3.2.45 \(\int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx\) [145]

3.2.45.1 Optimal result

Integrand size = 21, antiderivative size = 163 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{2 b d (a+b \text {arccosh}(c+d x))^2}+\frac {e}{2 b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e (c+d x)^2}{b^2 d (a+b \text {arccosh}(c+d x))}-\frac {e \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3 d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{b^3 d} \]

output

1/2*e/b^2/d/(a+b*arccosh(d*x+c))-e*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+c))+e* 
cosh(2*a/b)*Shi(2*(a+b*arccosh(d*x+c))/b)/b^3/d-e*Chi(2*(a+b*arccosh(d*x+c 
))/b)*sinh(2*a/b)/b^3/d-1/2*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/ 
(a+b*arccosh(d*x+c))^2
 

3.2.45.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.78 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx=\frac {e \left (-\frac {b^2 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{(a+b \text {arccosh}(c+d x))^2}+\frac {b \left (1-2 (c+d x)^2\right )}{a+b \text {arccosh}(c+d x)}-2 \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )\right )}{2 b^3 d} \]

input

Integrate[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^3,x]
 

output

(e*(-((b^2*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh[ 
c + d*x])^2) + (b*(1 - 2*(c + d*x)^2))/(a + b*ArcCosh[c + d*x]) - 2*CoshIn 
tegral[2*(a/b + ArcCosh[c + d*x])]*Sinh[(2*a)/b] + 2*Cosh[(2*a)/b]*SinhInt 
egral[2*(a/b + ArcCosh[c + d*x])]))/(2*b^3*d)
 

3.2.45.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {6411, 27, 6301, 6308, 6366, 6302, 25, 5971, 27, 3042, 26, 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.

\(\displaystyle \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx\)

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6301

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

\(\Big \downarrow \) 6308

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

\(\Big \downarrow \) 6366

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

\(\Big \downarrow \) 6302

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

input

Int[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^3,x]
 

output

(e*(-1/2*(Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*(a + b*ArcCos 
h[c + d*x])^2) + 1/(2*b^2*(a + b*ArcCosh[c + d*x])) + (-((c + d*x)^2/(b*(a 
 + b*ArcCosh[c + d*x]))) + (I*(I*CoshIntegral[(2*(a + b*ArcCosh[c + d*x])) 
/b]*Sinh[(2*a)/b] - I*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x 
]))/b]))/b^2)/b))/d
 

3.2.45.3.1 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 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 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6301

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + (-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCosh[c*x 
])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) 
 Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) 
), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
 

rule 6302

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
1/(b*c^(m + 1))   Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, 
 a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
 

rule 6308

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq 
rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + 
 c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ 
c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 
] && EqQ[e2, (-c)*d2] && NeQ[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 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]
 

3.2.45.4 Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(\frac {-\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b \,\operatorname {arccosh}\left (d x +c \right )+2 a -b \right )}{8 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{4 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{3}}}{d}\)	\(254\)
default	\(\frac {-\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b \,\operatorname {arccosh}\left (d x +c \right )+2 a -b \right )}{8 b^{2} \left (b^{2} \operatorname {arccosh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{8 b \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{4 b^{2} \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{3}}}{d}\)	\(254\)

input

int((d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x,method=_RETURNVERBOSE)
 

output

1/d*(-1/8*(-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e*(2* 
b*arccosh(d*x+c)+2*a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2 
)+1/2*e/b^3*exp(2*a/b)*Ei(1,2*arccosh(d*x+c)+2*a/b)-1/8/b*e*(2*(d*x+c)^2-1 
+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^2-1/4/b^2 
*e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh( 
d*x+c))-1/2/b^3*e*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*a/b))
 

3.2.45.5 Fricas [F]

\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]

input

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")
 

output

integral((d*e*x + c*e)/(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), x)
 

