\(\int (d-c^2 d x^2) (a+b \text {arccosh}(c x))^2 \, dx\) [9]

Optimal result

Integrand size = 22, antiderivative size = 136 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {14}{9} b^2 d x-\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c}+\frac {2 b d (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))}{9 c}+\frac {2}{3} d x (a+b \text {arccosh}(c x))^2+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2 \] Output:

14/9*b^2*d*x-2/27*b^2*c^2*d*x^3-4/3*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*a 
rccosh(c*x))/c+2/9*b*d*(c*x-1)^(3/2)*(c*x+1)^(3/2)*(a+b*arccosh(c*x))/c+2/ 
3*d*x*(a+b*arccosh(c*x))^2+1/3*d*x*(-c^2*x^2+1)*(a+b*arccosh(c*x))^2
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {d \left (-2 b^2 c x \left (-21+c^2 x^2\right )+6 a b \sqrt {-1+c x} \sqrt {1+c x} \left (-7+c^2 x^2\right )-9 a^2 c x \left (-3+c^2 x^2\right )+6 b \left (b \sqrt {-1+c x} \sqrt {1+c x} \left (-7+c^2 x^2\right )+a \left (9 c x-3 c^3 x^3\right )\right ) \text {arccosh}(c x)-9 b^2 c x \left (-3+c^2 x^2\right ) \text {arccosh}(c x)^2\right )}{27 c} \] Input:

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

Output:

(d*(-2*b^2*c*x*(-21 + c^2*x^2) + 6*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-7 + 
c^2*x^2) - 9*a^2*c*x*(-3 + c^2*x^2) + 6*b*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]* 
(-7 + c^2*x^2) + a*(9*c*x - 3*c^3*x^3))*ArcCosh[c*x] - 9*b^2*c*x*(-3 + c^2 
*x^2)*ArcCosh[c*x]^2))/(27*c)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6312, 6294, 6330, 24, 25, 39, 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.

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

\(\Big \downarrow \) 6312

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

\(\Big \downarrow \) 6294

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

\(\Big \downarrow \) 6330

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*x*(1 - c^2*x^2)*(a + b*ArcCosh[c*x])^2)/3 + (2*b*c*d*((b*(x - (c^2*x^3) 
/3))/(3*c) + ((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^ 
2)))/3 + (2*d*(x*(a + b*ArcCosh[c*x])^2 - 2*b*c*(-((b*x)/c) + (Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^2)))/3
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 2009

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

rule 6294

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A 
rcCosh[c*x])^n, x] - Simp[b*c*n   Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt 
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
 

rule 6312

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p 
)]   Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n 
 - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 
0] && GtQ[p, 0]
 

rule 6330

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p 
_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + 
e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 
*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ 
p]   Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 
 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E 
qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(\frac {-d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{2} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{2} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{3}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3}-\frac {40 c x}{27}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c x -1\right ) \left (c x +1\right )}{27}\right )-2 d a b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(171\)
default	\(\frac {-d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{2} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{2} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{3}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3}-\frac {40 c x}{27}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c x -1\right ) \left (c x +1\right )}{27}\right )-2 d a b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(171\)
parts	\(-d \,a^{2} \left (\frac {1}{3} c^{2} x^{3}-x \right )-\frac {d \,b^{2} \left (-\frac {2 \operatorname {arccosh}\left (c x \right )^{2} c x}{3}+\frac {\operatorname {arccosh}\left (c x \right )^{2} c x \left (c x -1\right ) \left (c x +1\right )}{3}+\frac {4 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{3}-\frac {40 c x}{27}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \left (c x -1\right )^{\frac {3}{2}} \left (c x +1\right )^{\frac {3}{2}}}{9}+\frac {2 c x \left (c x -1\right ) \left (c x +1\right )}{27}\right )}{c}-\frac {2 d a b \left (\frac {c^{3} x^{3} \operatorname {arccosh}\left (c x \right )}{3}-c x \,\operatorname {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2}-7\right )}{9}\right )}{c}\)	\(172\)
orering	\(\frac {x \left (19 c^{4} x^{4}-166 c^{2} x^{2}+27\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{27 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (2 c^{4} x^{4}-29 c^{2} x^{2}+7\right ) \left (-2 c^{2} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}+\frac {2 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{9 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {x \left (c^{2} x^{2}-21\right ) \left (-2 c^{2} d \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}-\frac {8 c^{3} d x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 \left (-c^{2} d \,x^{2}+d \right ) b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (-c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{27 c^{2}}\)	\(317\)

