\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d+e x^2)^3} \, dx\) [408]

Optimal result

Integrand size = 21, antiderivative size = 805 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}-\frac {b c e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \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 )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c e \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 d^{7/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}+\frac {3 e (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 d^4}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{2 \text {arccosh}(c x)}\right )}{d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}-\frac {3 b e \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^4} \] Output:

1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/x-1/8*b*c*e^2*x*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)/d^3/(c^2*d+e)/(e*x^2+d)-1/2*(a+b*arccosh(c*x))/d^3/x^2-1/4*e*(a+ 
b*arccosh(c*x))/d^2/(e*x^2+d)^2-e*(a+b*arccosh(c*x))/d^3/(e*x^2+d)+b*c*e*( 
c^2*x^2-1)^(1/2)*arctanh((c^2*d+e)^(1/2)*x/d^(1/2)/(c^2*x^2-1)^(1/2))/d^(7 
/2)/(c^2*d+e)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/8*b*c*e*(2*c^2*d+e)*(c^2 
*x^2-1)^(1/2)*arctanh((c^2*d+e)^(1/2)*x/d^(1/2)/(c^2*x^2-1)^(1/2))/d^(7/2) 
/(c^2*d+e)^(3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/2*e*(a+b*arccosh(c*x))*ln(1 
-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)) 
)/d^4+3/2*e*(a+b*arccosh(c*x))*ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 
2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/d^4+3/2*e*(a+b*arccosh(c*x))*ln(1-e^( 
1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^ 
4+3/2*e*(a+b*arccosh(c*x))*ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/ 
(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^4-3*e*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x 
-1)^(1/2)*(c*x+1)^(1/2))^2)/d^4+3/2*b*e*polylog(2,-e^(1/2)*(c*x+(c*x-1)^(1 
/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/d^4+3/2*b*e*polylog(2, 
e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2))) 
/d^4+3/2*b*e*polylog(2,-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^ 
(1/2)+(-c^2*d-e)^(1/2)))/d^4+3/2*b*e*polylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)* 
(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^4-3/2*b*e*polylog(2,-(c* 
x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^4
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.07 (sec) , antiderivative size = 1261, normalized size of antiderivative = 1.57 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*a/(d^3*x^2) - (a*e)/(4*d^2*(d + e*x^2)^2) - (a*e)/(d^3*(d + e*x^2)) - 
 (3*a*e*Log[x])/d^4 + (3*a*e*Log[d + e*x^2])/(2*d^4) + b*((c*x*Sqrt[-1 + c 
*x]*Sqrt[1 + c*x] - ArcCosh[c*x])/(2*d^3*x^2) + (((9*I)/16)*e*(ArcCosh[c*x 
]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*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]*(S 
qrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/d^(7/2) + (((9*I)/16)*e*(-(A 
rcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*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]))/d^(7/2) - (e^(3/2)*(( 
c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - 
 ArcCosh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4 
] + 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))]))/(Sqrt[e 
]*(c^2*d + e)^(3/2))))/(16*d^3) - (e^(3/2)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d 
] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sq 
rt[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))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*d^3) - ( 
3*e*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLo 
g[2, -E^(-2*ArcCosh[c*x])]))/(2*d^4) + (3*e*(ArcCosh[c*x]*(-ArcCosh[c*x...
 

Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b c x \left (1-c^2 x^2\right ) e^2}{8 d^3 \left (d c^2+e\right ) \sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )}-\frac {3 (a+b \text {arccosh}(c x))^2 e}{b d^4}-\frac {(a+b \text {arccosh}(c x)) e}{d^3 \left (e x^2+d\right )}-\frac {(a+b \text {arccosh}(c x)) e}{4 d^2 \left (e x^2+d\right )^2}+\frac {b c \left (2 d c^2+e\right ) \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e} x}{\sqrt {d} \sqrt {c^2 x^2-1}}\right ) e}{8 d^{7/2} \left (d c^2+e\right )^{3/2} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e} x}{\sqrt {d} \sqrt {c^2 x^2-1}}\right ) e}{d^{7/2} \sqrt {d c^2+e} \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right ) e}{d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d^3 x}\)

Input:

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

Output:

