\(\int \frac {(d-c^2 d x^2)^2 (a+b \text {arccosh}(c x))}{x^2} \, dx\) [16]

Optimal result

Integrand size = 25, antiderivative size = 140 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {16}{9} b c d^2 \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{9} b c^3 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}-\frac {d^2 (a+b \text {arccosh}(c x))}{x}-2 c^2 d^2 x (a+b \text {arccosh}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))+b c d^2 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \] Output:

16/9*b*c*d^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)-1/9*b*c^3*d^2*x^2*(c*x-1)^(1/2)*( 
c*x+1)^(1/2)-d^2*(a+b*arccosh(c*x))/x-2*c^2*d^2*x*(a+b*arccosh(c*x))+1/3*c 
^4*d^2*x^3*(a+b*arccosh(c*x))+b*c*d^2*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {d^2 \left (-9 a-18 a c^2 x^2+3 a c^4 x^4+16 b c x \sqrt {-1+c x} \sqrt {1+c x}-b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}+3 b \left (-3-6 c^2 x^2+c^4 x^4\right ) \text {arccosh}(c x)-9 b c x \arctan \left (\frac {1}{\sqrt {-1+c x} \sqrt {1+c x}}\right )\right )}{9 x} \] Input:

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

Output:

(d^2*(-9*a - 18*a*c^2*x^2 + 3*a*c^4*x^4 + 16*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + 
 c*x] - b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 3*b*(-3 - 6*c^2*x^2 + c^4 
*x^4)*ArcCosh[c*x] - 9*b*c*x*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/(9 
*x)
 

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6336, 27, 1905, 1578, 1192, 1467, 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.

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

\(\Big \downarrow \) 6336

\(\displaystyle -b c \int -\frac {d^2 \left (-c^4 x^4+6 c^2 x^2+3\right )}{3 x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-2 c^2 d^2 x (a+b \text {arccosh}(c x))-\frac {d^2 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} b c d^2 \int \frac {-c^4 x^4+6 c^2 x^2+3}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-2 c^2 d^2 x (a+b \text {arccosh}(c x))-\frac {d^2 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 1905

\(\displaystyle \frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {-c^4 x^4+6 c^2 x^2+3}{x \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-2 c^2 d^2 x (a+b \text {arccosh}(c x))-\frac {d^2 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {b c d^2 \sqrt {c^2 x^2-1} \int \frac {-c^4 x^4+6 c^2 x^2+3}{x^2 \sqrt {c^2 x^2-1}}dx^2}{6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} c^4 d^2 x^3 (a+b \text {arccosh}(c x))-2 c^2 d^2 x (a+b \text {arccosh}(c x))-\frac {d^2 (a+b \text {arccosh}(c x))}{x}\)

\(\Big \downarrow \) 1192

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

\(\Big \downarrow \) 1467

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d^2*(a + b*ArcCosh[c*x]))/x) - 2*c^2*d^2*x*(a + b*ArcCosh[c*x]) + (c^4* 
d^2*x^3*(a + b*ArcCosh[c*x]))/3 + (b*d^2*Sqrt[-1 + c^2*x^2]*(-1/3*(c^4*x^6 
) + 5*c^4*Sqrt[-1 + c^2*x^2] + 3*c^4*ArcTan[Sqrt[-1 + c^2*x^2]]))/(3*c^3*S 
qrt[-1 + c*x]*Sqrt[1 + c*x])
 

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 1192

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1)   Subst[Int[x^( 
2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + 
 c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && 
IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
 

rule 1467

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
 x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], 
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e 
 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(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] && Int 
egerQ[(m - 1)/2]
 