Input:

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

Output:

1/c*(-d*a^2*(1/3*c^3*x^3-c*x)-d*b^2*(-2/3*arccosh(c*x)^2*c*x+1/3*arccosh(c 
*x)^2*c*x*(c*x-1)*(c*x+1)+4/3*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-40/ 
27*c*x-2/9*arccosh(c*x)*(c*x-1)^(3/2)*(c*x+1)^(3/2)+2/27*c*x*(c*x-1)*(c*x+ 
1))-2*d*a*b*(1/3*c^3*x^3*arccosh(c*x)-c*x*arccosh(c*x)-1/9*(c*x-1)^(1/2)*( 
c*x+1)^(1/2)*(c^2*x^2-7)))
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.31 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=-\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} d x^{3} - 3 \, {\left (9 \, a^{2} + 14 \, b^{2}\right )} c d x + 9 \, {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (3 \, a b c^{3} d x^{3} - 9 \, a b c d x - {\left (b^{2} c^{2} d x^{2} - 7 \, b^{2} d\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (a b c^{2} d x^{2} - 7 \, a b d\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, c} \] Input:

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

Output:

-1/27*((9*a^2 + 2*b^2)*c^3*d*x^3 - 3*(9*a^2 + 14*b^2)*c*d*x + 9*(b^2*c^3*d 
*x^3 - 3*b^2*c*d*x)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 6*(3*a*b*c^3*d*x^3 - 
9*a*b*c*d*x - (b^2*c^2*d*x^2 - 7*b^2*d)*sqrt(c^2*x^2 - 1))*log(c*x + sqrt( 
c^2*x^2 - 1)) - 6*(a*b*c^2*d*x^2 - 7*a*b*d)*sqrt(c^2*x^2 - 1))/c
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.69 \[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=-\frac {1}{3} \, b^{2} c^{2} d x^{3} \operatorname {arcosh}\left (c x\right )^{2} - \frac {1}{3} \, a^{2} c^{2} d x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b c^{2} d + \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d}{c} \] Input:

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

Output:

-1/3*b^2*c^2*d*x^3*arccosh(c*x)^2 - 1/3*a^2*c^2*d*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))*a*b*c^2*d 
 + 2/27*(3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4)*arccosh 
(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*c^2*d + b^2*d*x*arccosh(c*x)^2 + 2*b^2*d* 
(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d*x + 2*(c*x*arccosh(c*x) - s 
qrt(c^2*x^2 - 1))*a*b*d/c
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {d \left (-6 \mathit {acosh} \left (c x \right ) a b \,c^{3} x^{3}+18 \mathit {acosh} \left (c x \right ) a b c x +2 \sqrt {c^{2} x^{2}-1}\, a b \,c^{2} x^{2}+4 \sqrt {c^{2} x^{2}-1}\, a b -18 \sqrt {c x +1}\, \sqrt {c x -1}\, a b +9 \left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) b^{2} c -9 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-3 a^{2} c^{3} x^{3}+9 a^{2} c x \right )}{9 c} \] Input:

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

Output:

(d*( - 6*acosh(c*x)*a*b*c**3*x**3 + 18*acosh(c*x)*a*b*c*x + 2*sqrt(c**2*x* 
*2 - 1)*a*b*c**2*x**2 + 4*sqrt(c**2*x**2 - 1)*a*b - 18*sqrt(c*x + 1)*sqrt( 
c*x - 1)*a*b + 9*int(acosh(c*x)**2,x)*b**2*c - 9*int(acosh(c*x)**2*x**2,x) 
*b**2*c**3 - 3*a**2*c**3*x**3 + 9*a**2*c*x))/(9*c)