\(\int \frac {x^2 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^3} \, dx\) [48]

Optimal result

Integrand size = 25, antiderivative size = 186 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b}{12 c^3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{8 c^3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {x (a+b \text {arccosh}(c x))}{8 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {(a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c^3 d^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c^3 d^3}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c^3 d^3} \] Output:

1/12*b/c^3/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+1/8*b/c^3/d^3/(c*x-1)^(1/2)/(c* 
x+1)^(1/2)+1/4*x*(a+b*arccosh(c*x))/c^2/d^3/(-c^2*x^2+1)^2-1/8*x*(a+b*arcc 
osh(c*x))/c^2/d^3/(-c^2*x^2+1)-1/4*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^ 
(1/2)*(c*x+1)^(1/2))/c^3/d^3-1/8*b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1 
/2))/c^3/d^3+1/8*b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^3/d^3
 

Mathematica [A] (warning: unable to verify)

Time = 1.08 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.54 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {b (-2+c x) \sqrt {1+c x}}{(-1+c x)^{3/2}}+\frac {b \sqrt {-1+c x} (2+c x)}{(1+c x)^{3/2}}+\frac {12 a c x}{\left (-1+c^2 x^2\right )^2}+\frac {6 a c x}{-1+c^2 x^2}+\frac {3 b \text {arccosh}(c x)}{(-1+c x)^2}-\frac {3 b \text {arccosh}(c x)}{(1+c x)^2}-3 b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )-3 b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )-\frac {3}{2} b \text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )+\frac {3}{2} b \text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )+3 a \log (1-c x)-3 a \log (1+c x)-6 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+6 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{48 c^3 d^3} \] Input:

Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]
 

Output:

(-((b*(-2 + c*x)*Sqrt[1 + c*x])/(-1 + c*x)^(3/2)) + (b*Sqrt[-1 + c*x]*(2 + 
 c*x))/(1 + c*x)^(3/2) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (6*a*c*x)/(-1 + c^2 
*x^2) + (3*b*ArcCosh[c*x])/(-1 + c*x)^2 - (3*b*ArcCosh[c*x])/(1 + c*x)^2 - 
 3*b*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) - 3*b*(Sqr 
t[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) - (3*b*ArcCosh[c*x]*(Arc 
Cosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]))/2 + (3*b*ArcCosh[c*x]*(ArcCosh[c*x 
] - 4*Log[1 + E^ArcCosh[c*x]]))/2 + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] - 
6*b*PolyLog[2, -E^ArcCosh[c*x]] + 6*b*PolyLog[2, E^ArcCosh[c*x]])/(48*c^3* 
d^3)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {6349, 27, 83, 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.

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

\(\Big \downarrow \) 6349

\(\displaystyle -\frac {\int \frac {a+b \text {arccosh}(c x)}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 c^2 d}-\frac {b \int \frac {x}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 c d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}-\frac {b \int \frac {x}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{4 c d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}\)

\(\Big \downarrow \) 83

\(\displaystyle -\frac {\int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 6316

\(\displaystyle -\frac {\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 83

\(\displaystyle -\frac {\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 6318

\(\displaystyle -\frac {-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}}{4 c^2 d^3}+\frac {x (a+b \text {arccosh}(c x))}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}\)

Input:

Int[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]
 

Output:

b/(12*c^3*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (x*(a + b*ArcCosh[c*x])) 
/(4*c^2*d^3*(1 - c^2*x^2)^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*Poly 
Log[2, E^ArcCosh[c*x]]))/c)/(4*c^2*d^3)
 

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 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 6349

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+3 c x \,\operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c^{3}}\)	\(256\)
default	\(\frac {-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}-\frac {1}{16 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}-\frac {1}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+3 c x \,\operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c^{3}}\)	\(256\)
parts	\(-\frac {a \left (\frac {1}{16 c^{3} \left (c x +1\right )^{2}}-\frac {1}{16 c^{3} \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16 c^{3}}-\frac {1}{16 c^{3} \left (c x -1\right )^{2}}-\frac {1}{16 c^{3} \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{16 c^{3}}\right )}{d^{3}}-\frac {b \left (-\frac {3 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}+3 c x \,\operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3} c^{3}}\)	\(273\)

