Integrand size = 21, antiderivative size = 786 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {sech}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \]
1/4*(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^(3/2)/(-d)^(1/2)-1/4*(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^(3/2)/(-d)^(1/2)+1/4*(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/ 2)/(-d)^(1/2)-1/4*(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^(3/2)/(-d)^(1/2)-1/4*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^(3/2)/(-d)^(1/2)+1/4*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^(3/2)/(-d)^( 1/2)-1/4*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^(3/2)/(-d)^(1/2)+1/4*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^( 3/2)/(-d)^(1/2)+1/4*(a+b*arcsech(c*x))/e/(-d/x+(-d)^(1/2)*e^(1/2))+1/4*(-a -b*arcsech(c*x))/e/(d/x+(-d)^(1/2)*e^(1/2))-1/2*b*arctan((1+1/c/x)^(1/2)*( c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x)^(1/2)/(c*d+(-d)^(1/2)*e^(1/2))^(1 /2))/e/(c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(c*d+(-d)^(1/2)*e^(1/2))^(1/2)-1/2*b *arctan((1+1/c/x)^(1/2)*(c*d+(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x)^(1/2)/(c *d-(-d)^(1/2)*e^(1/2))^(1/2))/e/(c*d-(-d)^(1/2)*e^(1/2))^(1/2)/(c*d+(-d)^( 1/2)*e^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 1.79 (sec) , antiderivative size = 1226, normalized size of antiderivative = 1.56 \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]
((-2*a*Sqrt[e]*x)/(d + e*x^2) + (b*ArcSech[c*x])/(I*Sqrt[d] - Sqrt[e]*x) - (b*ArcSech[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + (2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d ]])/Sqrt[d] - (4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTa nh[(((-I)*c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]])/Sqr t[d] + (4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTanh[((I* c*Sqrt[d] + Sqrt[e])*Tanh[ArcSech[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[d] - (I* b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSec h[c*x])])/Sqrt[d] - (2*b*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])])/Sqrt [d] + (I*b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d ]*E^ArcSech[c*x])])/Sqrt[d] + (2*b*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])])/Sqrt[d] + (I*b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])) /(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] - (2*b*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])])/Sqrt[d] - (I*b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2 *d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (2*b*ArcSin[Sqrt[1 + (I*S qrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*S qrt[d]*E^ArcSech[c*x])])/Sqrt[d] + (I*b*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)/S...
Time = 1.75 (sec) , antiderivative size = 842, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 6324, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6324 |
| \(\displaystyle -\int \left (-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 e \left (-\frac {d^2}{x^2}-e d\right )}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{4 e \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{4 e \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\frac {b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{2 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}\) |
(a + b*ArcCosh[1/(c*x)])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCosh[1 /(c*x)])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTan[(Sqrt[c*d - Sqrt[-d]*S qrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x) ])])/(2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) - (b* ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[- d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) + ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d ]*e^(3/2)) - ((a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c* x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -( (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*Sqrt[-d] *e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[ c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh [1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLo g[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt [-d]*e^(3/2))
3.2.20.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
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 = 46.70 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.16
| method | result | size |
| parts | \(-\frac {a x}{2 e \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+\frac {b \left (-\frac {c^{5} \operatorname {arcsech}\left (c x \right ) x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {c^{4} \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 {\textit {\_R1} \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 )}{4 e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 c \,d^{3} e}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}+2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) d^{3} c}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 c \,d^{3} e}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}-2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) d^{3} c}+\frac {c^{4} \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 {\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 )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{4 e}\right )}{c^{3}}\) | \(910\) |
| derivativedivides | \(\frac {-\frac {a \,c^{5} x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\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 {\textit {\_R1} \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}}{4 e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}+2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}-2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\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 {\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 )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}}{4 e}\right )}{c^{3}}\) | \(919\) |
| default | \(\frac {-\frac {a \,c^{5} x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\operatorname {arcsech}\left (c x \right ) c x}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {\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 {\textit {\_R1} \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}}{4 e}+\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {-\left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}+2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (-c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}-2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (c^{2} d -2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \,c^{5} d^{3}}-\frac {\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}\, \left (-c^{2} d \sqrt {e \left (c^{2} d +e \right )}+2 c^{2} d e +2 e^{2}-2 \sqrt {e \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {c d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )}{\sqrt {\left (c^{2} d +2 \sqrt {e \left (c^{2} d +e \right )}+2 e \right ) d}}\right )}{2 e \left (c^{2} d +e \right ) c^{5} d^{3}}+\frac {\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 {\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 )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}}{4 e}\right )}{c^{3}}\) | \(919\) |
-1/2*a/e*x/(e*x^2+d)+1/2*a/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b/c^3*(-1 /2*c^5*arcsech(c*x)/e*x/(c^2*e*x^2+c^2*d)-1/4/e*c^4*sum(_R1/(_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))+1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e) *d)^(1/2)*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*arctanh(c*d*(1/c/x+(-1+1/c/x)^ (1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/c/d^3 /e-1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d*(e*(c^2*d+e))^( 1/2)+2*c^2*d*e+2*e^2+2*(e*(c^2*d+e))^(1/2)*e)*arctanh(c*d*(1/c/x+(-1+1/c/x )^(1/2)*(1+1/c/x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/e/( c^2*d+e)/d^3/c+1/2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d-2*(e *(c^2*d+e))^(1/2)+2*e)*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)) /((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/c/d^3/e-1/2*((c^2*d+2*(e*(c^ 2*d+e))^(1/2)+2*e)*d)^(1/2)*(-c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^2-2* (e*(c^2*d+e))^(1/2)*e)*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)) /((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/e/(c^2*d+e)/d^3/c+1/4/e*c^4* sum(1/_R1/(_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)))
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]