Integrand size = 18, antiderivative size = 707 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \arctan \left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {b c \arctan \left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{4 d \sqrt {c^2 d-e} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \] Output:
-1/4*(a+b*arcsinh(c*x))/d/e^(1/2)/((-d)^(1/2)-e^(1/2)*x)+1/4*(a+b*arcsinh( c*x))/d/e^(1/2)/((-d)^(1/2)+e^(1/2)*x)-1/4*b*c*arctan((e^(1/2)-c^2*(-d)^(1 /2)*x)/(c^2*d-e)^(1/2)/(c^2*x^2+1)^(1/2))/d/(c^2*d-e)^(1/2)/e^(1/2)-1/4*b* c*arctan((e^(1/2)+c^2*(-d)^(1/2)*x)/(c^2*d-e)^(1/2)/(c^2*x^2+1)^(1/2))/d/( c^2*d-e)^(1/2)/e^(1/2)-1/4*(a+b*arcsinh(c*x))*ln(1-e^(1/2)*(c*x+(c^2*x^2+1 )^(1/2))/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)+1/4*(a+b*arcs inh(c*x))*ln(1+e^(1/2)*(c*x+(c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d+e)^(1 /2)))/(-d)^(3/2)/e^(1/2)-1/4*(a+b*arcsinh(c*x))*ln(1-e^(1/2)*(c*x+(c^2*x^2 +1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)+1/4*(a+b*ar csinh(c*x))*ln(1+e^(1/2)*(c*x+(c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d+e)^ (1/2)))/(-d)^(3/2)/e^(1/2)+1/4*b*polylog(2,-e^(1/2)*(c*x+(c^2*x^2+1)^(1/2) )/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-1/4*b*polylog(2,e^(1 /2)*(c*x+(c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^ (1/2)+1/4*b*polylog(2,-e^(1/2)*(c*x+(c^2*x^2+1)^(1/2))/(c*(-d)^(1/2)+(-c^2 *d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-1/4*b*polylog(2,e^(1/2)*(c*x+(c^2*x^2+1)^ (1/2))/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)
Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 622, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {1}{2} \left (\frac {a x}{d^2+d e x^2}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e}}+\frac {b \left (-2 \sqrt {d} \left (-\frac {\text {arcsinh}(c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \arctan \left (\frac {\sqrt {e}-i c^2 \sqrt {d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{\sqrt {c^2 d-e}}\right )+2 i \sqrt {d} \left (\frac {\text {arcsinh}(c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {c \text {arctanh}\left (\frac {i \sqrt {e}-c^2 \sqrt {d} x}{\sqrt {c^2 d-e} \sqrt {1+c^2 x^2}}\right )}{\sqrt {c^2 d-e}}\right )+i \left (\text {arcsinh}(c x) \left (-\text {arcsinh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )-i \left (\text {arcsinh}(c x) \left (-\text {arcsinh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d+e}}\right )\right )\right )}{4 d^{3/2} \sqrt {e}}\right ) \] Input:
Integrate[(a + b*ArcSinh[c*x])/(d + e*x^2)^2,x]
Output:
((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*(-2*Sqrt[d]*(-(ArcSinh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) + (c*ArcTan[(Sq rt[e] - I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*d - e]) + (2*I)*Sqrt[d]*(ArcSinh[c*x]/(Sqrt[d] + I*Sqrt[e]*x) + (c*ArcTanh[( I*Sqrt[e] - c^2*Sqrt[d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2* d - e]) + I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcSinh[c *x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) + e])] + Log[1 + (Sqrt[e]*E^ArcSinh[c*x] )/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcSinh[ c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + 2*PolyLog[2, -((Sqrt[e]*E^A rcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))]) - I*(ArcSinh[c*x]*(-Arc Sinh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c ^2*d) + e])] + Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2* d) + e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sq rt[-(c^2*d) + e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])))/(4*d^(3/2)*Sqrt[e]))/2
Time = 1.75 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6208, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 6208 |
| \(\displaystyle \int \left (-\frac {e (a+b \text {arcsinh}(c x))}{2 d \left (-d e-e^2 x^2\right )}-\frac {e (a+b \text {arcsinh}(c x))}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e (a+b \text {arcsinh}(c x))}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arcsinh}(c x)}{4 d \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b c \arctan \left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}-\frac {b c \arctan \left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {c^2 x^2+1} \sqrt {c^2 d-e}}\right )}{4 d \sqrt {e} \sqrt {c^2 d-e}}\) |
Input:
Int[(a + b*ArcSinh[c*x])/(d + e*x^2)^2,x]
Output:
-1/4*(a + b*ArcSinh[c*x])/(d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcS inh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*ArcTan[(Sqrt[e] - c^ 2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*S qrt[e]) - (b*c*ArcTan[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2 )*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqr t[-d] - Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSinh[c*x ])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(4 *(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c* x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLo g[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e]))])/(4*(- d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - S qrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^A rcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] )/(4*(-d)^(3/2)*Sqrt[e])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 51.42 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.20
