\(\int \frac {x^5 (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [404]

Optimal result

Integrand size = 21, antiderivative size = 727 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c d x \sqrt {-1+c x} \sqrt {1+c x}}{8 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3} \] Output:

-1/8*b*c*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/e^2/(c^2*d+e)/(e*x^2+d)-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-b*c*d^(1/2)*(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))/e^3/(c^2*d+e)^(1/2)/(c*x-1)^(1/2)/(c*x+ 
1)^(1/2)+1/8*b*c*d^(1/2)*(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))/e^3/(c^2*d+e)^(3/2)/(c*x-1)^(1/2)/(c*x+1 
)^(1/2)+1/2*(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)))/e^3+1/2*(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)))/e^3+ 
1/2*(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)))/e^3+1/2*(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)))/e^3+1/2*b*po 
lylog(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)))/e^3+1/2*b*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)))/e^3+1/2*b*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)))/e^3+1/2*b*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)))/ 
e^3
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.96 (sec) , antiderivative size = 1097, normalized size of antiderivative = 1.51 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] 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))])/(...
 

Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.01, 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)
 

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 = 2.53 (sec) , antiderivative size = 3551, normalized size of antiderivative = 4.88

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*d^2*c^4/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*(3/4*(2*c^2*d-2*(c^2*d*(c^2*d+e) 
)^(1/2)+e)/e^4/(c^2*d+e)*c^2*d*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 
))^2/(-2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)-e))+3/4*(c^2*d*(c^2*d+e))^(1/2)/e 
^3/(c^2*d+e)^2*c^2*d*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))-1/8*(c^2*d*(c^2*d+e))^(1/2)/e/(c^2*d 
+e)^2/c^2/d*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(c 
^2*d*(c^2*d+e))^(1/2)-e))-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)/e^5/(c^2*d 
+e)*c^4*d^2*arccosh(c*x)^2-3/2*(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)/e^4/( 
c^2*d+e)*c^2*d*arccosh(c*x)^2-1/4*(c^2*d*(c^2*d+e))^(1/2)/e^3/(c^2*d+e)^2* 
c^2*d*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(c^2*d*( 
c^2*d+e))^(1/2)-e))+1/8*d*c^2*(6*c^4*d^2*arccosh(c*x)+8*arccosh(c*x)*c^4*d 
*e*x^2-c^3*d*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)-c^3*e^2*x^3*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)+c^4*d^2+2*c^4*d*e*x^2+c^4*e^2*x^4+6*c^2*d*e*arccosh(c*x)+8*arcc 
osh(c*x)*e^2*c^2*x^2)/e^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)-(-2*(c^2*d*(c^2*d+ 
e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(c^2*d*(c^2*d+e))^(1/2)*e)*c^2*d/e^4/( 
c^4*d^2+2*c^2*d*e+e^2)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2 
*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)-e))-1/8*(-2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d 
+2*c^4*d^2+2*c^2*d*e-(c^2*d*(c^2*d+e))^(1/2)*e)/c^2/d/e^2/(c^4*d^2+2*c^2*d 
*e+e^2)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(c^...
 

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)
 

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
 

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)
 

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
                                                                                    
                                                                                    
 

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)
 

Reduce [F]

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

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

Output:

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