Integrand size = 21, antiderivative size = 523 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{d}-\frac {a}{d x}-\frac {b \text {sech}^{-1}(c x)}{d x}+\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}} \]
-a/d/x-b*arcsech(c*x)/d/x+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^(1/2)/(-d)^ (3/2)-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^(1/2)/(-d)^(3/2)+1/2*(a+b*arcse ch(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^(1/2)/(-d)^(3/2)-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^(1/2)/(-d)^(3/2)-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^(1/2)/(-d)^(3/2)+1/2*b*polyl og(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^(1/2)/(-d)^(3/2)-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^(1/2)/(-d)^(3/2)+ 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^(1/2)/(-d)^(3/2)+b*c*(-1+1/c/x)^(1/2)*(1+1/c/x)^( 1/2)/d
Result contains complex when optimal does not.
Time = 3.28 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.78 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {-4 a \sqrt {d}-4 a \sqrt {e} x \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (4 \sqrt {d} \sqrt {\frac {1-c x}{1+c x}} (1+c x)-4 \sqrt {d} \text {sech}^{-1}(c x)-2 i \sqrt {e} x \left (-4 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {\left (i c \sqrt {d}+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}-\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1+\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )\right )+2 i \sqrt {e} x \left (-4 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \text {arctanh}\left (\frac {\left (-i c \sqrt {d}+\sqrt {e}\right ) \tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )}{\sqrt {c^2 d+e}}\right )+\text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-\text {sech}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i \arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {c^2 d+e}\right ) e^{-\text {sech}^{-1}(c x)}}{c \sqrt {d}}\right )\right )\right )}{4 d^{3/2} x} \]
(-4*a*Sqrt[d] - 4*a*Sqrt[e]*x*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[d]*S qrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 4*Sqrt[d]*ArcSech[c*x] - (2*I)*Sqrt[e ]*x*((-4*I)*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]] + ArcSech[c*x] *Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^ 2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*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])] - ArcSech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c *Sqrt[d]*E^ArcSech[c*x])] - (2*I)*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 ])] + PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x ])] + PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c *x])]) + (2*I)*Sqrt[e]*x*((-4*I)*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]] + ArcSech[c*x]*Log[1 + E^(-2*ArcSech[c*x])] - ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + (2*I)*Ar cSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqr t[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - ArcSech[c*x]*Log[1 - (I*(Sqrt [e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + ...
Time = 1.44 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.10, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right ) x^2}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle -\int \left (\frac {a+b \text {arccosh}\left (\frac {1}{c x}\right )}{d}-\frac {e \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{d \left (\frac {d}{x^2}+e\right )}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 (-d)^{3/2}}-\frac {a}{d x}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 (-d)^{3/2}}-\frac {b \text {arccosh}\left (\frac {1}{c x}\right )}{d x}+\frac {b c \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{d}\) |
(b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d - a/(d*x) - (b*ArcCosh[1/(c*x )])/(d*x) + (Sqrt[e]*(a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCos h[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b *ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt [c^2*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcCosh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2 )) - (Sqrt[e]*(a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c* x)])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*(-d)^( 3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - S qrt[c^2*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^A rcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e ]*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])]) /(2*(-d)^(3/2))
3.2.14.3.1 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 = 52.02 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.71
| method | result | size |
| parts | \(-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d \sqrt {d e}}-\frac {a}{d x}+\frac {b c \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{d}-\frac {b \,\operatorname {arcsech}\left (c x \right )}{d x}+\frac {b c e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}-\frac {b c e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\) | \(372\) |
| derivativedivides | \(c \left (-\frac {a}{d c x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c d \sqrt {d e}}+\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{d}-\frac {b \,\operatorname {arcsech}\left (c x \right )}{d c x}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\right )\) | \(380\) |
| default | \(c \left (-\frac {a}{d c x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c d \sqrt {d e}}+\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{d}-\frac {b \,\operatorname {arcsech}\left (c x \right )}{d c x}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 d}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e \right )}\right )}{2 d}\right )\) | \(380\) |
-a*e/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-a/d/x+b*c/d*(-(c*x-1)/c/x)^(1/2 )*((c*x+1)/c/x)^(1/2)-b*arcsech(c*x)/d/x+1/2*b*c*e/d*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))/_ 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))-1/2*b*c*e/d*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))
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{2}} \,d x } \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,\left (e\,x^2+d\right )} \,d x \]