Input:

int(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/c^3*(-a/d^3*(-1/16/(c*x-1)^2-1/16/(c*x-1)-1/16*ln(c*x-1)+1/16/(c*x+1)^2- 
1/16/(c*x+1)+1/16*ln(c*x+1))-b/d^3*(-1/24*(3*c^3*x^3*arccosh(c*x)+3*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)*c^2*x^2+3*c*x*arccosh(c*x)-(c*x-1)^(1/2)*(c*x+1)^(1/ 
2))/(c^4*x^4-2*c^2*x^2+1)+1/8*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^ 
(1/2))+1/8*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-1/8*arccosh(c*x)*ln 
(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-1/8*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1 
)^(1/2))))
 

Fricas [F]

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

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral(-(b*x^2*arccosh(c*x) + a*x^2)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^ 
2*d^3*x^2 - d^3), x)
 

Sympy [F]

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

integrate(x**2*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**3,x)
 

Output:

-(Integral(a*x**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integr 
al(b*x**2*acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d**3
 

Maxima [F]

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

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
 

Output:

-1/2048*(6144*c^5*integrate(1/32*x^5*log(c*x - 1)/(c^8*d^3*x^6 - 3*c^6*d^3 
*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), x) - 16*c^4*(2*(5*c^2*x^3 - 3*x)/(c^10*d^ 
3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3) + 3*log(c*x + 1)/(c^7*d^3) - 3*log(c*x - 
1)/(c^7*d^3)) - 2048*c^4*integrate(1/32*x^4*log(c*x - 1)/(c^8*d^3*x^6 - 3* 
c^6*d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), x) + 6*(c*(2*(5*c^2*x^2 + 3*c*x - 
6)/(c^10*d^3*x^3 - c^9*d^3*x^2 - c^8*d^3*x + c^7*d^3) - 5*log(c*x + 1)/(c^ 
7*d^3) + 5*log(c*x - 1)/(c^7*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1)/(c^10 
*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3))*c^3 - 16*c^2*(2*(c^2*x^3 + x)/(c^8*d^ 
3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) - log(c*x + 1)/(c^5*d^3) + log(c*x - 1)/( 
c^5*d^3)) + 4096*c^2*integrate(1/32*x^2*log(c*x - 1)/(c^8*d^3*x^6 - 3*c^6* 
d^3*x^4 + 3*c^4*d^3*x^2 - c^2*d^3), x) + 3*(c*(2*(3*c^2*x^2 - 3*c*x - 2)/( 
c^8*d^3*x^3 - c^7*d^3*x^2 - c^6*d^3*x + c^5*d^3) - 3*log(c*x + 1)/(c^5*d^3 
) + 3*log(c*x - 1)/(c^5*d^3)) - 16*log(c*x - 1)/(c^8*d^3*x^4 - 2*c^6*d^3*x 
^2 + c^4*d^3))*c - 32*((c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1)^2 + 2*(c^4*x 
^4 - 2*c^2*x^2 + 1)*log(c*x + 1)*log(c*x - 1) + 4*(2*c^3*x^3 + 2*c*x - (c^ 
4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x - 
1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^7*d^3*x^4 - 2*c^5*d^3*x^2 + 
 c^3*d^3) + 2048*integrate(-1/16*(2*c^3*x^3 + 2*c*x - (c^4*x^4 - 2*c^2*x^2 
 + 1)*log(c*x + 1) + (c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x - 1))/(c^9*d^3*x^7 
- 3*c^7*d^3*x^5 + 3*c^5*d^3*x^3 - c^3*d^3*x + (c^8*d^3*x^6 - 3*c^6*d^3*...
 

Giac [F]

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

integrate(x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate(-(b*arccosh(c*x) + a)*x^2/(c^2*d*x^2 - d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:

int((x^2*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^3,x)
 

Output:

int((x^2*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-16 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{7} x^{4}+32 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{5} x^{2}-16 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b \,c^{3}+\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}+\mathrm {log}\left (c^{2} x -c \right ) a -\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) a +2 a \,c^{3} x^{3}+2 a c x}{16 c^{3} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:

int(x^2*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^3,x)
 

Output:

( - 16*int((acosh(c*x)*x**2)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x 
)*b*c**7*x**4 + 32*int((acosh(c*x)*x**2)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2 
*x**2 - 1),x)*b*c**5*x**2 - 16*int((acosh(c*x)*x**2)/(c**6*x**6 - 3*c**4*x 
**4 + 3*c**2*x**2 - 1),x)*b*c**3 + log(c**2*x - c)*a*c**4*x**4 - 2*log(c** 
2*x - c)*a*c**2*x**2 + log(c**2*x - c)*a - log(c**2*x + c)*a*c**4*x**4 + 2 
*log(c**2*x + c)*a*c**2*x**2 - log(c**2*x + c)*a + 2*a*c**3*x**3 + 2*a*c*x 
)/(16*c**3*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))