3.6.6 \(\int \frac {x^5 (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [506]

3.6.6.1 Optimal result
3.6.6.2 Mathematica [C] (warning: unable to verify)
3.6.6.3 Rubi [A] (verified)
3.6.6.4 Maple [C] (warning: unable to verify)
3.6.6.5 Fricas [F]
3.6.6.6 Sympy [F(-1)]
3.6.6.7 Maxima [F]
3.6.6.8 Giac [F(-2)]
3.6.6.9 Mupad [F(-1)]

3.6.6.1 Optimal result

Integrand size = 21, antiderivative size = 737 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3} \]

output
-1/4*d^2*(a+b*arccosh(c*x))/e^3/(e*x^2+d)^2+d*(a+b*arccosh(c*x))/e^3/(e*x^ 
2+d)-1/2*(a+b*arccosh(c*x))^2/b/e^3+1/2*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x- 
1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e^3+1/2*( 
a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^( 
1/2)-(-c^2*d-e)^(1/2)))/e^3+1/2*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2) 
*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e^3+1/2*(a+b*arcc 
osh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c 
^2*d-e)^(1/2)))/e^3+1/2*b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^( 
1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e^3+1/2*b*polylog(2,(c*x+(c*x-1)^(1/ 
2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e^3+1/2*b*polyl 
og(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^( 
1/2)))/e^3+1/2*b*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(- 
d)^(1/2)+(-c^2*d-e)^(1/2)))/e^3+1/8*b*c*d*x*(-c^2*x^2+1)/e^2/(c^2*d+e)/(e* 
x^2+d)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/8*b*c*(2*c^2*d+e)*arctanh(x*(c^2*d+e) 
^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*d^(1/2)*(c^2*x^2-1)^(1/2)/e^3/(c^2*d+e)^ 
(3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c*arctanh(x*(c^2*d+e)^(1/2)/d^(1/2)/(c 
^2*x^2-1)^(1/2))*d^(1/2)*(c^2*x^2-1)^(1/2)/e^3/(c^2*d+e)^(1/2)/(c*x-1)^(1/ 
2)/(c*x+1)^(1/2)
 
3.6.6.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.63 (sec) , antiderivative size = 1097, normalized size of antiderivative = 1.49 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {4 a d^2}{\left (d+e x^2\right )^2}+\frac {16 a d}{d+e x^2}+8 a \log \left (d+e x^2\right )+b \left (-\frac {c d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {c d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}+\frac {7 \sqrt {d} \text {arccosh}(c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {d \text {arccosh}(c x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {7 \sqrt {d} \text {arccosh}(c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {d \text {arccosh}(c x)}{\left (i \sqrt {d}+\sqrt {e} x\right )^2}-8 \text {arccosh}(c x)^2+8 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+8 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )-\frac {7 i c \sqrt {d} \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}+\frac {7 i c \sqrt {d} \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-\frac {c^3 d^{3/2} \log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}+\frac {c^3 d^{3/2} \log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}+8 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+8 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{16 e^3} \]

input
Integrate[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]
 
output
((-4*a*d^2)/(d + e*x^2)^2 + (16*a*d)/(d + e*x^2) + 8*a*Log[d + e*x^2] + b* 
(-((c*d*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + 
 Sqrt[e]*x))) - (c*d*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I 
*Sqrt[d] + Sqrt[e]*x)) + (7*Sqrt[d]*ArcCosh[c*x])/(Sqrt[d] - I*Sqrt[e]*x) 
- (d*ArcCosh[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (7*Sqrt[d]*ArcCosh[c*x])/(S 
qrt[d] + I*Sqrt[e]*x) + (d*ArcCosh[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 - 8*Arc 
Cosh[c*x]^2 + 8*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] 
 - Sqrt[-(c^2*d) - e])] + 8*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/ 
((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 8*ArcCosh[c*x]*Log[1 - (Sqrt[e]*E 
^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 8*ArcCosh[c*x]*Log[1 
+ (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] - ((7*I)*c* 
Sqrt[d]*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 
 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/S 
qrt[-(c^2*d) - e] + ((7*I)*c*Sqrt[d]*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x 
+ Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]* 
(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (c^3*d^(3/2)*Log[(e*Sqrt[c 
^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*S 
qrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e)^(3/2) + (c^3* 
d^(3/2)*Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d 
+ e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))])/(...
 
