Integrand size = 21, antiderivative size = 631 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}-\frac {b d \sqrt {-1+\frac {1}{c^2 x^2}} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3} \]
1/2*d*(a+b*arcsech(c*x))/e^2/(e+d/x^2)+1/2*x^2*(a+b*arcsech(c*x))/e^2+2*d* (a+b*arcsech(c*x))^2/b/e^3+2*d*(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-d*(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-d*(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-d*(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-d*(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-b*d*polylog(2,-1/(1/c/x+(-1+1/c/x)^(1/2)*(1 +1/c/x)^(1/2))^2)/e^3-b*d*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-b*d*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-b *d*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-b*d*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*d*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)-1/2*b*x*(-1+1/c/x)^(1/2)*(1+ 1/c/x)^(1/2)/c/e^2
Result contains complex when optimal does not.
Time = 5.54 (sec) , antiderivative size = 1278, normalized size of antiderivative = 2.03 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]
-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*((2*e* Sqrt[(1 - c*x)/(1 + c*x)])/c^2 + (2*e*x*Sqrt[(1 - c*x)/(1 + c*x)])/c - 2*e *x^2*ArcSech[c*x] + (d^(3/2)*ArcSech[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3 /2)*ArcSech[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (16*I)*d*ArcSin[Sqrt[1 - (I*Sq rt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcS ech[c*x]/2])/Sqrt[c^2*d + e]] + (16*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sq rt[d])]/Sqrt[2]]*ArcTanh[((I*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sq rt[c^2*d + e]] - 8*d*ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] + 4*d*ArcSe ch[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x]) ] - (8*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*( Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 4*d*ArcSech[c*x] *Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (8 *I)*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[ e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 4*d*ArcSech[c*x]*Log[ 1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (8*I)*d* ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sq rt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 4*d*ArcSech[c*x]*Log[1 + (I* (Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (8*I)*d*ArcSin[ Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2* d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + 2*d*Log[x] - 2*d*Log[1 + Sqrt[(1...
Time = 1.82 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {x^3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{\left (\frac {d}{x^2}+e\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle -\int \left (\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) x^3}{e^2}-\frac {2 d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) x}{e^3}+\frac {2 d^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{e^3 \left (\frac {d}{x^2}+e\right ) x}+\frac {d^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{e^2 \left (\frac {d}{x^2}+e\right )^2 x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{e^3}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{e^3}+\frac {2 d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )^2}{b e^3}+\frac {2 d \log \left (e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}+1\right ) \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 )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {x^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 e^2}-\frac {b d \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 )}{e^3}-\frac {b d \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 )}{e^3}-\frac {b d \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 )}{e^3}-\frac {b d \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 )}{e^3}-\frac {b d \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}\right )}{e^3}-\frac {b d \sqrt {\frac {1}{c^2 x^2}-1} \text {arctanh}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 e^{5/2} \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}}-\frac {b x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{2 c e^2}\) |
-1/2*(b*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(c*e^2) + (d*(a + b*ArcCos h[1/(c*x)]))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*ArcCosh[1/(c*x)]))/(2*e^2) + (2*d*(a + b*ArcCosh[1/(c*x)])^2)/(b*e^3) - (b*d*Sqrt[-1 + 1/(c^2*x^2)]*A rcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*e^(5/2)*S qrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (2*d*(a + b*ArcCosh [1/(c*x)])*Log[1 + E^(-2*ArcCosh[1/(c*x)])])/e^3 - (d*(a + b*ArcCosh[1/(c* x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])]) /e^3 - (d*(a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)]) /(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcCosh[1/(c*x)])*Log[1 - ( c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sq rt[c^2*d + e])])/e^3 - (b*d*PolyLog[2, -E^(-2*ArcCosh[1/(c*x)])])/e^3 - (b *d*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e] ))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt [c^2*d + e])])/e^3 - (b*d*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sq rt[e] + Sqrt[c^2*d + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/ (c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3
