\(\int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx\) [149]

Optimal result

Integrand size = 18, antiderivative size = 485 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}} \] Output:

1/2*(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)^(1/2)/e^(1/2)-1/2*(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)^(1/2)/e^(1/2 
)+1/2*(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)^(1/2)/e^(1/2)-1/2*(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)^(1/2)/e^(1 
/2)-1/2*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)^(1/2)/e^(1/2)+1/2*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)^(1/2)/e^(1/2)-1/2*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)^(1/2 
)/e^(1/2)+1/2*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)^(1/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\frac {2 a \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {arcsinh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d^2} \sqrt {e}} \] Input:

Integrate[(a + b*ArcSinh[c*x])/(d + e*x^2),x]
 

Output:

(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])
 

Rubi [A] (verified)

Time = 1.92 (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

\(\displaystyle \frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \text {arcsinh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arcsinh}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}\)

Input:

Int[(a + b*ArcSinh[c*x])/(d + e*x^2),x]
 

Output:

((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])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6208

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])
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 34.33 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.46

method	result	size
parts	\(\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {b c \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}\)	\(224\)
derivativedivides	\(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}\right )}{c}\)	\(231\)
default	\(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+b \,c^{2} \left (\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x c -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}\right )}{c}\)	\(231\)

Input:

int((a+b*arcsinh(x*c))/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

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(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*b*c*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)+dil 
og((_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))
 

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x^{2} + d} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*arcsinh(c*x) + a)/(e*x^2 + d), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{d + e x^{2}}\, dx \] Input:

integrate((a+b*asinh(c*x))/(e*x**2+d),x)
 

Output:

Integral((a + b*asinh(c*x))/(d + e*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arcsinh(c*x))/(e*x^2+d),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
 

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{e x^{2} + d} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/(e*x^2+d),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)/(e*x^2 + d), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{e\,x^2+d} \,d x \] Input:

int((a + b*asinh(c*x))/(d + e*x^2),x)
 

Output:

int((a + b*asinh(c*x))/(d + e*x^2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a +\left (\int \frac {\mathit {asinh} \left (c x \right )}{e \,x^{2}+d}d x \right ) b d e}{d e} \] Input:

int((a+b*asinh(c*x))/(e*x^2+d),x)
 

Output:

(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a + int(asinh(c*x)/(d + e*x 
**2),x)*b*d*e)/(d*e)