Integrand size = 19, antiderivative size = 459 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{b e}-\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )}{e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e} \]
-(a+b*arcsech(c*x))^2/b/e-(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+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*(a+b*arcs ech(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*(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*(a+b*arcs ech(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*b*polylog(2,-1/(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/ x)^(1/2))^2)/e+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*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*b*po lylog(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*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
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.87 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=\frac {4 i b \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 )+4 i b \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 )-2 b \text {sech}^{-1}(c x) \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )+b \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 b \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 )+b \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 b \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 )+b \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 b \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 )+b \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 b \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 )+a \log \left (d+e x^2\right )+b \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )-b \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 )-b \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 )-b \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 )-b \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 )}{2 e} \]
((4*I)*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]] + (4*I)*b*ArcS in[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]] - 2*b*ArcSech[c*x]*Log[1 + E^( -2*ArcSech[c*x])] + b*ArcSech[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])) /(c*Sqrt[d]*E^ArcSech[c*x])] - (2*I)*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^ArcSec h[c*x])] + b*ArcSech[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt [d]*E^ArcSech[c*x])] - (2*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sq rt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x]) ] + b*ArcSech[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^Ar cSech[c*x])] + (2*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*L og[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] + b*Arc Sech[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x ])] + (2*I)*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])] + a*Log[d + e*x^ 2] + b*PolyLog[2, -E^(-2*ArcSech[c*x])] - b*PolyLog[2, ((-I)*(-Sqrt[e] + S qrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b*PolyLog[2, ((-I)*(Sqrt[ e] + Sqrt[c^2*d + e]))/(c*Sqrt[d]*E^ArcSech[c*x])] - b*PolyLog[2, (I*(S...
Time = 1.50 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.
| \(\displaystyle \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \frac {x \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{\frac {d}{x^2}+e}d\frac {1}{x}\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle -\int \left (\frac {x \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{e}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{e \left (\frac {d}{x^2}+e\right ) x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{2 e}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 e}+\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{2 e}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )^2}{b e}-\frac {\log \left (e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}+1\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}\left (\frac {1}{c x}\right )}\right )}{2 e}\) |
-((a + b*ArcCosh[1/(c*x)])^2/(b*e)) - ((a + b*ArcCosh[1/(c*x)])*Log[1 + E^ (-2*ArcCosh[1/(c*x)])])/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*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*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*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*e) + (b*PolyLog[2, -E^(-2*ArcCosh[1/(c*x)])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[- d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e) + (b*PolyLog[2 , (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + (b *PolyLog[2, -((c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c^2*d + e])) ])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] + Sqrt[c ^2*d + e])])/(2*e)
3.2.11.3.1 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 = 1.45 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.10
| method | result | size |
| parts | \(\frac {a \ln \left (e \,x^{2}+d \right )}{2 e}-\frac {b \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \,\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {b \operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}+\frac {b \,c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{4 e}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{4 e}\) | \(506\) |
| derivativedivides | \(\frac {\frac {a \,c^{2} \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 e}+b \,c^{2} \left (\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{4 e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}}{4 e}\right )}{c^{2}}\) | \(520\) |
| default | \(\frac {\frac {a \,c^{2} \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 e}+b \,c^{2} \left (\frac {c^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{4 e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}-\frac {\operatorname {dilog}\left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\operatorname {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}}{4 e}\right )}{c^{2}}\) | \(520\) |
1/2*a/e*ln(e*x^2+d)-b/e*arcsech(c*x)*ln(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c /x)^(1/2)))-b/e*arcsech(c*x)*ln(1-I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2 )))-b/e*dilog(1+I*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))-b/e*dilog(1-I* (1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)))+1/4*b*c^2*d/e*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))+1/4*b/e*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))
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{e x^{2} + d} \,d x } \]
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{e x^{2} + d} \,d x } \]
b*integrate(x*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e*x^2 + d), x) + 1/2*a*log(e*x^2 + d)/e
\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{e x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \]