\(\int \frac {a+b \text {arccosh}(c x)}{x^2 (d-c^2 d x^2)^2} \, dx\) [43]

Optimal result

Integrand size = 25, antiderivative size = 158 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{d^2 x}+\frac {c^2 x (a+b \text {arccosh}(c x))}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b c \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {3 c (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{d^2}+\frac {3 b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 d^2}-\frac {3 b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d^2} \] Output:

-1/2*b*c/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-(a+b*arccosh(c*x))/d^2/x+1/2*c^2* 
x*(a+b*arccosh(c*x))/d^2/(-c^2*x^2+1)+b*c*arctan((c*x-1)^(1/2)*(c*x+1)^(1/ 
2))/d^2+3*c*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^ 
2+3/2*b*c*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2-3/2*b*c*polylog( 
2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2
 

Mathematica [A] (warning: unable to verify)

Time = 0.50 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.79 \[ \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {-\frac {4 a}{x}+b c \sqrt {\frac {-1+c x}{1+c x}}+\frac {b c \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c^2 x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a c^2 x}{-1+c^2 x^2}-\frac {4 b \text {arccosh}(c x)}{x}+\frac {b c \text {arccosh}(c x)}{1-c x}-\frac {b c \text {arccosh}(c x)}{1+c x}+\frac {4 b c \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-6 b c \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+6 b c \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )-3 a c \log (1-c x)+3 a c \log (1+c x)+6 b c \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-6 b c \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 d^2} \] Input:

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

Output:

((-4*a)/x + b*c*Sqrt[(-1 + c*x)/(1 + c*x)] + (b*c*Sqrt[(-1 + c*x)/(1 + c*x 
)])/(1 - c*x) + (b*c^2*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a*c^2* 
x)/(-1 + c^2*x^2) - (4*b*ArcCosh[c*x])/x + (b*c*ArcCosh[c*x])/(1 - c*x) - 
(b*c*ArcCosh[c*x])/(1 + c*x) + (4*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + 
c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 6*b*c*ArcCosh[c*x]*Log[1 - E^A 
rcCosh[c*x]] + 6*b*c*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] - 3*a*c*Log[1 - 
c*x] + 3*a*c*Log[1 + c*x] + 6*b*c*PolyLog[2, -E^ArcCosh[c*x]] - 6*b*c*Poly 
Log[2, E^ArcCosh[c*x]])/(4*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.24, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6347, 27, 115, 27, 103, 218, 6316, 83, 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.

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

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 6316

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {3 c^2 \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2}-\frac {a+b \text {arccosh}(c x)}{d^2 x \left (1-c^2 x^2\right )}-\frac {b c \left (-\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}\right )}{d^2}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((a + b*ArcCosh[c*x])/(d^2*x*(1 - c^2*x^2))) - (b*c*(-(1/(Sqrt[-1 + c*x]* 
Sqrt[1 + c*x])) - ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]))/d^2 + (3*c^2*(-1/ 
2*b/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(2*(1 - c^ 
2*x^2)) - ((I/2)*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]] + I*b 
*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/c))/d^2
 

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 83

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

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 115

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

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 6316

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + 
 c*x)^p*(-1 + c*x)^p)]   Int[x*(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}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
 

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 = 0.38 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(c \left (\frac {a \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c x}+2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {3 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\right )\)	\(207\)
default	\(c \left (\frac {a \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b \left (-\frac {3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c x}+2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {3 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\right )\)	\(207\)
parts	\(\frac {a \left (-\frac {1}{x}-\frac {c}{4 \left (c x +1\right )}+\frac {3 c \ln \left (c x +1\right )}{4}-\frac {c}{4 \left (c x -1\right )}-\frac {3 c \ln \left (c x -1\right )}{4}\right )}{d^{2}}+\frac {b c \left (-\frac {3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -2 \,\operatorname {arccosh}\left (c x \right )}{2 \left (c^{2} x^{2}-1\right ) c x}+2 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\frac {3 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2}\right )}{d^{2}}\)	\(207\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*arccosh(c*x) + a)/(c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2), x)
 

Sympy [F]

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

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

Output:

(Integral(a/(c**4*x**6 - 2*c**2*x**4 + x**2), x) + Integral(b*acosh(c*x)/( 
c**4*x**6 - 2*c**2*x**4 + x**2), x))/d**2
 

Maxima [F]

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

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

Output:

1/64*(576*c^5*integrate(1/8*x^3*log(c*x - 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 
+ d^2), x) - 24*c^4*(2*x/(c^4*d^2*x^2 - c^2*d^2) + log(c*x + 1)/(c^3*d^2) 
- log(c*x - 1)/(c^3*d^2)) - 192*c^4*integrate(1/8*x^2*log(c*x - 1)/(c^4*d^ 
2*x^4 - 2*c^2*d^2*x^2 + d^2), x) + 9*(c*(2/(c^4*d^2*x - c^3*d^2) - log(c*x 
 + 1)/(c^3*d^2) + log(c*x - 1)/(c^3*d^2)) + 4*log(c*x - 1)/(c^4*d^2*x^2 - 
c^2*d^2))*c^3 + 16*c^2*(2*x/(c^2*d^2*x^2 - d^2) - log(c*x + 1)/(c*d^2) + l 
og(c*x - 1)/(c*d^2)) + 192*c^2*integrate(1/8*log(c*x - 1)/(c^4*d^2*x^4 - 2 
*c^2*d^2*x^2 + d^2), x) - 4*(3*(c^3*x^3 - c*x)*log(c*x + 1)^2 + 6*(c^3*x^3 
 - c*x)*log(c*x + 1)*log(c*x - 1) + 4*(6*c^2*x^2 - 3*(c^3*x^3 - c*x)*log(c 
*x + 1) + 3*(c^3*x^3 - c*x)*log(c*x - 1) - 4)*log(c*x + sqrt(c*x + 1)*sqrt 
(c*x - 1)))/(c^2*d^2*x^3 - d^2*x) + 64*integrate(-1/4*(6*c^3*x^2 - 3*(c^4* 
x^3 - c^2*x)*log(c*x + 1) + 3*(c^4*x^3 - c^2*x)*log(c*x - 1) - 4*c)/(c^5*d 
^2*x^6 - 2*c^3*d^2*x^4 + c*d^2*x^2 + (c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x) 
*sqrt(c*x + 1)*sqrt(c*x - 1)), x))*b - 1/4*a*(2*(3*c^2*x^2 - 2)/(c^2*d^2*x 
^3 - d^2*x) - 3*c*log(c*x + 1)/d^2 + 3*c*log(c*x - 1)/d^2)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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