\(\int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx\) [29]

Optimal result

Integrand size = 25, antiderivative size = 140 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c^3 d}+\frac {b \text {arccosh}(c x)}{4 c^4 d}-\frac {x^2 (a+b \text {arccosh}(c x))}{2 c^2 d}+\frac {(a+b \text {arccosh}(c x))^2}{2 b c^4 d}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c^4 d}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 c^4 d} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=-\frac {2 c^2 x^2 (a+b \text {arccosh}(c x))-\frac {2 (a+b \text {arccosh}(c x))^2}{b}-b \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+4 (a+b \text {arccosh}(c x)) \log \left (1-e^{\text {arccosh}(c x)}\right )+4 (a+b \text {arccosh}(c x)) \log \left (1+e^{\text {arccosh}(c x)}\right )+4 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+4 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c^4 d} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {6353, 27, 101, 43, 6328, 3042, 26, 4199, 25, 2620, 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.

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

\(\Big \downarrow \) 6353

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6328

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/2*(x^2*(a + b*ArcCosh[c*x]))/(c^2*d) + (b*((x*Sqrt[-1 + c*x]*Sqrt[1 + c 
*x])/(2*c^2) + ArcCosh[c*x]/(2*c^3)))/(2*c*d) + (I*(((-1/2*I)*(a + b*ArcCo 
sh[c*x])^2)/b - (2*I)*(-1/2*((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x 
])]) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/4)))/(c^4*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6328

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Coth[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 = 0.29 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(\frac {-\frac {a \left (\frac {c^{2} x^{2}}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (c x \right )}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{4}}\)	\(217\)
default	\(\frac {-\frac {a \left (\frac {c^{2} x^{2}}{2}+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}+\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 d}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (c x \right )}{4 d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{4}}\)	\(217\)
parts	\(-\frac {a \,x^{2}}{2 d \,c^{2}}-\frac {a \ln \left (c^{2} x^{2}-1\right )}{2 d \,c^{4}}+\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 d \,c^{4}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{2}}{2 d \,c^{2}}+\frac {b x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c^{3} d}+\frac {b \,\operatorname {arccosh}\left (c x \right )}{4 c^{4} d}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}-\frac {b \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}-\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}-\frac {b \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d \,c^{4}}\)	\(233\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*a*(x^2/(c^2*d) + log(c^2*x^2 - 1)/(c^4*d)) + 1/8*b*((2*c^2*x^2 - 4*(c 
^2*x^2 + log(c*x + 1) + log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1 
)) + 2*(log(c*x - 1) + 1)*log(c*x + 1) + log(c*x + 1)^2 + log(c*x - 1)^2 + 
 2*log(c*x - 1))/(c^4*d) - 8*integrate(1/2*(c^2*x^2 + log(c*x + 1) + log(c 
*x - 1))/(c^6*d*x^3 - c^4*d*x + (c^5*d*x^2 - c^3*d)*e^(1/2*log(c*x + 1) + 
1/2*log(c*x - 1))), x))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d-c^2 d x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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

Output:

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