\(\int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx\) [44]

Optimal result

Integrand size = 23, antiderivative size = 192 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=-\frac {(a+b \text {arccosh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arccosh}(c+d x))^4 \log \left (1+e^{2 \text {arccosh}(c+d x)}\right )}{d e}+\frac {2 b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(c+d x)}\right )}{d e}+\frac {3 b^3 (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{2 \text {arccosh}(c+d x)}\right )}{d e}-\frac {3 b^4 \operatorname {PolyLog}\left (5,-e^{2 \text {arccosh}(c+d x)}\right )}{2 d e} \] Output:

-1/5*(a+b*arccosh(d*x+c))^5/b/d/e+(a+b*arccosh(d*x+c))^4*ln(1+(d*x+c+(d*x+ 
c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e+2*b*(a+b*arccosh(d*x+c))^3*polylog(2,-( 
d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e-3*b^2*(a+b*arccosh(d*x+c))^2 
*polylog(3,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e+3*b^3*(a+b*arcc 
osh(d*x+c))*polylog(4,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e-3/2* 
b^4*polylog(5,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\frac {2 a^3 b \text {arccosh}(c+d x)^2+2 a^2 b^2 \text {arccosh}(c+d x)^3+a b^3 \text {arccosh}(c+d x)^4+\frac {1}{5} b^4 \text {arccosh}(c+d x)^5+4 a^3 b \text {arccosh}(c+d x) \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+6 a^2 b^2 \text {arccosh}(c+d x)^2 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+4 a b^3 \text {arccosh}(c+d x)^3 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+b^4 \text {arccosh}(c+d x)^4 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+a^4 \log (c+d x)-2 b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )-3 b^2 (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(c+d x)}\right )-3 a b^3 \operatorname {PolyLog}\left (4,-e^{-2 \text {arccosh}(c+d x)}\right )-3 b^4 \text {arccosh}(c+d x) \operatorname {PolyLog}\left (4,-e^{-2 \text {arccosh}(c+d x)}\right )-\frac {3}{2} b^4 \operatorname {PolyLog}\left (5,-e^{-2 \text {arccosh}(c+d x)}\right )}{d e} \] Input:

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

Output:

(2*a^3*b*ArcCosh[c + d*x]^2 + 2*a^2*b^2*ArcCosh[c + d*x]^3 + a*b^3*ArcCosh 
[c + d*x]^4 + (b^4*ArcCosh[c + d*x]^5)/5 + 4*a^3*b*ArcCosh[c + d*x]*Log[1 
+ E^(-2*ArcCosh[c + d*x])] + 6*a^2*b^2*ArcCosh[c + d*x]^2*Log[1 + E^(-2*Ar 
cCosh[c + d*x])] + 4*a*b^3*ArcCosh[c + d*x]^3*Log[1 + E^(-2*ArcCosh[c + d* 
x])] + b^4*ArcCosh[c + d*x]^4*Log[1 + E^(-2*ArcCosh[c + d*x])] + a^4*Log[c 
 + d*x] - 2*b*(a + b*ArcCosh[c + d*x])^3*PolyLog[2, -E^(-2*ArcCosh[c + d*x 
])] - 3*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[3, -E^(-2*ArcCosh[c + d*x]) 
] - 3*a*b^3*PolyLog[4, -E^(-2*ArcCosh[c + d*x])] - 3*b^4*ArcCosh[c + d*x]* 
PolyLog[4, -E^(-2*ArcCosh[c + d*x])] - (3*b^4*PolyLog[5, -E^(-2*ArcCosh[c 
+ d*x])])/2)/(d*e)
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6411, 27, 6297, 25, 3042, 26, 4201, 2620, 3011, 7163, 7163, 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.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{e (c+d x)}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^4}{c+d x}d(c+d x)}{d e}\)

\(\Big \downarrow \) 6297

\(\displaystyle \frac {\int -(a+b \text {arccosh}(c+d x))^4 \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int (a+b \text {arccosh}(c+d x))^4 \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int -i (a+b \text {arccosh}(c+d x))^4 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int (a+b \text {arccosh}(c+d x))^4 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 4201

