\(\int (d+e x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [373]

Optimal result

Integrand size = 18, antiderivative size = 162 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{15 c^5}-\frac {2 b e \left (5 c^2 d+3 e\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{45 c^5}-\frac {b e^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{25 c^5}+d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x)) \] Output:

-1/15*b*(15*c^4*d^2+10*c^2*d*e+3*e^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5-2/45 
*b*e*(5*c^2*d+3*e)*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^5-1/25*b*e^2*(c*x-1)^(5/2 
)*(c*x+1)^(5/2)/c^5+d^2*x*(a+b*arccosh(c*x))+2/3*d*e*x^3*(a+b*arccosh(c*x) 
)+1/5*e^2*x^5*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.80 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{225} \left (15 a x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )}{c^5}+15 b x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {arccosh}(c x)\right ) \] Input:

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

Output:

(15*a*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x 
^4)))/c^5 + 15*b*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x])/225
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6323, 27, 1905, 1576, 1140, 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.

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

\(\Big \downarrow \) 6323

\(\displaystyle -b c \int \frac {x \left (3 e^2 x^4+10 d e x^2+15 d^2\right )}{15 \sqrt {c x-1} \sqrt {c x+1}}dx+d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{15} b c \int \frac {x \left (3 e^2 x^4+10 d e x^2+15 d^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1905

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x \left (3 e^2 x^4+10 d e x^2+15 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{15 \sqrt {c x-1} \sqrt {c x+1}}+d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1576

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {3 e^2 x^4+10 d e x^2+15 d^2}{\sqrt {c^2 x^2-1}}dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1140

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {3 \left (c^2 x^2-1\right )^{3/2} e^2}{c^4}+\frac {2 \left (5 d c^2+3 e\right ) \sqrt {c^2 x^2-1} e}{c^4}+\frac {15 d^2 c^4+10 d e c^2+3 e^2}{c^4 \sqrt {c^2 x^2-1}}\right )dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle d^2 x (a+b \text {arccosh}(c x))+\frac {2}{3} d e x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} e^2 x^5 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {4 e \left (c^2 x^2-1\right )^{3/2} \left (5 c^2 d+3 e\right )}{3 c^6}+\frac {6 e^2 \left (c^2 x^2-1\right )^{5/2}}{5 c^6}+\frac {2 \sqrt {c^2 x^2-1} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right )}{c^6}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/30*(b*c*Sqrt[-1 + c^2*x^2]*((2*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*Sqrt[- 
1 + c^2*x^2])/c^6 + (4*e*(5*c^2*d + 3*e)*(-1 + c^2*x^2)^(3/2))/(3*c^6) + ( 
6*e^2*(-1 + c^2*x^2)^(5/2))/(5*c^6)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2 
*x*(a + b*ArcCosh[c*x]) + (2*d*e*x^3*(a + b*ArcCosh[c*x]))/3 + (e^2*x^5*(a 
 + b*ArcCosh[c*x]))/5
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 2009

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

rule 6323

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] 
, x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] 
|| ILtQ[p + 1/2, 0])
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85

method	result	size
parts	\(a \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \,\operatorname {arccosh}\left (c x \right ) d e \,x^{3}}{3}+\operatorname {arccosh}\left (c x \right ) c x \,d^{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 c^{2} e^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225 c^{4}}\right )}{c}\)	\(138\)
derivativedivides	\(\frac {\frac {a \left (c^{5} d^{2} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 c^{2} e^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225}\right )}{c^{4}}}{c}\)	\(157\)
default	\(\frac {\frac {a \left (c^{5} d^{2} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 c^{2} e^{2} x^{2}+100 c^{2} d e +24 e^{2}\right )}{225}\right )}{c^{4}}}{c}\)	\(157\)
orering	\(\frac {x \left (81 c^{6} e^{3} x^{6}+395 c^{6} d \,e^{2} x^{4}+1275 c^{6} d^{2} e \,x^{2}+12 c^{4} e^{3} x^{4}+225 c^{6} d^{3}+200 c^{4} d \,e^{2} x^{2}-900 c^{4} d^{2} e +48 c^{2} e^{3} x^{2}-400 c^{2} d \,e^{2}-96 e^{3}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{225 \left (e \,x^{2}+d \right ) c^{6}}-\frac {\left (9 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+225 c^{4} d^{2}+12 c^{2} e^{2} x^{2}+100 c^{2} d e +24 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (4 \left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e x +\frac {\left (e \,x^{2}+d \right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{225 c^{6} \left (e \,x^{2}+d \right )^{2}}\)	\(249\)