3.2.15.3.1 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 = 7.18 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.25
| method | result | size |
| parts | \(\frac {a \,x^{2}}{2 e^{2}}-\frac {a d \ln \left (e \,x^{2}+d \right )}{e^{3}}-\frac {a \,d^{2}}{2 e^{3} \left (e \,x^{2}+d \right )}+\frac {b \left (\frac {c^{4} \left (2 \,\operatorname {arcsech}\left (c x \right ) c^{4} d \,x^{2}+e \,\operatorname {arcsech}\left (c x \right ) c^{4} x^{4}-\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} d x -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e \,c^{3} x^{3}+c^{2} d +e \,c^{2} x^{2}\right )}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e^{2}}+\frac {\sqrt {e \left (c^{2} d +e \right )}\, d \,c^{6} \operatorname {arctanh}\left (\frac {2 c^{2} d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right )}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 d \,c^{6} \operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 d \,c^{6} \operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 d \,c^{6} \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 d \,c^{6} \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}-\frac {d \,c^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}-\frac {d^{2} c^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}\right )}{c^{6}}\) | \(786\) |
| derivativedivides | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{6} d \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{e^{3}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (\frac {2 \,\operatorname {arcsech}\left (c x \right ) c^{4} d \,x^{2}+e \,\operatorname {arcsech}\left (c x \right ) c^{4} x^{4}-\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} d x -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e \,c^{3} x^{3}+c^{2} d +e \,c^{2} x^{2}}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e^{2}}+\frac {\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d \,\operatorname {arctanh}\left (\frac {2 c^{2} d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right )}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 c^{2} d \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \,\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}-\frac {c^{4} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}\right )}{c^{6}}\) | \(810\) |
| default | \(\frac {\frac {a \,c^{6} x^{2}}{2 e^{2}}-\frac {a \,c^{6} d \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{e^{3}}-\frac {a \,c^{8} d^{2}}{2 e^{3} \left (e \,c^{2} x^{2}+c^{2} d \right )}+b \,c^{4} \left (\frac {2 \,\operatorname {arcsech}\left (c x \right ) c^{4} d \,x^{2}+e \,\operatorname {arcsech}\left (c x \right ) c^{4} x^{4}-\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} d x -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e \,c^{3} x^{3}+c^{2} d +e \,c^{2} x^{2}}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e^{2}}+\frac {\sqrt {e \left (c^{2} d +e \right )}\, c^{2} d \,\operatorname {arctanh}\left (\frac {2 c^{2} d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right )}{2 e^{3} \left (c^{2} d +e \right )}+\frac {2 c^{2} d \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \,\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 c^{2} d \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}-\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}-\frac {c^{4} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}\right )}{c^{6}}\) | \(810\) |
1/2*a*x^2/e^2-a*d/e^3*ln(e*x^2+d)-1/2*a*d^2/e^3/(e*x^2+d)+b/c^6*(1/2*c^4*( 2*arcsech(c*x)*c^4*d*x^2+e*arcsech(c*x)*c^4*x^4-(-(c*x-1)/c/x)^(1/2)*((c*x +1)/c/x)^(1/2)*c^3*d*x-(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*e*c^3*x^3+ c^2*d+e*c^2*x^2)/(c^2*e*x^2+c^2*d)/e^2+1/2*(e*(c^2*d+e))^(1/2)/e^3/(c^2*d+ e)*d*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))+2/e^3*d*c^6*arcsech(c*x)*ln(1+I*(1/c/x+(- 1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))+2/e^3*d*c^6*arcsech(c*x)*ln(1-I*(1/c/x+(- 1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))+2/e^3*d*c^6*dilog(1+I*(1/c/x+(-1+1/c/x)^( 1/2)*(1+1/c/x)^(1/2)))+2/e^3*d*c^6*dilog(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/ c/x)^(1/2)))-1/2/e^3*d*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)+di log((_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))-1/2/e^3*d^2*c^8*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))/_R 1)+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)))
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
-1/2*a*(d^2/(e^4*x^2 + d*e^3) - x^2/e^2 + 2*d*log(e*x^2 + d)/e^3) + b*inte grate(x^5*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e^2*x^4 + 2* d*e*x^2 + d^2), x)
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]