\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx\) [28]

Optimal result

Integrand size = 25, antiderivative size = 158 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x (a+b \text {arccosh}(c x))}{c^4 d}-\frac {x^3 (a+b \text {arccosh}(c x))}{3 c^2 d}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^5 d}+\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^5 d}-\frac {b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^5 d} \] Output:

11/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d+1/9*b*x^2*(c*x-1)^(1/2)*(c*x+1)^( 
1/2)/c^3/d-x*(a+b*arccosh(c*x))/c^4/d-1/3*x^3*(a+b*arccosh(c*x))/c^2/d+2*( 
a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d+b*polylog 
(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^5/d-b*polylog(2,c*x+(c*x-1)^(1/2)*( 
c*x+1)^(1/2))/c^5/d
 

Mathematica [A] (warning: unable to verify)

Time = 0.37 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=-\frac {18 a c x+6 a c^3 x^3-18 b \sqrt {\frac {-1+c x}{1+c x}}-18 b c x \sqrt {\frac {-1+c x}{1+c x}}-4 b \sqrt {-1+c x} \sqrt {1+c x}-2 b c^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}+18 b c x \text {arccosh}(c x)+6 b c^3 x^3 \text {arccosh}(c x)-9 b \text {arccosh}(c x)^2-18 b \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+18 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+9 a \log (1-c x)-9 a \log (1+c x)+18 b \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )+18 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{18 c^5 d} \] Input:

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

Output:

-1/18*(18*a*c*x + 6*a*c^3*x^3 - 18*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 18*b*c*x 
*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^2*x 
^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 18*b*c*x*ArcCosh[c*x] + 6*b*c^3*x^3*ArcC 
osh[c*x] - 9*b*ArcCosh[c*x]^2 - 18*b*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x] 
)] + 18*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 9*a*Log[1 - c*x] - 9*a*Lo 
g[1 + c*x] + 18*b*PolyLog[2, -E^(-ArcCosh[c*x])] + 18*b*PolyLog[2, E^ArcCo 
sh[c*x]])/(c^5*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.19 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.18, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {6353, 27, 111, 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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(b*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 
+ c*x])/(3*c^2)))/(3*c*d) - (x^3*(a + b*ArcCosh[c*x]))/(3*c^2*d) + ((b*Sqr 
t[-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)/(c^2*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 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 111

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

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 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 = 0.35 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.47

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{3} x^{3}}{3}+c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}}{9 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{3} x^{3}}{3 d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{5}}\)	\(232\)
default	\(\frac {-\frac {a \left (\frac {c^{3} x^{3}}{3}+c x +\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c x}{d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}}{9 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{3} x^{3}}{3 d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{5}}\)	\(232\)
parts	\(-\frac {a \left (\frac {\frac {1}{3} c^{2} x^{3}+x}{c^{4}}-\frac {\ln \left (c x +1\right )}{2 c^{5}}+\frac {\ln \left (c x -1\right )}{2 c^{5}}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{3}}{3 d \,c^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{5}}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{5} d}+\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}+\frac {b \,x^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3} d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{5}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x}{d \,c^{4}}\)	\(254\)

Input:

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

Output:

1/c^5*(-a/d*(1/3*c^3*x^3+c*x+1/2*ln(c*x-1)-1/2*ln(c*x+1))+11/9*b/d*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)-b/d*arccosh(c*x)*c*x+1/9*b/d*(c*x+1)^(1/2)*(c*x-1)^(1 
/2)*c^2*x^2-1/3*b/d*arccosh(c*x)*c^3*x^3+b/d*polylog(2,-c*x-(c*x-1)^(1/2)* 
(c*x+1)^(1/2))-b/d*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-b/d*arccosh( 
c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+b/d*arccosh(c*x)*ln(1+c*x+(c*x- 
1)^(1/2)*(c*x+1)^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/72*(4*c^4*(2*(c^2*x^3 + 3*x)/(c^8*d) - 3*log(c*x + 1)/(c^9*d) + 3*log(c* 
x - 1)/(c^9*d)) + 36*c^2*(2*x/(c^6*d) - log(c*x + 1)/(c^7*d) + log(c*x - 1 
)/(c^7*d)) + 648*c*integrate(1/12*x*log(c*x - 1)/(c^6*d*x^2 - c^4*d), x) - 
 3*(4*(2*c^3*x^3 + 6*c*x - 3*log(c*x + 1) + 3*log(c*x - 1))*log(c*x + sqrt 
(c*x + 1)*sqrt(c*x - 1)) + 3*log(c*x + 1)^2 + 6*log(c*x + 1)*log(c*x - 1)) 
/(c^5*d) + 72*integrate(-1/6*(2*c^3*x^3 + 6*c*x - 3*log(c*x + 1) + 3*log(c 
*x - 1))/(c^7*d*x^3 - c^5*d*x + (c^6*d*x^2 - c^4*d)*sqrt(c*x + 1)*sqrt(c*x 
 - 1)), x) - 216*integrate(1/12*log(c*x - 1)/(c^6*d*x^2 - c^4*d), x))*b - 
1/6*a*(2*(c^2*x^3 + 3*x)/(c^4*d) - 3*log(c*x + 1)/(c^5*d) + 3*log(c*x - 1) 
/(c^5*d))
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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