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 6336

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && E 
qQ[c^2*d + e, 0] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99

method	result	size
parts	\(d^{2} a \left (\frac {c^{4} x^{3}}{3}-2 c^{2} x -\frac {1}{x}\right )+d^{2} b c \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-2 c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-16 \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )\)	\(139\)
derivativedivides	\(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-2 c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-16 \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(141\)
default	\(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-2 c x \,\operatorname {arccosh}\left (c x \right )-\frac {\operatorname {arccosh}\left (c x \right )}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} \sqrt {c^{2} x^{2}-1}+9 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-16 \sqrt {c^{2} x^{2}-1}\right )}{9 \sqrt {c^{2} x^{2}-1}}\right )\right )\)	\(141\)

Input:

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

Output:

d^2*a*(1/3*c^4*x^3-2*c^2*x-1/x)+d^2*b*c*(1/3*c^3*x^3*arccosh(c*x)-2*c*x*ar 
ccosh(c*x)-arccosh(c*x)/c/x-1/9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(c^2*x^2*(c^2* 
x^2-1)^(1/2)+9*arctan(1/(c^2*x^2-1)^(1/2))-16*(c^2*x^2-1)^(1/2))/(c^2*x^2- 
1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.44 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {3 \, a c^{4} d^{2} x^{4} - 18 \, a c^{2} d^{2} x^{2} + 18 \, b c d^{2} x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 3 \, {\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 9 \, a d^{2} + 3 \, {\left (b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} - {\left (b c^{4} - 6 \, b c^{2} - 3 \, b\right )} d^{2} x - 3 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{3} d^{2} x^{3} - 16 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, x} \] Input:

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

Output:

1/9*(3*a*c^4*d^2*x^4 - 18*a*c^2*d^2*x^2 + 18*b*c*d^2*x*arctan(-c*x + sqrt( 
c^2*x^2 - 1)) - 3*(b*c^4 - 6*b*c^2 - 3*b)*d^2*x*log(-c*x + sqrt(c^2*x^2 - 
1)) - 9*a*d^2 + 3*(b*c^4*d^2*x^4 - 6*b*c^2*d^2*x^2 - (b*c^4 - 6*b*c^2 - 3* 
b)*d^2*x - 3*b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^3*d^2*x^3 - 16*b*c 
*d^2*x)*sqrt(c^2*x^2 - 1))/x
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {1}{3} \, a c^{4} d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} - 2 \, a c^{2} d^{2} x - 2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b c d^{2} - {\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d^{2} - \frac {a d^{2}}{x} \] Input:

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

Output:

1/3*a*c^4*d^2*x^3 + 1/9*(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^4*d^2 - 2*a*c^2*d^2*x - 2*(c*x*arccosh(c* 
x) - sqrt(c^2*x^2 - 1))*b*c*d^2 - (c*arcsin(1/(c*abs(x))) + arccosh(c*x)/x 
)*b*d^2 - a*d^2/x
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d-c^2 d x^2\right )^2 (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {d^{2} \left (3 \mathit {acosh} \left (c x \right ) b \,c^{4} x^{4}-18 \mathit {acosh} \left (c x \right ) b \,c^{2} x^{2}-9 \mathit {acosh} \left (c x \right ) b -18 \mathit {atan} \left (\sqrt {c^{2} x^{2}-1}+c x \right ) b c x -\sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3}-2 \sqrt {c^{2} x^{2}-1}\, b c x +18 \sqrt {c x +1}\, \sqrt {c x -1}\, b c x +3 a \,c^{4} x^{4}-18 a \,c^{2} x^{2}-9 a \right )}{9 x} \] Input:

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

Output:

(d**2*(3*acosh(c*x)*b*c**4*x**4 - 18*acosh(c*x)*b*c**2*x**2 - 9*acosh(c*x) 
*b - 18*atan(sqrt(c**2*x**2 - 1) + c*x)*b*c*x - sqrt(c**2*x**2 - 1)*b*c**3 
*x**3 - 2*sqrt(c**2*x**2 - 1)*b*c*x + 18*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c*x 
 + 3*a*c**4*x**4 - 18*a*c**2*x**2 - 9*a))/(9*x)