\(\int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx\) [126]

Optimal result

Integrand size = 14, antiderivative size = 481 \[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}} \] Output:

1/2*arccosh(a*x)*ln(1-d^(1/2)*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a*(-c)^(1 
/2)-(-a^2*c-d)^(1/2)))/(-c)^(1/2)/d^(1/2)-1/2*arccosh(a*x)*ln(1+d^(1/2)*(a 
*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a*(-c)^(1/2)-(-a^2*c-d)^(1/2)))/(-c)^(1/2 
)/d^(1/2)+1/2*arccosh(a*x)*ln(1-d^(1/2)*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/ 
(a*(-c)^(1/2)+(-a^2*c-d)^(1/2)))/(-c)^(1/2)/d^(1/2)-1/2*arccosh(a*x)*ln(1+ 
d^(1/2)*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a*(-c)^(1/2)+(-a^2*c-d)^(1/2))) 
/(-c)^(1/2)/d^(1/2)-1/2*polylog(2,-d^(1/2)*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2 
))/(a*(-c)^(1/2)-(-a^2*c-d)^(1/2)))/(-c)^(1/2)/d^(1/2)+1/2*polylog(2,d^(1/ 
2)*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a*(-c)^(1/2)-(-a^2*c-d)^(1/2)))/(-c) 
^(1/2)/d^(1/2)-1/2*polylog(2,-d^(1/2)*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a 
*(-c)^(1/2)+(-a^2*c-d)^(1/2)))/(-c)^(1/2)/d^(1/2)+1/2*polylog(2,d^(1/2)*(a 
*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/(a*(-c)^(1/2)+(-a^2*c-d)^(1/2)))/(-c)^(1/2 
)/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.78 \[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\frac {-\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )+\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\text {arccosh}(a x) \log \left (1-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )-\text {arccosh}(a x) \log \left (1+\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}-\sqrt {-a^2 c-d}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{-a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )-\operatorname {PolyLog}\left (2,-\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {d} e^{\text {arccosh}(a x)}}{a \sqrt {-c}+\sqrt {-a^2 c-d}}\right )}{2 \sqrt {-c} \sqrt {d}} \] Input:

Integrate[ArcCosh[a*x]/(c + d*x^2),x]
 

Output:

(-(ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2* 
c) - d])]) + ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(-(a*Sqrt[-c]) 
+ Sqrt[-(a^2*c) - d])] + ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a* 
Sqrt[-c] + Sqrt[-(a^2*c) - d])] - ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[ 
a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])] + PolyLog[2, (Sqrt[d]*E^ArcCosh[a 
*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])] - PolyLog[2, (Sqrt[d]*E^ArcCosh[a* 
x])/(-(a*Sqrt[-c]) + Sqrt[-(a^2*c) - d])] - PolyLog[2, -((Sqrt[d]*E^ArcCos 
h[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d]))] + PolyLog[2, (Sqrt[d]*E^ArcCos 
h[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d])
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6324, 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.

\(\displaystyle \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx\)

\(\Big \downarrow \) 6324

\(\displaystyle \int \left (\frac {\sqrt {-c} \text {arccosh}(a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \text {arccosh}(a x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[ArcCosh[a*x]/(c + d*x^2),x]
 

Output:

(ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) 
 - d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a 
*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCosh[a 
*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])])/( 
2*Sqrt[-c]*Sqrt[d]) - (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sq 
rt[-c] + Sqrt[-(a^2*c) - d])])/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d 
]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d]))]/(2*Sqrt[-c]*Sqrt[d]) 
 + PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[-(a^2*c) - d])]/ 
(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + 
 Sqrt[-(a^2*c) - d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*E^ArcCos 
h[a*x])/(a*Sqrt[-c] + Sqrt[-(a^2*c) - d])]/(2*Sqrt[-c]*Sqrt[d])
 

Defintions of rubi rules used

rule 2009

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

rule 6324

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && 
(p > 0 || IGtQ[n, 0])
 

Maple [C] (verified)

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

Time = 10.98 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.46

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 a^{2} c +d}\right )}{2}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 a^{2} c +d \right )}\right )}{2}}{a}\)	\(222\)
default	\(\frac {\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 a^{2} c +d}\right )}{2}-\frac {a^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\operatorname {arccosh}\left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -a x -\sqrt {a x -1}\, \sqrt {a x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 a^{2} c +d \right )}\right )}{2}}{a}\)	\(222\)

Input:

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

Output:

1/a*(1/2*a^2*sum(_R1/(_R1^2*d+2*a^2*c+d)*(arccosh(a*x)*ln((_R1-a*x-(a*x-1) 
^(1/2)*(a*x+1)^(1/2))/_R1)+dilog((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1 
)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d)*_Z^2+d))-1/2*a^2*sum(1/_R1/(_R1^2*d+2*a 
^2*c+d)*(arccosh(a*x)*ln((_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1)+dilog( 
(_R1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))/_R1)),_R1=RootOf(d*_Z^4+(4*a^2*c+2*d 
)*_Z^2+d)))
 

Fricas [F]

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

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

Output:

integral(arccosh(a*x)/(d*x^2 + c), x)
 

Sympy [F]

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

integrate(acosh(a*x)/(d*x**2+c),x)
 

Output:

Integral(acosh(a*x)/(c + d*x**2), x)
 

Maxima [F]

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

integrate(arccosh(a*x)/(d*x^2+c),x, algorithm="maxima")
 

Output:

integrate(arccosh(a*x)/(d*x^2 + c), x)
 

Giac [F]

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

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

Output:

integrate(arccosh(a*x)/(d*x^2 + c), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(acosh(a*x)/(c + d*x^2),x)
 

Output:

int(acosh(a*x)/(c + d*x^2), x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}(a x)}{c+d x^2} \, dx=\int \frac {\mathit {acosh} \left (a x \right )}{d \,x^{2}+c}d x \] Input:

int(acosh(a*x)/(d*x^2+c),x)
 

Output:

int(acosh(a*x)/(c + d*x**2),x)