\(\int \frac {a+b \text {arccosh}(c x)}{x^2 (d-c^2 d x^2)} \, dx\) [34]

Optimal result

Integrand size = 25, antiderivative size = 95 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {a+b \text {arccosh}(c x)}{d x}+\frac {b c \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d}+\frac {b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{d}-\frac {b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d} \] Output:

-(a+b*arccosh(c*x))/d/x+b*c*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d+2*c*(a+b 
*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d+b*c*polylog(2,-c 
*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/d-b*c*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^ 
(1/2))/d
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.39 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=\frac {-\frac {a+b \text {arccosh}(c x)}{x}+\frac {b c \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-c (a+b \text {arccosh}(c x)) \log \left (1-e^{\text {arccosh}(c x)}\right )+c (a+b \text {arccosh}(c x)) \log \left (1+e^{\text {arccosh}(c x)}\right )+b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{d} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6347, 27, 103, 218, 6318, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c \int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d}-\frac {a+b \text {arccosh}(c x)}{d x}+\frac {b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i c \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{d}-\frac {a+b \text {arccosh}(c x)}{d x}+\frac {b c \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d}\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((a + b*ArcCosh[c*x])/(d*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]) 
/d - (I*c*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b*PolyLo 
g[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/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 27

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

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6318

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[-(c*d)^(-1)   Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x 
]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 6347

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 
)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp 
[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*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*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.51

method	result	size
parts	\(-\frac {a \left (-\frac {c \ln \left (c x +1\right )}{2}+\frac {c \ln \left (c x -1\right )}{2}+\frac {1}{x}\right )}{d}-\frac {b c \left (\frac {\operatorname {arccosh}\left (c x \right )}{c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(143\)
derivativedivides	\(c \left (-\frac {a \left (\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right )}{c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(146\)
default	\(c \left (-\frac {a \left (\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right )}{c x}-2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(146\)

Input:

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

Output:

-a/d*(-1/2*c*ln(c*x+1)+1/2*c*ln(c*x-1)+1/x)-b/d*c*(arccosh(c*x)/c/x-2*arct 
an(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-dilog(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) 
-dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-arccosh(c*x)*ln(1+c*x+(c*x-1)^(1 
/2)*(c*x+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral(-(b*arccosh(c*x) + a)/(c^2*d*x^4 - d*x^2), x)
 

Sympy [F]

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

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

Output:

-(Integral(a/(c**2*x**4 - x**2), x) + Integral(b*acosh(c*x)/(c**2*x**4 - x 
**2), x))/d
 

Maxima [F]

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

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

Output:

1/8*(24*c^3*integrate(1/4*x*log(c*x - 1)/(c^2*d*x^2 - d), x) - 4*c^2*(log( 
c*x + 1)/(c*d) - log(c*x - 1)/(c*d)) - 8*c^2*integrate(1/4*log(c*x - 1)/(c 
^2*d*x^2 - d), x) - (c*x*log(c*x + 1)^2 + 2*c*x*log(c*x + 1)*log(c*x - 1) 
- 4*(c*x*log(c*x + 1) - c*x*log(c*x - 1) - 2)*log(c*x + sqrt(c*x + 1)*sqrt 
(c*x - 1)))/(d*x) + 8*integrate(1/2*(c^2*x*log(c*x + 1) - c^2*x*log(c*x - 
1) - 2*c)/(c^3*d*x^4 - c*d*x^2 + (c^2*d*x^3 - d*x)*sqrt(c*x + 1)*sqrt(c*x 
- 1)), x))*b + 1/2*a*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*int(acosh(c*x)/(c**2*x**4 - x**2),x)*b*x - log(c**2*x - c)*a*c*x + l 
og(c**2*x + c)*a*c*x - 2*a)/(2*d*x)