\(\displaystyle \frac {i \left (2 i \int \frac {e^{\frac {2 (a-c-d x)}{b}} (a+b \text {arccosh}(c+d x))^4}{1+e^{\frac {2 (a-c-d x)}{b}}}d(a+b \text {arccosh}(c+d x))-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {i \left (2 i \left (2 b \int (a+b \text {arccosh}(c+d x))^3 \log \left (1+e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^4 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {3}{2} b \int (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^4 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {3}{2} b \left (b \int (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^4 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {3}{2} b \left (b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (4,-e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))-\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^4 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {3}{2} b \left (b \left (-\frac {1}{4} b^2 \int e^{-\frac {2 (a-c-d x)}{b}} \operatorname {PolyLog}(4,-c-d x)de^{\frac {2 (a-c-d x)}{b}}-\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^4 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {3}{2} b \left (b \left (-\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {1}{4} b^2 \operatorname {PolyLog}(5,-c-d x)\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^4 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{5} i (a+b \text {arccosh}(c+d x))^5\right )}{b d e}\)

Input:

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

Output:

(I*((-1/5*I)*(a + b*ArcCosh[c + d*x])^5 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c 
+ d*x])^4*Log[1 + E^((2*(a - c - d*x))/b)]) + 2*b*((b*(a + b*ArcCosh[c + d 
*x])^3*PolyLog[2, -E^((2*(a - c - d*x))/b)])/2 - (3*b*(-1/2*(b*(a + b*ArcC 
osh[c + d*x])^2*PolyLog[3, -E^((2*(a - c - d*x))/b)]) + b*(-1/2*(b*(a + b* 
ArcCosh[c + d*x])*PolyLog[4, -E^((2*(a - c - d*x))/b)]) - (b^2*PolyLog[5, 
-c - d*x])/4)))/2))))/(b*d*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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]
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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]
 

rule 6297

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]
 

rule 6411

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]
 

rule 7143

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]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs. \(2(258)=516\).

Time = 0.42 (sec) , antiderivative size = 582, normalized size of antiderivative = 3.03

method	result	size
derivativedivides	\(\frac {\frac {a^{4} \ln \left (d x +c \right )}{e}+\frac {b^{4} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{5}}{5}+\operatorname {arccosh}\left (d x +c \right )^{4} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+2 \operatorname {arccosh}\left (d x +c \right )^{3} \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-3 \operatorname {arccosh}\left (d x +c \right )^{2} \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+3 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (5, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{4}}{4}+\operatorname {arccosh}\left (d x +c \right )^{3} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{4}\right )}{e}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{3}+\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 )+\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 )-\frac {\operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}+\frac {4 a^{3} b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\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 {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}}{d}\)	\(582\)
default	\(\frac {\frac {a^{4} \ln \left (d x +c \right )}{e}+\frac {b^{4} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{5}}{5}+\operatorname {arccosh}\left (d x +c \right )^{4} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+2 \operatorname {arccosh}\left (d x +c \right )^{3} \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-3 \operatorname {arccosh}\left (d x +c \right )^{2} \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+3 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (5, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{4}}{4}+\operatorname {arccosh}\left (d x +c \right )^{3} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{4}\right )}{e}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{3}+\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 )+\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 )-\frac {\operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}+\frac {4 a^{3} b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\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 {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}}{d}\)	\(582\)
parts	\(\frac {a^{4} \ln \left (d x +c \right )}{e d}+\frac {b^{4} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{5}}{5}+\operatorname {arccosh}\left (d x +c \right )^{4} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+2 \operatorname {arccosh}\left (d x +c \right )^{3} \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-3 \operatorname {arccosh}\left (d x +c \right )^{2} \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+3 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (5, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e d}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{4}}{4}+\operatorname {arccosh}\left (d x +c \right )^{3} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{4}\right )}{e d}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{3}+\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 )+\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 )-\frac {\operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e d}+\frac {4 a^{3} b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\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 {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e d}\)	\(593\)

