Integrand size = 18, antiderivative size = 485 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]
1/2*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)- (-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^ 2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2 )+1/2*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2 )+(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arcsinh(c*x))*ln(1+(c*x+( c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1 /2)-1/2*b*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d +e)^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2*b*polylog(2,(c*x+(c^2*x^2+1)^(1/2))*e^( 1/2)/(c*(-d)^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*b*polylog(2,- (c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(1/2 )/e^(1/2)+1/2*b*polylog(2,(c*x+(c^2*x^2+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(- c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)
Time = 0.33 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\frac {2 a \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d^2} \sqrt {e}} \]
(2*a*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] + b*Sqrt[d]*A rcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d ) + e])] + b*Sqrt[d]*ArcSinh[c*x]*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt [-d] + Sqrt[-(c^2*d) + e])] - b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^Ar cSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + b*Sqrt[d]*PolyLog[2, (Sqr t[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] - b*Sqrt[d]*PolyLo g[2, (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] - b*Sq rt[d]*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))] + b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[- (c^2*d) + e])])/(2*Sqrt[-d^2]*Sqrt[e])
Time = 1.20 (sec) , antiderivative size = 485, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 6208 |
| \(\displaystyle \int \left (\frac {\sqrt {-d} (a+b \text {arcsinh}(c x))}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \text {arcsinh}(c x))}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}\) |
((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[ -(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqr t[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[ e]) + ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d] *Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-( c^2*d) + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c* x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[ 2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))])/(2*Sqrt [-d]*Sqrt[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[ -(c^2*d) + e])])/(2*Sqrt[-d]*Sqrt[e])
3.7.11.3.1 Defintions of rubi rules used
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 = 7.91 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.46
| method | result | size |
| parts | \(\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}\) | \(224\) |
| derivativedivides | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}\right )}{c}\) | \(231\) |
| default | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}\right )}{c}\) | \(231\) |
a/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/2*b*c*sum(1/_R1/(_R1^2*e+2*c^2*d-e )*(arcsinh(c*x)*ln((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-c*x-(c^2*x^ 2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d-2*e)*_Z^2+e))+1/2*b*c*sum(_R1 /(_R1^2*e+2*c^2*d-e)*(arcsinh(c*x)*ln((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)+dil og((_R1-c*x-(c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d-2*e)*_Z^2+ e))
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x^{2} + d} \,d x } \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x^{2} + d} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{e\,x^2+d} \,d x \]