3.2.45.6 Sympy [F]

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

input

integrate((d*e*x+c*e)/(a+b*acosh(d*x+c))**3,x)
 

output

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

3.2.45.7 Maxima [F]

\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]

input

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")
 

output

-1/2*((2*a*d^8*e + b*d^8*e)*x^8 + 8*(2*a*c*d^7*e + b*c*d^7*e)*x^7 + (2*(28 
*c^2*d^6*e - 3*d^6*e)*a + (28*c^2*d^6*e - 3*d^6*e)*b)*x^6 + 2*(2*(28*c^3*d 
^5*e - 9*c*d^5*e)*a + (28*c^3*d^5*e - 9*c*d^5*e)*b)*x^5 + (2*(70*c^4*d^4*e 
 - 45*c^2*d^4*e + 3*d^4*e)*a + (70*c^4*d^4*e - 45*c^2*d^4*e + 3*d^4*e)*b)* 
x^4 + ((2*a*d^5*e + b*d^5*e)*x^5 + 5*(2*a*c*d^4*e + b*c*d^4*e)*x^4 + (2*(1 
0*c^2*d^3*e - d^3*e)*a + (10*c^2*d^3*e - d^3*e)*b)*x^3 + (2*(10*c^3*d^2*e 
- 3*c*d^2*e)*a + (10*c^3*d^2*e - 3*c*d^2*e)*b)*x^2 + 2*(c^5*e - c^3*e)*a + 
 (c^5*e - c^3*e)*b + (2*(5*c^4*d*e - 3*c^2*d*e)*a + (5*c^4*d*e - 3*c^2*d*e 
)*b)*x)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 4*(2*(14*c^5*d^3*e - 15* 
c^3*d^3*e + 3*c*d^3*e)*a + (14*c^5*d^3*e - 15*c^3*d^3*e + 3*c*d^3*e)*b)*x^ 
3 + (3*(2*a*d^6*e + b*d^6*e)*x^6 + 18*(2*a*c*d^5*e + b*c*d^5*e)*x^5 + 5*(2 
*(9*c^2*d^4*e - d^4*e)*a + (9*c^2*d^4*e - d^4*e)*b)*x^4 + 20*(2*(3*c^3*d^3 
*e - c*d^3*e)*a + (3*c^3*d^3*e - c*d^3*e)*b)*x^3 + (5*(18*c^4*d^2*e - 12*c 
^2*d^2*e + d^2*e)*a + (45*c^4*d^2*e - 30*c^2*d^2*e + 2*d^2*e)*b)*x^2 + (6* 
c^6*e - 10*c^4*e + 5*c^2*e - e)*a + (3*c^6*e - 5*c^4*e + 2*c^2*e)*b + 2*(( 
18*c^5*d*e - 20*c^3*d*e + 5*c*d*e)*a + (9*c^5*d*e - 10*c^3*d*e + 2*c*d*e)* 
b)*x)*(d*x + c + 1)*(d*x + c - 1) + (2*(28*c^6*d^2*e - 45*c^4*d^2*e + 18*c 
^2*d^2*e - d^2*e)*a + (28*c^6*d^2*e - 45*c^4*d^2*e + 18*c^2*d^2*e - d^2*e) 
*b)*x^2 + (3*(2*a*d^7*e + b*d^7*e)*x^7 + 21*(2*a*c*d^6*e + b*c*d^6*e)*x^6 
+ 7*(2*(9*c^2*d^5*e - d^5*e)*a + (9*c^2*d^5*e - d^5*e)*b)*x^5 + 35*(2*(...
 

3.2.45.8 Giac [F]

\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}} \,d x } \]

input

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")
 

output

integrate((d*e*x + c*e)/(b*arccosh(d*x + c) + a)^3, x)
 

3.2.45.9 Mupad [F(-1)]

Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^3} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \]

input

int((c*e + d*e*x)/(a + b*acosh(c + d*x))^3,x)
 

output

int((c*e + d*e*x)/(a + b*acosh(c + d*x))^3, x)