\(\int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx\) [30]

Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.84 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {-\frac {a^2}{(c+d x)^3}+a b \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{(c+d x)^2}-\frac {2 \text {arccosh}(c+d x)}{(c+d x)^3}+2 \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(c+d x)\right )\right )\right )+b^2 \left (\frac {1}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{(c+d x)^2}-\frac {\text {arccosh}(c+d x)^2}{(c+d x)^3}-i \text {arccosh}(c+d x) \log \left (1-i e^{-\text {arccosh}(c+d x)}\right )+i \text {arccosh}(c+d x) \log \left (1+i e^{-\text {arccosh}(c+d x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c+d x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c+d x)}\right )\right )}{3 d e^4} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^4,x]
 

Output:

(-(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)
 

Rubi [A] (verified)

Time = 1.23 (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

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

Input:

Int[(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^4,x]
 

Output:

(-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)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -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 2715

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]
 

rule 2838

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]
 

rule 3042

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

rule 4668

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]
 

rule 6298

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]
 

rule 6348

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]
 

rule 6362

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]
 

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]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {-\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\)	\(314\)
default	\(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {-\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\)	\(314\)
parts	\(-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{2} \left (-\frac {-\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4} d}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4} d}\)	\(319\)

Input:

int((a+b*arccosh(d*x+c))^2/(d*e*x+c*e)^4,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

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

Fricas [F]

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

integrate((a+b*arccosh(d*x+c))^2/(d*e*x+c*e)^4,x, algorithm="fricas")
 

Output:

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)
 

Sympy [F]

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

integrate((a+b*acosh(d*x+c))**2/(d*e*x+c*e)**4,x)
 

Output:

(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
 

Maxima [F]

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

integrate((a+b*arccosh(d*x+c))^2/(d*e*x+c*e)^4,x, algorithm="maxima")
 

Output:

-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)
 

Giac [F]

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

integrate((a+b*arccosh(d*x+c))^2/(d*e*x+c*e)^4,x, algorithm="giac")
 

Output:

integrate((b*arccosh(d*x + c) + a)^2/(d*e*x + c*e)^4, x)
 

Mupad [F(-1)]

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

int((a + b*acosh(c + d*x))^2/(c*e + d*e*x)^4,x)
 

Output:

int((a + b*acosh(c + d*x))^2/(c*e + d*e*x)^4, x)
 

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {6 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) a b \,c^{3} d +18 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) a b \,c^{2} d^{2} x +18 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) a b c \,d^{3} x^{2}+6 \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) a b \,d^{4} x^{3}+3 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) b^{2} c^{3} d +9 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) b^{2} c^{2} d^{2} x +9 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) b^{2} c \,d^{3} x^{2}+3 \left (\int \frac {\mathit {acosh} \left (d x +c \right )^{2}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) b^{2} d^{4} x^{3}-a^{2}}{3 d \,e^{4} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:

int((a+b*acosh(d*x+c))^2/(d*e*x+c*e)^4,x)
 

Output:

(6*int(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)*a*b*c**3*d + 18*int(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)*a*b*c**2*d**2*x + 18*int( 
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)*a*b*c*d**3*x**2 + 6*int(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)*a*b*d**4*x**3 + 3*int(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)*b**2*c**3*d + 9*int(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)*b**2*c**2*d**2*x + 9*int(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)*b**2*c*d**3*x**2 + 3*int(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)*b**2*d**4*x**3 - a**2)/(3*d* 
e**4*(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))