Input:

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e),x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^4/e*ln(d*x+c)+b^4/e*(-1/5*arccosh(d*x+c)^5+arccosh(d*x+c)^4*ln(1+(d 
*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+2*arccosh(d*x+c)^3*polylog(2,-(d* 
x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3*arccosh(d*x+c)^2*polylog(3,-(d*x 
+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3*arccosh(d*x+c)*polylog(4,-(d*x+c+ 
(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3/2*polylog(5,-(d*x+c+(d*x+c-1)^(1/2)* 
(d*x+c+1)^(1/2))^2))+4*a*b^3/e*(-1/4*arccosh(d*x+c)^4+arccosh(d*x+c)^3*ln( 
1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/2*arccosh(d*x+c)^2*polylog( 
2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-3/2*arccosh(d*x+c)*polylog(3 
,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+3/4*polylog(4,-(d*x+c+(d*x+c- 
1)^(1/2)*(d*x+c+1)^(1/2))^2))+6*a^2*b^2/e*(-1/3*arccosh(d*x+c)^3+arccosh(d 
*x+c)^2*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+arccosh(d*x+c)*pol 
ylog(2,-(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-1/2*polylog(3,-(d*x+c+( 
d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2))+4*a^3*b/e*(-1/2*arccosh(d*x+c)^2+arcco 
sh(d*x+c)*ln(1+(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+1/2*polylog(2,-( 
d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)))
 

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \] Input:

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

Output:

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*e*x + c*e), x)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\frac {\int \frac {a^{4}}{c + d x}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \] Input:

integrate((a+b*acosh(d*x+c))**4/(d*e*x+c*e),x)
 

Output:

(Integral(a**4/(c + d*x), x) + Integral(b**4*acosh(c + d*x)**4/(c + d*x), 
x) + Integral(4*a*b**3*acosh(c + d*x)**3/(c + d*x), x) + Integral(6*a**2*b 
**2*acosh(c + d*x)**2/(c + d*x), x) + Integral(4*a**3*b*acosh(c + d*x)/(c 
+ d*x), x))/e
 

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \] Input:

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

Output:

a^4*log(d*e*x + c*e)/(d*e) + integrate(b^4*log(d*x + sqrt(d*x + c + 1)*sqr 
t(d*x + c - 1) + c)^4/(d*e*x + c*e) + 4*a*b^3*log(d*x + sqrt(d*x + c + 1)* 
sqrt(d*x + c - 1) + c)^3/(d*e*x + c*e) + 6*a^2*b^2*log(d*x + sqrt(d*x + c 
+ 1)*sqrt(d*x + c - 1) + c)^2/(d*e*x + c*e) + 4*a^3*b*log(d*x + sqrt(d*x + 
 c + 1)*sqrt(d*x + c - 1) + c)/(d*e*x + c*e), x)
 

Giac [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \] Input:

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

Output:

integrate((b*arccosh(d*x + c) + a)^4/(d*e*x + c*e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{c\,e+d\,e\,x} \,d x \] Input:

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x),x)
 

Output:

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x), x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^4}{c e+d e x} \, dx=\frac {4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d x +c}d x \right ) a^{3} b d +\left (\int \frac {\mathit {acosh} \left (d x +c \right )^{4}}{d x +c}d x \right ) b^{4} d +4 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{3}}{d x +c}d x \right ) a \,b^{3} d +6 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d x +c}d x \right ) a^{2} b^{2} d +\mathrm {log}\left (d x +c \right ) a^{4}}{d e} \] Input:

int((a+b*acosh(d*x+c))^4/(d*e*x+c*e),x)
                                                                                    
                                                                                    
 

Output:

(4*int(acosh(c + d*x)/(c + d*x),x)*a**3*b*d + int(acosh(c + d*x)**4/(c + d 
*x),x)*b**4*d + 4*int(acosh(c + d*x)**3/(c + d*x),x)*a*b**3*d + 6*int(acos 
h(c + d*x)**2/(c + d*x),x)*a**2*b**2*d + log(c + d*x)*a**4)/(d*e)