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

Optimal result

Integrand size = 22, antiderivative size = 143 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{15 c}+\frac {4 b d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{45 c}-\frac {b d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{25 c}+d^2 x (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x)) \] Output:

-8/15*b*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+4/45*b*d^2*(c*x-1)^(3/2)*(c*x+1) 
^(3/2)/c-1/25*b*d^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+d^2*x*(a+b*arccosh(c*x)) 
-2/3*c^2*d^2*x^3*(a+b*arccosh(c*x))+1/5*c^4*d^2*x^5*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-149+38 c^2 x^2-9 c^4 x^4\right )+15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+15 b c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \text {arccosh}(c x)\right )}{225 c} \] Input:

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

Output:

(d^2*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-149 + 38*c^2*x^2 - 9*c^4*x^4) + 15* 
a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^ 
4)*ArcCosh[c*x]))/(225*c)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6309, 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.

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

\(\Big \downarrow \) 6309

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1576

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

\(\Big \downarrow \) 1140

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 x^2-1} \int \left (3 \left (c^2 x^2-1\right )^{3/2}-4 \sqrt {c^2 x^2-1}+\frac {8}{\sqrt {c^2 x^2-1}}\right )dx^2}{30 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} c^4 d^2 x^5 (a+b \text {arccosh}(c x))-\frac {2}{3} c^2 d^2 x^3 (a+b \text {arccosh}(c x))+d^2 x (a+b \text {arccosh}(c x))-\frac {b c d^2 \sqrt {c^2 x^2-1} \left (\frac {6 \left (c^2 x^2-1\right )^{5/2}}{5 c^2}-\frac {8 \left (c^2 x^2-1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2-1}}{c^2}\right )}{30 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/30*(b*c*d^2*Sqrt[-1 + c^2*x^2]*((16*Sqrt[-1 + c^2*x^2])/c^2 - (8*(-1 + 
c^2*x^2)^(3/2))/(3*c^2) + (6*(-1 + c^2*x^2)^(5/2))/(5*c^2)))/(Sqrt[-1 + c* 
x]*Sqrt[1 + c*x]) + d^2*x*(a + b*ArcCosh[c*x]) - (2*c^2*d^2*x^3*(a + b*Arc 
Cosh[c*x]))/3 + (c^4*d^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 6309

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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69

method	result	size
parts	\(d^{2} a \left (\frac {1}{5} c^{4} x^{5}-\frac {2}{3} c^{2} x^{3}+x \right )+\frac {d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\)	\(99\)
derivativedivides	\(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\)	\(102\)
default	\(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}+c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right )}{225}\right )}{c}\)	\(102\)
orering	\(\frac {x \left (81 c^{4} x^{4}-302 c^{2} x^{2}+821\right ) \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{225 \left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-1\right )}-\frac {\left (9 c^{4} x^{4}-38 c^{2} x^{2}+149\right ) \left (-4 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d x +\frac {\left (-c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{225 c^{2} \left (c x -1\right ) \left (c x +1\right )}\)	\(163\)

Input:

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

Output:

d^2*a*(1/5*c^4*x^5-2/3*c^2*x^3+x)+d^2*b/c*(1/5*arccosh(c*x)*c^5*x^5-2/3*c^ 
3*x^3*arccosh(c*x)+c*x*arccosh(c*x)-1/225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(9*c 
^4*x^4-38*c^2*x^2+149))
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {45 \, a c^{5} d^{2} x^{5} - 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} - 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} d^{2} x^{4} - 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c} \] Input:

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

Output:

1/225*(45*a*c^5*d^2*x^5 - 150*a*c^3*d^2*x^3 + 225*a*c*d^2*x + 15*(3*b*c^5* 
d^2*x^5 - 10*b*c^3*d^2*x^3 + 15*b*c*d^2*x)*log(c*x + sqrt(c^2*x^2 - 1)) - 
(9*b*c^4*d^2*x^4 - 38*b*c^2*d^2*x^2 + 149*b*d^2)*sqrt(c^2*x^2 - 1))/c
 

Sympy [F]

\[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=d^{2} \left (\int a\, dx + \int b \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 a c^{2} x^{2}\right )\, dx + \int a c^{4} x^{4}\, dx + \int \left (- 2 b c^{2} x^{2} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{4} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \] Input:

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

Output:

d**2*(Integral(a, x) + Integral(b*acosh(c*x), x) + Integral(-2*a*c**2*x**2 
, x) + Integral(a*c**4*x**4, x) + Integral(-2*b*c**2*x**2*acosh(c*x), x) + 
 Integral(b*c**4*x**4*acosh(c*x), x))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.36 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} \, a c^{4} d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{2} - \frac {2}{3} \, a c^{2} d^{2} x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{2}}{c} \] Input:

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

Output:

1/5*a*c^4*d^2*x^5 + 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*c^4*d^2 - 
 2/3*a*c^2*d^2*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*c^2*d^2 + a*d^2*x + (c*x*arccosh(c*x) - sq 
rt(c^2*x^2 - 1))*b*d^2/c
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93 \[ \int \left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {d^{2} \left (45 \mathit {acosh} \left (c x \right ) b \,c^{5} x^{5}-150 \mathit {acosh} \left (c x \right ) b \,c^{3} x^{3}+225 \mathit {acosh} \left (c x \right ) b c x -9 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} x^{4}+38 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} x^{2}+76 \sqrt {c^{2} x^{2}-1}\, b -225 \sqrt {c x +1}\, \sqrt {c x -1}\, b +45 a \,c^{5} x^{5}-150 a \,c^{3} x^{3}+225 a c x \right )}{225 c} \] Input:

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

Output:

(d**2*(45*acosh(c*x)*b*c**5*x**5 - 150*acosh(c*x)*b*c**3*x**3 + 225*acosh( 
c*x)*b*c*x - 9*sqrt(c**2*x**2 - 1)*b*c**4*x**4 + 38*sqrt(c**2*x**2 - 1)*b* 
c**2*x**2 + 76*sqrt(c**2*x**2 - 1)*b - 225*sqrt(c*x + 1)*sqrt(c*x - 1)*b + 
 45*a*c**5*x**5 - 150*a*c**3*x**3 + 225*a*c*x))/(225*c)