\(\int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\) [388]

Optimal result

Integrand size = 21, antiderivative size = 521 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{4 c e}-\frac {b \text {arccosh}(c x)}{4 c^2 e}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 e}+\frac {d (a+b \text {arccosh}(c x))^2}{2 b e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {-2 a c^2 e x^2+b c e x \sqrt {-1+c x} \sqrt {1+c x}-2 b c^2 e x^2 \text {arccosh}(c x)-2 b c^2 d \text {arccosh}(c x)^2+2 b e \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )+2 b c^2 d \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )+2 b c^2 d \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b c^2 d \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b c^2 d \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 a c^2 d \log \left (d+e x^2\right )+2 b c^2 d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )+2 b c^2 d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b c^2 d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )+2 b c^2 d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 c^2 e^2} \] Input:

Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]
 

Output:

-1/4*(-2*a*c^2*e*x^2 + b*c*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^2*e*x^ 
2*ArcCosh[c*x] - 2*b*c^2*d*ArcCosh[c*x]^2 + 2*b*e*ArcTanh[Sqrt[(-1 + c*x)/ 
(1 + c*x)]] + 2*b*c^2*d*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*S 
qrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E 
^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*ArcCosh[c 
*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] + 
2*b*c^2*d*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt 
[-(c^2*d) - e])] + 2*a*c^2*d*Log[d + e*x^2] + 2*b*c^2*d*PolyLog[2, (Sqrt[e 
]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*PolyLog[2 
, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b*c^2 
*d*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]) 
)] + 2*b*c^2*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^ 
2*d) - e])])/(c^2*e^2)
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

\(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\)

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {d (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^2}-\frac {d (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^2}+\frac {d (a+b \text {arccosh}(c x))^2}{2 b e^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{2 e}-\frac {b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^2}-\frac {b d \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^2}-\frac {b \text {arccosh}(c x)}{4 c^2 e}-\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{4 c e}\)

Input:

Int[(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]
 

Output:

-1/4*(b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e) - (b*ArcCosh[c*x])/(4*c^2*e) 
 + (x^2*(a + b*ArcCosh[c*x]))/(2*e) + (d*(a + b*ArcCosh[c*x])^2)/(2*b*e^2) 
 - (d*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - 
Sqrt[-(c^2*d) - e])])/(2*e^2) - (d*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E 
^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^2) - (d*(a + b*Arc 
Cosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - 
e])])/(2*e^2) - (d*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/( 
c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^2) - (b*d*PolyLog[2, -((Sqrt[e]*E^ 
ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e^2) - (b*d*PolyLog[ 
2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^2) - 
(b*d*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e 
]))])/(2*e^2) - (b*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqr 
t[-(c^2*d) - e])])/(2*e^2)
 

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 [C] (warning: unable to verify)

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

Time = 1.45 (sec) , antiderivative size = 2111, normalized size of antiderivative = 4.05

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2111\)
default	\(\text {Expression too large to display}\)	\(2111\)
parts	\(\text {Expression too large to display}\)	\(2118\)

Input:

int(x^3*(a+b*arccosh(c*x))/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

1/c^4*(1/2*a*c^4/e*x^2-1/2*a*c^4*d/e^2*ln(c^2*e*x^2+c^2*d)+b*c^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)* 
d*c^2/e^3/(c^2*d+e)*arccosh(c*x)^2+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)*d^2*c^4/e^4/(c^2*d+e)*polylo 
g(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(c^2*d*(c^2*d+e))^(1 
/2)-e))-(-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)*d^2*c^4/e^4/(c^2*d+e)*arccosh(c*x)^2-(2*c^2*d-2*(c^2*d*(c^2 
*d+e))^(1/2)+e)/e^4*c^4*d^2*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2 
*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-1/2*(2*c^2*d-2*(c^2*d*(c 
^2*d+e))^(1/2)+e)/e^3*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d 
-2*(c^2*d*(c^2*d+e))^(1/2)-e))*c^2*d*arccosh(c*x)+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)*d*c^2/e^3/(c^ 
2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(c^2*d* 
(c^2*d+e))^(1/2)-e))-1/2*(c^2*d*(c^2*d+e))^(1/2)*d*c^2/e^2/(c^2*d+e)*arcco 
sh(c*x)^2+1/4*(c^2*d*(c^2*d+e))^(1/2)*d*c^2/e^2/(c^2*d+e)*polylog(2,e*(c*x 
+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e))+(- 
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)*d*c^2/e^3/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^ 
2*d-2*(c^2*d*(c^2*d+e))^(1/2)-e))*arccosh(c*x)+(-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)*d^2*c^4/e^4/(c^2*...
 

Fricas [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{e x^{2} + d} \,d x } \] Input:

integrate(x^3*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*x^3*arccosh(c*x) + a*x^3)/(e*x^2 + d), x)
 

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \] Input:

integrate(x**3*(a+b*acosh(c*x))/(e*x**2+d),x)
 

Output:

Integral(x**3*(a + b*acosh(c*x))/(d + e*x**2), x)
 

Maxima [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{e x^{2} + d} \,d x } \] Input:

integrate(x^3*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="maxima")
 

Output:

1/2*a*(x^2/e - d*log(e*x^2 + d)/e^2) + b*integrate(x^3*log(c*x + sqrt(c*x 
+ 1)*sqrt(c*x - 1))/(e*x^2 + d), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(a+b*arccosh(c*x))/(e*x^2+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \] Input:

int((x^3*(a + b*acosh(c*x)))/(d + e*x^2),x)
 

Output:

int((x^3*(a + b*acosh(c*x)))/(d + e*x^2), x)
 

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\frac {2 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{3}}{e \,x^{2}+d}d x \right ) b \,e^{2}-\mathrm {log}\left (e \,x^{2}+d \right ) a d +a e \,x^{2}}{2 e^{2}} \] Input:

int(x^3*(a+b*acosh(c*x))/(e*x^2+d),x)
 

Output:

(2*int((acosh(c*x)*x**3)/(d + e*x**2),x)*b*e**2 - log(d + e*x**2)*a*d + a* 
e*x**2)/(2*e**2)