| method | result | size |
| parts | \(\frac {a x}{2 d \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b \left (\frac {c^{3} \operatorname {arcsinh}\left (x c \right ) x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{4 d}+\frac {c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{4 d}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) c^{2} \arctan \left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \arctan \left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right ) c^{2}}{2 d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) c^{2} \operatorname {arctanh}\left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \operatorname {arctanh}\left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right ) c^{2}}{2 d \,e^{3}}\right )}{c}\) | \(848\) |
| derivativedivides | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}}{4 c^{2} d}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}}{4 c^{2} d}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \arctan \left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \arctan \left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \operatorname {arctanh}\left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}\right )}{c}\) | \(863\) |
| default | \(\frac {\frac {a \,c^{3} x}{2 d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+b \,c^{4} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x}{2 c d \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}}{4 c^{2} d}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}}{4 c^{2} d}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \arctan \left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \arctan \left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}}\right )}{2 c^{2} d \,e^{3}}+\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 \sqrt {c^{2} d \left (c^{2} d -e \right )}\, c^{2} d +2 c^{4} d^{2}-2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d -e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \left (c^{2} d -e \right ) e^{3}}-\frac {\sqrt {-\left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) e}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}-e \right ) \operatorname {arctanh}\left (\frac {e \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d -e \right )}+e \right ) e}}\right )}{2 c^{2} d \,e^{3}}\right )}{c}\) | \(863\) |
Input:
int((a+b*arcsinh(x*c))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/2*a*x/d/(e*x^2+d)+1/2*a/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+b/c*(1/2*c ^3*arcsinh(x*c)*x/d/(c^2*e*x^2+c^2*d)+1/4/d*c^2*sum(1/_R1/(_R1^2*e+2*c^2*d -e)*(arcsinh(x*c)*ln((_R1-x*c-(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-x*c-(c^2* x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d-2*e)*_Z^2+e))+1/4/d*c^2*sum (_R1/(_R1^2*e+2*c^2*d-e)*(arcsinh(x*c)*ln((_R1-x*c-(c^2*x^2+1)^(1/2))/_R1) +dilog((_R1-x*c-(c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d-2*e)*_ Z^2+e))+1/2*((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*(-2*(c^2*d*(c^ 2*d-e))^(1/2)*c^2*d+2*c^4*d^2-2*c^2*d*e+(c^2*d*(c^2*d-e))^(1/2)*e)*c^2*arc tan(e*(x*c+(c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1 /2))/d/(c^2*d-e)/e^3-1/2*((2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*( 2*c^2*d-2*(c^2*d*(c^2*d-e))^(1/2)-e)*arctan(e*(x*c+(c^2*x^2+1)^(1/2))/((2* c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2))*c^2/d/e^3+1/2*(-(2*c^2*d-2*(c ^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*(2*(c^2*d*(c^2*d-e))^(1/2)*c^2*d+2*c^4*d ^2-2*c^2*d*e-(c^2*d*(c^2*d-e))^(1/2)*e)*c^2*arctanh(e*(x*c+(c^2*x^2+1)^(1/ 2))/((-2*c^2*d+2*(c^2*d*(c^2*d-e))^(1/2)+e)*e)^(1/2))/d/(c^2*d-e)/e^3-1/2* (-(2*c^2*d-2*(c^2*d*(c^2*d-e))^(1/2)-e)*e)^(1/2)*(2*c^2*d+2*(c^2*d*(c^2*d- e))^(1/2)-e)*arctanh(e*(x*c+(c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d- e))^(1/2)+e)*e)^(1/2))*c^2/d/e^3)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*arcsinh(c*x) + a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate((a+b*asinh(c*x))/(e*x**2+d)**2,x)
Output:
Integral((a + b*asinh(c*x))/(d + e*x**2)**2, x)
Exception generated. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="maxima")
Output:
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
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/(e*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)/(e*x^2 + d)^2, x)
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((a + b*asinh(c*x))/(d + e*x^2)^2,x)
Output:
int((a + b*asinh(c*x))/(d + e*x^2)^2, x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d +\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a e \,x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{3} e +2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{2} e^{2} x^{2}+a d e x}{2 d^{2} e \left (e \,x^{2}+d \right )} \] Input:
int((a+b*asinh(c*x))/(e*x^2+d)^2,x)
Output:
(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d + sqrt(e)*sqrt(d)*atan( (e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 + 2*int(asinh(c*x)/(d**2 + 2*d*e*x**2 + e**2*x**4),x)*b*d**3*e + 2*int(asinh(c*x)/(d**2 + 2*d*e*x**2 + e**2*x**4), x)*b*d**2*e**2*x**2 + a*d*e*x)/(2*d**2*e*(d + e*x**2))