right )}{e^{3}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2} \left (-3 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+3 \left (d x +c \right )^{2}+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+3 \operatorname {arccosh}\left (d x +c \right )^{2}-3 \,\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e^{3}}+\frac {6 a^{2} b^{2} \left (2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {\operatorname {arccosh}\left (d x +c \right ) \left (2 \left (d x +c \right )^{2}-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-\ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )\right )}{e^{3}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3}}}{d}\) | \(492\) |
| parts | \(-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{4} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3} \left (4 \left (d x +c \right )^{2}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+4 \operatorname {arccosh}\left (d x +c \right )^{3}-6 \operatorname {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )\right )}{e^{3} d}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2} \left (-3 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+3 \left (d x +c \right )^{2}+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+3 \operatorname {arccosh}\left (d x +c \right )^{2}-3 \,\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e^{3} d}+\frac {6 a^{2} b^{2} \left (2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {\operatorname {arccosh}\left (d x +c \right ) \left (2 \left (d x +c \right )^{2}-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-\ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )\right )}{e^{3} d}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2 d x +2 c}\right )}{e^{3} d}\) | \(503\) |
1/d*(-1/2*a^4/e^3/(d*x+c)^2+b^4/e^3*(-1/2*arccosh(d*x+c)^3*(4*(d*x+c)^2-4* (d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+arccosh(d*x+c))/(d*x+c)^2+4*arccos h(d*x+c)^3-6*arccosh(d*x+c)^2*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)) ^2)-6*arccosh(d*x+c)*polylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2) +3*polylog(3,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2))+4*a*b^3/e^3*(-1/ 2*arccosh(d*x+c)^2*(-3*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+3*(d*x+c)^2 +arccosh(d*x+c))/(d*x+c)^2+3*arccosh(d*x+c)^2-3*arccosh(d*x+c)*ln(1+(d*x+c +(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3/2*polylog(2,-(d*x+c+(d*x+c-1)^(1/2) *(d*x+c+1)^(1/2))^2))+6*a^2*b^2/e^3*(2*arccosh(d*x+c)-1/2*arccosh(d*x+c)*( 2*(d*x+c)^2-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+arccosh(d*x+c))/(d*x +c)^2-ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2))+4*b*a^3/e^3*(-1/2/( d*x+c)^2*arccosh(d*x+c)+1/2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)))
\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
integral((b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2* arccosh(d*x + c)^2 + 4*a^3*b*arccosh(d*x + c) + a^4)/(d^3*e^3*x^3 + 3*c*d^ 2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)
\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{4}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
(Integral(a**4/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integ ral(b**4*acosh(c + d*x)**4/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3) , x) + Integral(4*a*b**3*acosh(c + d*x)**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x **2 + d**3*x**3), x) + Integral(6*a**2*b**2*acosh(c + d*x)**2/(c**3 + 3*c* *2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(4*a**3*b*acosh(c + d*x) /(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/e**3
\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
-1/2*b^4*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + 6*(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d*arc cosh(d*x + c)/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c)/(d*e^3))*a^2*b^2 + 2* a^3*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d/(d^3*e^3*x + c*d^2*e^3) - arcco sh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)) - 3*a^2*b^2*arccosh (d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^4/(d^3*e^3*x ^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + integrate(2*(2*(c^3 - c)*a*b^3 + (c^3 - c)*b^4 + (2*a*b^3*d^3 + b^4*d^3)*x^3 + 3*(2*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (b^4*c^2 + 2*(c^2 - 1)*a*b^3 + (2*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(2*a*b^3*c *d + b^4*c*d)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (2*(3*c^2*d - d)*a* b^3 + (3*c^2*d - d)*b^4)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/(d^6*e^3*x^6 + 6*c*d^5*e^3*x^5 + c^6*e^3 - c^4*e^3 + (15*c^2*d^4*e^ 3 - d^4*e^3)*x^4 + 4*(5*c^3*d^3*e^3 - c*d^3*e^3)*x^3 + 3*(5*c^4*d^2*e^3 - 2*c^2*d^2*e^3)*x^2 + (d^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + c^5*e^3 - c^3*e^3 + (10*c^2*d^3*e^3 - d^3*e^3)*x^3 + (10*c^3*d^2*e^3 - 3*c*d^2*e^3)*x^2 + (5*c ^4*d*e^3 - 3*c^2*d*e^3)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 2*(3*c^5* d*e^3 - 2*c^3*d*e^3)*x), x)
\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]