\(\int \frac {x^5 (a+b \text {sech}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [122]

Optimal result

Integrand size = 21, antiderivative size = 749 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b c d \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{8 e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \text {sech}^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {b \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b \left (c^2 d+2 e\right ) \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{2 e^3} \] Output:

-1/8*b*c*d*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/e^2/(c^2*d+e)/(e+d/x^2)/x-1/4* 
(a+b*arcsech(c*x))/e/(e+d/x^2)^2-1/2*(a+b*arcsech(c*x))/e^2/(e+d/x^2)+1/2* 
b*(-1+1/c^2/x^2)^(1/2)*arctanh((c^2*d+e)^(1/2)/c/e^(1/2)/(-1+1/c^2/x^2)^(1 
/2)/x)/e^(5/2)/(c^2*d+e)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+1/8*b*(c^2 
*d+2*e)*(-1+1/c^2/x^2)^(1/2)*arctanh((c^2*d+e)^(1/2)/c/e^(1/2)/(-1+1/c^2/x 
^2)^(1/2)/x)/e^(5/2)/(c^2*d+e)^(3/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+1/2* 
(a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/ 
2))/(e^(1/2)-(c^2*d+e)^(1/2)))/e^3+1/2*(a+b*arcsech(c*x))*ln(1+c*(-d)^(1/2 
)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/e^3+ 
1/2*(a+b*arcsech(c*x))*ln(1-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x) 
^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/e^3+1/2*(a+b*arcsech(c*x))*ln(1+c*(-d)^ 
(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/ 
e^3-(a+b*arcsech(c*x))*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)/e^ 
3+1/2*b*polylog(2,-c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/( 
e^(1/2)-(c^2*d+e)^(1/2)))/e^3+1/2*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(-1+1/c/ 
x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/e^3+1/2*b*polylog(2,- 
c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(e^(1/2)+(c^2*d+e)^( 
1/2)))/e^3+1/2*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^ 
(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/e^3-1/2*b*polylog(2,-(1/c/x+(-1+1/c/x)^( 
1/2)*(1+1/c/x)^(1/2))^2)/e^3
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.27 (sec) , antiderivative size = 2000, normalized size of antiderivative = 2.67 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
 

Output:

-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e* 
x^2])/(2*e^3) + b*(-1/16*(d*(((-I)*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + 
c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqr 
t[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 
- c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d 
 + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[ 
c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/ 
(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^( 
3/2))))/e^(5/2) - (d*((I*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqr 
t[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[ 
d] + Sqrt[e]*x)^2) + Log[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x) 
] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d* 
Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[ 
(1 - c*x)/(1 + c*x)] + c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2 
*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*e^(5/2) 
) - (((7*I)/16)*Sqrt[d]*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(L 
og[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 
+ c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - c*x)/(1 + c*x)]*( 
1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e]))/(I*Sqrt[d] + Sq 
rt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/e^(5/2) + (((7*I)/16)*Sqrt[d]*(-(...
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 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.

\(\displaystyle \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6857

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )^2}{b e^3}-\frac {\log \left (1+e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{e^3}+\frac {\log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 e^3}+\frac {\log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 e^3}+\frac {\log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 e^3}+\frac {\log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 e^3}-\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}-\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{4 e \left (\frac {d}{x^2}+e\right )^2}+\frac {b \left (d c^2+2 e\right ) \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e}}{c \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} x}\right )}{8 e^{5/2} \left (d c^2+e\right )^{3/2} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}+\frac {b \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e}}{c \sqrt {e} \sqrt {\frac {1}{c^2 x^2}-1} x}\right )}{2 e^{5/2} \sqrt {d c^2+e} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e^3}+\frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (d c^2+e\right ) \left (\frac {d}{x^2}+e\right ) \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} x}\)

Input:

Int[(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
 

Output:

