\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^2} \, dx\) [37]

Optimal result

Integrand size = 25, antiderivative size = 177 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=-\frac {b x^2}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{2 c^5 d^2}+\frac {3 x (a+b \text {arccosh}(c x))}{2 c^4 d^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d^2}-\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2}+\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 c^5 d^2} \] Output:

-1/2*b*x^2/c^3/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*b*(c*x-1)^(1/2)*(c*x+1) 
^(1/2)/c^5/d^2+3/2*x*(a+b*arccosh(c*x))/c^4/d^2+1/2*x^3*(a+b*arccosh(c*x)) 
/c^2/d^2/(-c^2*x^2+1)-3*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))/c^5/d^2-3/2*b*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^ 
2+3/2*b*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d^2
 

Mathematica [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^2} \, dx=\frac {4 a c x-3 b \sqrt {\frac {-1+c x}{1+c x}}-4 b c x \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a c x}{-1+c^2 x^2}+4 b c x \text {arccosh}(c x)+\frac {b \text {arccosh}(c x)}{1-c x}-\frac {b \text {arccosh}(c x)}{1+c x}+6 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-6 b \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+3 a \log (1-c x)-3 a \log (1+c x)-6 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+6 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c^5 d^2} \] Input:

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

Output:

(4*a*c*x - 3*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*b*c*x*Sqrt[(-1 + c*x)/(1 + c 
*x)] + (b*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + (b*c*x*Sqrt[(-1 + c*x)/( 
1 + c*x)])/(1 - c*x) - (2*a*c*x)/(-1 + c^2*x^2) + 4*b*c*x*ArcCosh[c*x] + ( 
b*ArcCosh[c*x])/(1 - c*x) - (b*ArcCosh[c*x])/(1 + c*x) + 6*b*ArcCosh[c*x]* 
Log[1 - E^ArcCosh[c*x]] - 6*b*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 3*a*L 
og[1 - c*x] - 3*a*Log[1 + c*x] - 6*b*PolyLog[2, -E^ArcCosh[c*x]] + 6*b*Pol 
yLog[2, E^ArcCosh[c*x]])/(4*c^5*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.88 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {6349, 27, 109, 27, 83, 6353, 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.

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

\(\Big \downarrow \) 6349

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(b*(-(x^2/(c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (2*Sqrt[-1 + c*x]*Sqrt[1 + 
 c*x])/c^4))/(2*c*d^2) + (x^3*(a + b*ArcCosh[c*x]))/(2*c^2*d^2*(1 - c^2*x^ 
2)) - (3*((b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^3 - (x*(a + b*ArcCosh[c*x]))/ 
c^2 - (I*((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^3))/(2*c^2*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 109

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

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 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]
 

rule 6353

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/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 
*p + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] 
- Simp[b*f*(n/(c*(m + 2*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*Ar 
cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* 
d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c^{5}}\)	\(254\)
default	\(\frac {\frac {a \left (c x -\frac {1}{4 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {3 b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c^{5}}\)	\(254\)
parts	\(\frac {a \left (\frac {x}{c^{4}}-\frac {1}{4 c^{5} \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{4 c^{5}}-\frac {1}{4 c^{5} \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{4 c^{5}}\right )}{d^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{d^{2} c^{4}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{c^{5} d^{2}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{2 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2} c^{5}}-\frac {3 b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}+\frac {3 b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2} c^{5}}+\frac {3 b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{5} d^{2}}\)	\(286\)

Input:

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

Output:

1/c^5*(a/d^2*(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*(c*x-1)^(1/2)*(c*x+1)^(1/2)+b/d^2*arccosh(c*x)*c*x-1/2*b/d^2/(c^2*x^2-1 
)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/2*b/d^2/(c^2*x^2-1)*arccosh(c*x)*c*x-3/2*b 
/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/2*b/d^2*polylog( 
2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/2*b/d^2*arccosh(c*x)*ln(1-c*x-(c*x-1 
)^(1/2)*(c*x+1)^(1/2))+3/2*b/d^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) 
))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/64*(16*c^4*(2*x/(c^10*d^2*x^2 - c^8*d^2) - 4*x/(c^8*d^2) + 3*log(c*x + 1 
)/(c^9*d^2) - 3*log(c*x - 1)/(c^9*d^2)) - 576*c^3*integrate(1/8*x^3*log(c* 
x - 1)/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x) - 24*c^2*(2*x/(c^8*d^2* 
x^2 - c^6*d^2) + log(c*x + 1)/(c^7*d^2) - log(c*x - 1)/(c^7*d^2)) + 192*c^ 
2*integrate(1/8*x^2*log(c*x - 1)/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), 
x) - 9*(c*(2/(c^8*d^2*x - c^7*d^2) - log(c*x + 1)/(c^7*d^2) + log(c*x - 1) 
/(c^7*d^2)) + 4*log(c*x - 1)/(c^8*d^2*x^2 - c^6*d^2))*c + 4*(3*(c^2*x^2 - 
1)*log(c*x + 1)^2 + 6*(c^2*x^2 - 1)*log(c*x + 1)*log(c*x - 1) + 4*(4*c^3*x 
^3 - 6*c*x - 3*(c^2*x^2 - 1)*log(c*x + 1) + 3*(c^2*x^2 - 1)*log(c*x - 1))* 
log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^7*d^2*x^2 - c^5*d^2) - 64*integ 
rate(-1/4*(4*c^3*x^3 - 6*c*x - 3*(c^2*x^2 - 1)*log(c*x + 1) + 3*(c^2*x^2 - 
 1)*log(c*x - 1))/(c^9*d^2*x^5 - 2*c^7*d^2*x^3 + c^5*d^2*x + (c^8*d^2*x^4 
- 2*c^6*d^2*x^2 + c^4*d^2)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) - 192*integrat 
e(1/8*log(c*x - 1)/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x))*b - 1/4*a* 
(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c^5*d^2) - 
3*log(c*x - 1)/(c^5*d^2))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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