\(\int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\) [390]

Optimal result

Integrand size = 19, antiderivative size = 449 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \] Output:

-1/2*(a+b*arccosh(c*x))^2/b/e+1/2*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c* 
x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e+1/2*(a+b*arcc 
osh(c*x))*ln(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)-(-c 
^2*d-e)^(1/2)))/e+1/2*(a+b*arccosh(c*x))*ln(1-e^(1/2)*(c*x+(c*x-1)^(1/2)*( 
c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e+1/2*(a+b*arccosh(c*x))*ln 
(1+e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2 
)))/e+1/2*b*polylog(2,-e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^( 
1/2)-(-c^2*d-e)^(1/2)))/e+1/2*b*polylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+ 
1)^(1/2))/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/e+1/2*b*polylog(2,-e^(1/2)*(c*x 
+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/e+1/2*b*pol 
ylog(2,e^(1/2)*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*(-d)^(1/2)+(-c^2*d-e)^ 
(1/2)))/e
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {b \text {arccosh}(c x)^2}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {a \log \left (d+e x^2\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \] Input:

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

Output:

-1/2*(b*ArcCosh[c*x]^2)/e + (b*ArcCosh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x 
])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCosh[c*x]*Log[1 + (Sq 
rt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCo 
sh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]) 
])/(2*e) + (b*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + 
Sqrt[-(c^2*d) - e])])/(2*e) + (a*Log[d + e*x^2])/(2*e) + (b*PolyLog[2, -(( 
Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e) + (b*Po 
lyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e 
) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - 
 e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[ 
-(c^2*d) - e])])/(2*e)
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(a + b*ArcCosh[c*x])^2/(b*e) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e] 
*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + ((a + b*ArcCo 
sh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e] 
)])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt 
[-d] + Sqrt[-(c^2*d) - e])])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e 
]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e) + (b*PolyLog[2 
, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e) + 
(b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])]) 
/(2*e) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2 
*d) - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + 
Sqrt[-(c^2*d) - e])])/(2*e)
 

Defintions of rubi rules used

rule 2009

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

rule 6374

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]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {a \ln \left (e \,x^{2}+d \right )}{2 e}-\frac {b \operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{2 e}+\frac {b \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{2 e}+\frac {b \operatorname {dilog}\left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{4 e}+\frac {b \operatorname {dilog}\left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{4 e}\)	\(386\)
derivativedivides	\(\frac {\frac {a \,c^{2} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e}+b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{4 e}+\frac {\operatorname {dilog}\left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{4 e}\right )}{c^{2}}\)	\(401\)
default	\(\frac {\frac {a \,c^{2} \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 e}+b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{2 e}+\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {-2 c^{2} d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}{-2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}-e}\right )}{4 e}+\frac {\operatorname {dilog}\left (\frac {2 c^{2} d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}+2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}{2 c^{2} d +2 \sqrt {c^{4} d^{2}+c^{2} d e}+e}\right )}{4 e}\right )}{c^{2}}\)	\(401\)

Input:

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

Output:

1/2*a/e*ln(e*x^2+d)-1/2*b*arccosh(c*x)^2/e+1/2*b/e*arccosh(c*x)*ln((-2*c^2 
*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*(c^4*d^2+c^2*d*e)^(1/2)-e)/(-2* 
c^2*d+2*(c^4*d^2+c^2*d*e)^(1/2)-e))+1/2*b/e*arccosh(c*x)*ln((2*c^2*d+e*(c* 
x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*(c^4*d^2+c^2*d*e)^(1/2)+e)/(2*c^2*d+2*( 
c^4*d^2+c^2*d*e)^(1/2)+e))+1/4*b/e*dilog((-2*c^2*d-e*(c*x+(c*x-1)^(1/2)*(c 
*x+1)^(1/2))^2+2*(c^4*d^2+c^2*d*e)^(1/2)-e)/(-2*c^2*d+2*(c^4*d^2+c^2*d*e)^ 
(1/2)-e))+1/4*b/e*dilog((2*c^2*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*( 
c^4*d^2+c^2*d*e)^(1/2)+e)/(2*c^2*d+2*(c^4*d^2+c^2*d*e)^(1/2)+e))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

b*integrate(x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e*x^2 + d), x) + 1/2 
*a*log(e*x^2 + d)/e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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