(b*d*(c^2 - x^(-2)))/(8*c*e^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*S 
qrt[1 + 1/(c*x)]*x) - (a + b*ArcCosh[1/(c*x)])/(4*e*(e + d/x^2)^2) - (a + 
b*ArcCosh[1/(c*x)])/(2*e^2*(e + d/x^2)) - (a + b*ArcCosh[1/(c*x)])^2/(b*e^ 
3) + (b*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 
+ 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 
 1/(c*x)]) + (b*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + 
e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*e^(5/2)*(c^2*d + e)^(3/2)*Sqr 
t[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcCosh[1/(c*x)])*Log[1 + E^( 
-2*ArcCosh[1/(c*x)])])/e^3 + ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d] 
*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcCo 
sh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d 
 + e])])/(2*e^3) + ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh 
[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcCosh[1/(c*x) 
])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/( 
2*e^3) + (b*PolyLog[2, -E^(-2*ArcCosh[1/(c*x)])])/(2*e^3) + (b*PolyLog[2, 
-((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^3) + 
 (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]) 
])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sq 
rt[c^2*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/ 
(Sqrt[e] + 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]
 

rule 6857

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCosh[x/c])^n/x 
^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 
] && IntegersQ[m, p]
 

Maple [C] (warning: unable to verify)

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

Time = 5.55 (sec) , antiderivative size = 1549, normalized size of antiderivative = 2.07

method	result	size
parts	\(\text {Expression too large to display}\)	\(1549\)
derivativedivides	\(\text {Expression too large to display}\)	\(1562\)
default	\(\text {Expression too large to display}\)	\(1562\)

Input:

int(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

a*(1/e^3*d/(e*x^2+d)+1/2/e^3*ln(e*x^2+d)-1/4*d^2/e^3/(e*x^2+d)^2)+b/c^6*(- 
1/8*c^6*(4*arcsech(c*x)*c^6*d^2*x^2+6*e*arcsech(c*x)*c^6*d*x^4+(-(c*x-1)/c 
/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c^5*d^2*x+(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x) 
^(1/2)*c^5*d*e*x^3+4*c^4*d*e*arcsech(c*x)*x^2+6*arcsech(c*x)*e^2*c^4*x^4-c 
^4*d^2-2*c^4*d*e*x^2-c^4*e^2*x^4)/e^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2-3/4*(e 
*(c^2*d+e))^(1/2)/(c^2*d+e)^2/e^2*c^6*arctanh(1/4*(2*c^2*d*(1/c/x+(-1+1/c/ 
x)^(1/2)*(1+1/c/x)^(1/2))^2+2*c^2*d+4*e)/(c^2*d*e+e^2)^(1/2))-1/(c^2*d+e)/ 
e^2*c^6*arcsech(c*x)*ln(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-1/(c 
^2*d+e)/e^2*c^6*arcsech(c*x)*ln(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2 
)))-1/(c^2*d+e)/e^2*c^6*dilog(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)) 
)-1/(c^2*d+e)/e^2*c^6*dilog(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))+ 
1/4/(c^2*d+e)/e^2*c^6*sum((_R1^2*c^2*d+c^2*d+4*e)/(_R1^2*c^2*d+c^2*d+2*e)* 
(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)+dilog(( 
_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(2 
*c^2*d+4*e)*_Z^2+c^2*d))-5/8*(e*(c^2*d+e))^(1/2)/(c^2*d+e)^2/e^3*c^8*d*arc 
tanh(1/4*(2*c^2*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2+2*c^2*d+4*e)/ 
(c^2*d*e+e^2)^(1/2))+1/4/(c^2*d+e)/e^2*c^8*d*sum((_R1^2+1)/(_R1^2*c^2*d+c^ 
2*d+2*e)*(arcsech(c*x)*ln((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1 
)+dilog((_R1-1/c/x-(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/_R1)),_R1=RootOf(c^2* 
d*_Z^4+(2*c^2*d+4*e)*_Z^2+c^2*d))-1/(c^2*d+e)/e^3*c^8*d*arcsech(c*x)*ln...
 

Fricas [F]

\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*x^5*arcsech(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. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**5*(a+b*asech(c*x))/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

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

Giac [F]

\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x^5*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((x^5*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3,x)
 

Output:

int((x^5*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{5}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{2} e^{3}+8 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{5}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b d \,e^{4} x^{2}+4 \left (\int \frac {\mathit {asech} \left (c x \right ) x^{5}}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,e^{5} x^{4}+2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2}+4 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e \,x^{2}+2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{2} x^{4}+a \,d^{2}-2 a \,e^{2} x^{4}}{4 e^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int((asech(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((asech(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((asech(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))