3.6.6.3 Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6374, 2009}

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.

\(\displaystyle \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^3}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{8 e^3 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right )^{3/2}}-\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{e^3 \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}+\frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )}\)

input
Int[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]
 
output
(b*c*d*x*(1 - c^2*x^2))/(8*e^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d 
 + e*x^2)) - (d^2*(a + b*ArcCosh[c*x]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b* 
ArcCosh[c*x]))/(e^3*(d + e*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*e^3) - (b*c 
*Sqrt[d]*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + 
 c^2*x^2])])/(e^3*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqr 
t[d]*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d] 
*Sqrt[-1 + c^2*x^2])])/(8*e^3*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c* 
x]) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - 
 Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ 
ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCos 
h[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]) 
])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqr 
t[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh 
[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[ 
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + (b*PolyLo 
g[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e^ 
3) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - 
e])])/(2*e^3)
 

3.6.6.3.1 Defintions of rubi rules used

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6374
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, 
 (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d 
 + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 
3.6.6.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.52 (sec) , antiderivative size = 3551, normalized size of antiderivative = 4.82

method result size
derivativedivides \(\text {Expression too large to display}\) \(3551\)
default \(\text {Expression too large to display}\) \(3551\)
parts \(\text {Expression too large to display}\) \(3556\)

input
int(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 
output
1/c^6*(a*c^6*(-1/4*c^4*d^2/e^3/(c^2*e*x^2+c^2*d)^2+1/2/e^3*ln(c^2*e*x^2+c^ 
2*d)+1/e^3*d*c^2/(c^2*e*x^2+c^2*d))+b*c^6*((2*c^2*d-2*(d*c^2*(c^2*d+e))^(1 
/2)+e)/e^5/(c^2*d+e)*c^4*d^2*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(- 
2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-1/4*(d*c^2*(c^2*d+e))^( 
1/2)/e/(c^2*d+e)^2/d/c^2*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1 
/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))-1/2*(d*c^2*(c^2*d+e))^(1/2) 
/e^3/(c^2*d+e)^2*d*c^2*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 
))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))+3/2*(2*c^2*d-2*(d*c^2*(c^2*d+ 
e))^(1/2)+e)/e^4/(c^2*d+e)*c^2*d*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^ 
2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-(-2*(d*c^2*(c^2*d+e 
))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^4*d^2/e^5/ 
(c^4*d^2+2*c^2*d*e+e^2)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2 
*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-2*(-2*(d*c^2*(c^2*d+e))^(1/2 
)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*d/e^4/(c^4*d^2+ 
2*c^2*d*e+e^2)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c 
^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-1/4*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d 
+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/c^2/d/e^2/(c^4*d^2+2*c^2*d 
*e+e^2)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2 
*d+e))^(1/2)-e))*arccosh(c*x)+(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+ 
2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^4*d^2/e^5/(c^4*d^2+2*c^2*d*e+e^2...
 
3.6.6.5 Fricas [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

input
integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 
output
integral((b*x^5*arccosh(c*x) + a*x^5)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 
 + d^3), x)
 
3.6.6.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

input
integrate(x**5*(a+b*acosh(c*x))/(e*x**2+d)**3,x)
 
output
Timed out
 
3.6.6.7 Maxima [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

input
integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
 
output
1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 
 + d)/e^3) + b*integrate(x^5*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e^3*x 
^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
 
3.6.6.8 Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]

input
integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,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
 
3.6.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

input
int((x^5*(a + b*acosh(c*x)))/(d + e*x^2)^3,x)
 
output
int((x^5*(a + b*acosh(c*x)))/(d + e*x^2)^3, x)