Input:

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

Output:

a*(1/5*e^2*x^5+2/3*d*e*x^3+d^2*x)+b/c*(1/5*c*arccosh(c*x)*e^2*x^5+2/3*c*ar 
ccosh(c*x)*d*e*x^3+arccosh(c*x)*c*x*d^2-1/225/c^4*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)*(9*c^4*e^2*x^4+50*c^4*d*e*x^2+225*c^4*d^2+12*c^2*e^2*x^2+100*c^2*d*e+2 
4*e^2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {45 \, a c^{5} e^{2} x^{5} + 150 \, a c^{5} d e x^{3} + 225 \, a c^{5} d^{2} x + 15 \, {\left (3 \, b c^{5} e^{2} x^{5} + 10 \, b c^{5} d e x^{3} + 15 \, b c^{5} d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} e^{2} x^{4} + 225 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 24 \, b e^{2} + 2 \, {\left (25 \, b c^{4} d e + 6 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c^{5}} \] Input:

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

Output:

1/225*(45*a*c^5*e^2*x^5 + 150*a*c^5*d*e*x^3 + 225*a*c^5*d^2*x + 15*(3*b*c^ 
5*e^2*x^5 + 10*b*c^5*d*e*x^3 + 15*b*c^5*d^2*x)*log(c*x + sqrt(c^2*x^2 - 1) 
) - (9*b*c^4*e^2*x^4 + 225*b*c^4*d^2 + 100*b*c^2*d*e + 24*b*e^2 + 2*(25*b* 
c^4*d*e + 6*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^5
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.11 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} \, a e^{2} x^{5} + \frac {2}{3} \, a d e x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d e + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} \] Input:

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

Output:

1/5*a*e^2*x^5 + 2/3*a*d*e*x^3 + 2/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 
- 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d*e + 1/75*(15*x^5*arccosh(c*x) 
 - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2 
*x^2 - 1)/c^6)*c)*b*e^2 + a*d^2*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1)) 
*b*d^2/c
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.26 \[ \int \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {225 \mathit {acosh} \left (c x \right ) b \,c^{5} d^{2} x +150 \mathit {acosh} \left (c x \right ) b \,c^{5} d e \,x^{3}+45 \mathit {acosh} \left (c x \right ) b \,c^{5} e^{2} x^{5}-50 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d e \,x^{2}-9 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{2} x^{4}-100 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d e -12 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{2} x^{2}-24 \sqrt {c^{2} x^{2}-1}\, b \,e^{2}-225 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{4} d^{2}+225 a \,c^{5} d^{2} x +150 a \,c^{5} d e \,x^{3}+45 a \,c^{5} e^{2} x^{5}}{225 c^{5}} \] Input:

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

Output:

(225*acosh(c*x)*b*c**5*d**2*x + 150*acosh(c*x)*b*c**5*d*e*x**3 + 45*acosh( 
c*x)*b*c**5*e**2*x**5 - 50*sqrt(c**2*x**2 - 1)*b*c**4*d*e*x**2 - 9*sqrt(c* 
*2*x**2 - 1)*b*c**4*e**2*x**4 - 100*sqrt(c**2*x**2 - 1)*b*c**2*d*e - 12*sq 
rt(c**2*x**2 - 1)*b*c**2*e**2*x**2 - 24*sqrt(c**2*x**2 - 1)*b*e**2 - 225*s 
qrt(c*x + 1)*sqrt(c*x - 1)*b*c**4*d**2 + 225*a*c**5*d**2*x + 150*a*c**5*d* 
e*x**3 + 45*a*c**5*e**2*x**5)/(225*c**5)