\(\int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx\) [70]

Optimal result

Integrand size = 12, antiderivative size = 174 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b d (a+b \text {arccosh}(c+d x))^3}-\frac {c+d x}{6 b^2 d (a+b \text {arccosh}(c+d x))^2}-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 b^3 d (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{6 b^4 d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{6 b^4 d} \] Output:

-1/3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^3-1/6*(d*x+c 
)/b^2/d/(a+b*arccosh(d*x+c))^2-1/6*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/( 
a+b*arccosh(d*x+c))+1/6*cosh(a/b)*Chi((a+b*arccosh(d*x+c))/b)/b^4/d-1/6*si 
nh(a/b)*Shi((a+b*arccosh(d*x+c))/b)/b^4/d
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=-\frac {\frac {2 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^3}+\frac {b^2 (c+d x)}{(a+b \text {arccosh}(c+d x))^2}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{a+b \text {arccosh}(c+d x)}-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c+d x)\right )}{6 b^4 d} \] Input:

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

Output:

-1/6*((2*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x] 
)^3 + (b^2*(c + d*x))/(a + b*ArcCosh[c + d*x])^2 + (b*Sqrt[-1 + c + d*x]*S 
qrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x]) - Cosh[a/b]*CoshIntegral[a/b + 
ArcCosh[c + d*x]] + Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]])/(b^4*d 
)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6410, 6295, 6366, 6295, 6368, 3042, 3784, 26, 3042, 26, 3779, 3782}

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.

\(\displaystyle \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx\)

\(\Big \downarrow \) 6410

\(\displaystyle \frac {\int \frac {1}{(a+b \text {arccosh}(c+d x))^4}d(c+d x)}{d}\)

\(\Big \downarrow \) 6295

\(\displaystyle \frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3}d(c+d x)}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}}{d}\)

\(\Big \downarrow \) 6366

\(\displaystyle \frac {\frac {\frac {\int \frac {1}{(a+b \text {arccosh}(c+d x))^2}d(c+d x)}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}}{d}\)

\(\Big \downarrow \) 6295

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {c+d x}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}d(c+d x)}{b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}}{d}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}+\frac {-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{3 b}}{d}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}+\frac {-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-i \sinh \left (\frac {a}{b}\right ) \int -\frac {i \sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{3 b}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {\frac {\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\cosh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int \frac {\sinh \left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}+\frac {-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))-\sinh \left (\frac {a}{b}\right ) \int -\frac {i \sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{3 b}}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}+\frac {-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {i \sinh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{3 b}}{d}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}+\frac {-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}+\frac {-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}+\frac {-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )+\cosh \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c+d x)}d(a+b \text {arccosh}(c+d x))}{b^2}}{2 b}}{3 b}}{d}\)

\(\Big \downarrow \) 3782

\(\displaystyle \frac {\frac {\frac {\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{b^2}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{b (a+b \text {arccosh}(c+d x))}}{2 b}-\frac {c+d x}{2 b (a+b \text {arccosh}(c+d x))^2}}{3 b}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b (a+b \text {arccosh}(c+d x))^3}}{d}\)

Input:

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

Output:

(-1/3*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x])^3 
) + (-1/2*(c + d*x)/(b*(a + b*ArcCosh[c + d*x])^2) + (-((Sqrt[-1 + c + d*x 
]*Sqrt[1 + c + d*x])/(b*(a + b*ArcCosh[c + d*x]))) + (Cosh[a/b]*CoshIntegr 
al[(a + b*ArcCosh[c + d*x])/b] - Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + 
 d*x])/b])/b^2)/(2*b))/(3*b))/d
 

Defintions of rubi rules used

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 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 6295

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c* 
x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c 
/(b*(n + 1))   Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + 
 c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
 

rule 6366

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 
_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a 
 + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x 
]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp 
[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Int[ 
(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 
1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

rule 6410

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d 
   Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d 
, n}, x]
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(\frac {\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{12 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{12 b^{4}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{6 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{12 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{12 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{12 b^{4}}}{d}\)	\(295\)
default	\(\frac {\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )-b^{2} \operatorname {arccosh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{12 b^{3} \left (b^{3} \operatorname {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{12 b^{4}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{6 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{12 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{12 b^{3} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{12 b^{4}}}{d}\)	\(295\)

Input:

int(1/(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/12*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*(b^2*arccosh(d*x+c)^2+2 
*a*b*arccosh(d*x+c)-b^2*arccosh(d*x+c)+a^2-a*b+2*b^2)/b^3/(b^3*arccosh(d*x 
+c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x+c)+a^3)-1/12/b^4*exp(a/ 
b)*Ei(1,arccosh(d*x+c)+a/b)-1/6/b*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/ 
(a+b*arccosh(d*x+c))^3-1/12/b^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a 
+b*arccosh(d*x+c))^2-1/12/b^3*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b 
*arccosh(d*x+c))-1/12/b^4*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b))
 

Fricas [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(1/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")
 

Output:

integral(1/(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), x)
 

Sympy [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{4}}\, dx \] Input:

integrate(1/(a+b*acosh(d*x+c))**4,x)
 

Output:

Integral((a + b*acosh(c + d*x))**(-4), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \] Input:

integrate(1/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \] Input:

int(1/(a + b*acosh(c + d*x))^4,x)
 

Output:

int(1/(a + b*acosh(c + d*x))^4, x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \text {arccosh}(c+d x))^4} \, dx=\int \frac {1}{\mathit {acosh} \left (d x +c \right )^{4} b^{4}+4 \mathit {acosh} \left (d x +c \right )^{3} a \,b^{3}+6 \mathit {acosh} \left (d x +c \right )^{2} a^{2} b^{2}+4 \mathit {acosh} \left (d x +c \right ) a^{3} b +a^{4}}d x \] Input:

int(1/(a+b*acosh(d*x+c))^4,x)
 

Output:

int(1/(acosh(c + d*x)**4*b**4 + 4*acosh(c + d*x)**3*a*b**3 + 6*acosh(c + d 
*x)**2*a**2*b**2 + 4*acosh(c + d*x)*a**3*b + a**4),x)