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d^3*x) + (b*c*e^2*x*(1 - c^2*x^2))/( 
8*d^3*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (a + b*ArcCo 
sh[c*x])/(2*d^3*x^2) - (e*(a + b*ArcCosh[c*x]))/(4*d^2*(d + e*x^2)^2) - (e 
*(a + b*ArcCosh[c*x]))/(d^3*(d + e*x^2)) - (3*e*(a + b*ArcCosh[c*x])^2)/(b 
*d^4) + (b*c*e*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqr 
t[-1 + c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) 
+ (b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqr 
t[d]*Sqrt[-1 + c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqr 
t[1 + c*x]) - (3*e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^4 
+ (3*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - 
 Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e 
]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*e*(a + 
b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2* 
d) - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[ 
c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*b*e*PolyLog[2, -E^( 
-2*ArcCosh[c*x])])/(2*d^4) + (3*b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/ 
(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*d^4) + (3*b*e*PolyLog[2, (Sqrt[e]* 
E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*b*e*PolyL 
og[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*d 
^4) + (3*b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c...
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 1.08 (sec) , antiderivative size = 1455, normalized size of antiderivative = 1.81

method	result	size
parts	\(\text {Expression too large to display}\)	\(1455\)
derivativedivides	\(\text {Expression too large to display}\)	\(1515\)
default	\(\text {Expression too large to display}\)	\(1515\)

Input:

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

Output:

-a*e/d^3/(e*x^2+d)-1/4*a*e/d^2/(e*x^2+d)^2+3/2*a*e/d^4*ln(e*x^2+d)-1/2*a/d 
^3/x^2-3*a/d^4*e*ln(x)+b*c^2*(-1/8*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^7*d^3 
*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^7*d^2*e*x^3-4*(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)*c^7*d*e^2*x^5+4*c^8*d^3*x^2+8*c^8*d^2*e*x^4+4*c^8*d*e^2*x^6+4*c^6*d^3*a 
rccosh(c*x)+18*arccosh(c*x)*c^6*d^2*e*x^2+12*arccosh(c*x)*c^6*d*e^2*x^4-4* 
c^5*d^2*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)-7*c^5*d*e^2*x^3*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)-3*c^5*e^3*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)+3*c^6*d^2*e*x^2+6*c^6* 
d*e^2*x^4+3*c^6*e^3*x^6+4*c^4*d^2*e*arccosh(c*x)+18*arccosh(c*x)*c^4*d*e^2 
*x^2+12*arccosh(c*x)*e^3*c^4*x^4)/c^2/x^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2/d^ 
3-9/8*(c^2*d*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^4/c^2*e^2*arctanh(1/4*(4*c^2*d 
+2*e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))-5/4 
*(c^2*d*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^3*e*arctanh(1/4*(4*c^2*d+2*e*(c*x+( 
c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))+3/4/(c^2*d+e)/ 
d^4*e^2/c^2*sum((_R1^2*e+4*c^2*d+e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln(( 
_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c* 
x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-3/(c^2*d+e)/d^4 
*e^2/c^2*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e 
)/d^4*e^2/c^2*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^ 
2*d+e)/d^4*e^2/c^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e 
)/d^4*e^2/c^2*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3/4/(c^2*d+e...
 

Fricas [F]

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

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

Output:

integral((b*arccosh(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x 
^3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x 
^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*log(x)/d^4) + b*integrate(log(c*x + sq 
rt(c*x + 1)*sqrt(c*x - 1))/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^3) 
, x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{6} x^{2}+8 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{5} e \,x^{4}+4 \left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{4} e^{2} x^{6}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2} e \,x^{2}+12 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d \,e^{2} x^{4}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{3} x^{6}-12 \,\mathrm {log}\left (x \right ) a \,d^{2} e \,x^{2}-24 \,\mathrm {log}\left (x \right ) a d \,e^{2} x^{4}-12 \,\mathrm {log}\left (x \right ) a \,e^{3} x^{6}-2 a \,d^{3}-6 a \,d^{2} e \,x^{2}+3 a \,e^{3} x^{6}}{4 d^{4} x^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int(acosh(c*x)/(d**3*x**3 + 3*d**2*e*x**5 + 3*d*e**2*x**7 + e**3*x**9), 
x)*b*d**6*x**2 + 8*int(acosh(c*x)/(d**3*x**3 + 3*d**2*e*x**5 + 3*d*e**2*x* 
*7 + e**3*x**9),x)*b*d**5*e*x**4 + 4*int(acosh(c*x)/(d**3*x**3 + 3*d**2*e* 
x**5 + 3*d*e**2*x**7 + e**3*x**9),x)*b*d**4*e**2*x**6 + 6*log(d + e*x**2)* 
a*d**2*e*x**2 + 12*log(d + e*x**2)*a*d*e**2*x**4 + 6*log(d + e*x**2)*a*e** 
3*x**6 - 12*log(x)*a*d**2*e*x**2 - 24*log(x)*a*d*e**2*x**4 - 12*log(x)*a*e 
**3*x**6 - 2*a*d**3 - 6*a*d**2*e*x**2 + 3*a*e**3*x**6)/(4*d**4*x**2*(d**2 
+ 2*d*e*x**2 + e**2*x**4))