Integrand size = 21, antiderivative size = 778 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b d \left (c^2-\frac {1}{x^2}\right )}{8 c e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} 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 {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\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+e^{-2 \text {sech}^{-1}(c x)}\right )}{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+\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}+\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} \]
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 )-(a+b*arcsech(c*x))^2/b/e^3-(a+b*arcsech(c*x))*ln(1+1/(1/c/x+(-1+1/c/x)^( 1/2)*(1+1/c/x)^(1/2))^2)/e^3+1/2*(a+b*arcsech(c*x))*ln(1-c*(1/c/x+(-1+1/c/ x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/e^3+1/2*(a +b*arcsech(c*x))*ln(1+c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2 )/(e^(1/2)-(c^2*d+e)^(1/2)))/e^3+1/2*(a+b*arcsech(c*x))*ln(1-c*(1/c/x+(-1+ 1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^3+1/ 2*(a+b*arcsech(c*x))*ln(1+c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^ (1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^3+1/2*b*polylog(2,-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*(1/c/x+(-1+1/c/x)^(1/2)*( 1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/e^3+1/2*b*polylog(2, c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^( 1/2)))/e^3+1/2*b*polylog(2,-c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d )^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^3+1/2*b*polylog(2,c*(1/c/x+(-1+1/c/x) ^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^3+1/8*b*d* (c^2-1/x^2)/c/e^2/(c^2*d+e)/(e+d/x^2)/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+1 /8*b*(c^2*d+2*e)*arctanh((c^2*d+e)^(1/2)/c/x/e^(1/2)/(-1+1/c^2/x^2)^(1/2)) *(-1+1/c^2/x^2)^(1/2)/e^(5/2)/(c^2*d+e)^(3/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^( 1/2)+1/2*b*arctanh((c^2*d+e)^(1/2)/c/x/e^(1/2)/(-1+1/c^2/x^2)^(1/2))*(-1+1 /c^2/x^2)^(1/2)/e^(5/2)/(c^2*d+e)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2...
Result contains complex when optimal does not.
Time = 7.93 (sec) , antiderivative size = 2000, normalized size of antiderivative = 2.57 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \]
-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]*(-(...
Time = 1.95 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.09, 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}\) |
(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)
3.2.23.3.1 Defintions of rubi rules used
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.87 (sec) , antiderivative size = 1549, normalized size of antiderivative = 1.99
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1549\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1562\) |
| default | \(\text {Expression too large to display}\) | \(1562\) |
a*(1/2/e^3*ln(e*x^2+d)+d/e^3/(e*x^2+d)-1/4*d^2/e^3/(e*x^2+d)^2)+b/c^6*(-1/ 8*c^6*((-(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*arcsech(c*x)*c^6*d^2*x^2+6*c^6*d*e* arcsech(c*x)*x^4+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))-5/8*(e*(c^2*d+ e))^(1/2)/(c^2*d+e)^2/e^3*c^8*d*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)*(arcse ch(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))+1/4/(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(1...
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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)
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]