\(\int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^2} \, dx\) [28]

Optimal result

Integrand size = 23, antiderivative size = 110 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^2} \, dx=-\frac {(a+b \text {arccosh}(c+d x))^2}{d e^2 (c+d x)}+\frac {4 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{d e^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{d e^2}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{d e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^2} \, dx=\frac {-\frac {a^2}{c+d x}+2 a b \left (-\frac {\text {arccosh}(c+d x)}{c+d x}+2 \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(c+d x)\right )\right )\right )-i b^2 \left (\text {arccosh}(c+d x) \left (-\frac {i \text {arccosh}(c+d x)}{c+d x}+2 \log \left (1-i e^{-\text {arccosh}(c+d x)}\right )-2 \log \left (1+i e^{-\text {arccosh}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c+d x)}\right )\right )}{d e^2} \] Input:

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

Output:

(-(a^2/(c + d*x)) + 2*a*b*(-(ArcCosh[c + d*x]/(c + d*x)) + 2*ArcTan[Tanh[A 
rcCosh[c + d*x]/2]]) - I*b^2*(ArcCosh[c + d*x]*(((-I)*ArcCosh[c + d*x])/(c 
 + d*x) + 2*Log[1 - I/E^ArcCosh[c + d*x]] - 2*Log[1 + I/E^ArcCosh[c + d*x] 
]) + 2*PolyLog[2, (-I)/E^ArcCosh[c + d*x]] - 2*PolyLog[2, I/E^ArcCosh[c + 
d*x]]))/(d*e^2)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6411, 27, 6298, 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.

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

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^2}{e^2 (c+d x)^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^2}{(c+d x)^2}d(c+d x)}{d e^2}\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6362

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 = 0.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.24

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{d x +c}-2 i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )+2 i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{d x +c}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{2}}}{d}\)	\(246\)
default	\(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{d x +c}-2 i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )+2 i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{d x +c}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{2}}}{d}\)	\(246\)
parts	\(-\frac {a^{2}}{e^{2} \left (d x +c \right ) d}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{d x +c}-2 i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )+2 i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )\right )}{e^{2} d}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{d x +c}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{2} d}\)	\(251\)

Input:

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

Output:

1/d*(-a^2/e^2/(d*x+c)+b^2/e^2*(-1/(d*x+c)*arccosh(d*x+c)^2-2*I*arccosh(d*x 
+c)*ln(1+I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))+2*I*arccosh(d*x+c)*ln( 
1-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)))-2*I*dilog(1+I*(d*x+c+(d*x+c-1 
)^(1/2)*(d*x+c+1)^(1/2)))+2*I*dilog(1-I*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^( 
1/2))))+2*a*b/e^2*(-1/(d*x+c)*arccosh(d*x+c)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/ 
2)/((d*x+c)^2-1)^(1/2)*arctan(1/((d*x+c)^2-1)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int(acosh(c + d*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a*b*c**2 + 2*int(aco 
sh(c + d*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a*b*c*d*x + int(acosh(c + d*x) 
**2/(c**2 + 2*c*d*x + d**2*x**2),x)*b**2*c**2 + int(acosh(c + d*x)**2/(c** 
2 + 2*c*d*x + d**2*x**2),x)*b**2*c*d*x + a**2*x)/(c*e**2*(c + d*x))