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

Optimal result

Integrand size = 14, antiderivative size = 215 \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=2 d^2 x+\frac {4 e^2 x}{9 c^2}+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{9 c}-\frac {d^3 \text {arccosh}(c x)^2}{3 e}-\frac {d e \text {arccosh}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \text {arccosh}(c x)^2}{3 e} \] Output:

2*d^2*x+4/9*e^2*x/c^2+1/2*d*e*x^2+2/27*e^2*x^3-2*d^2*(c*x-1)^(1/2)*(c*x+1) 
^(1/2)*arccosh(c*x)/c-4/9*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c^3 
-d*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c-2/9*e^2*x^2*(c*x-1)^(1/2 
)*(c*x+1)^(1/2)*arccosh(c*x)/c-1/3*d^3*arccosh(c*x)^2/e-1/2*d*e*arccosh(c* 
x)^2/c^2+1/3*(e*x+d)^3*arccosh(c*x)^2/e
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.61 \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\frac {c x \left (24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )-6 \sqrt {-1+c x} \sqrt {1+c x} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right ) \text {arccosh}(c x)+9 \left (-3 c d e+2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \text {arccosh}(c x)^2}{54 c^3} \] Input:

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

Output:

(c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*Sqrt[-1 + c*x]*Sq 
rt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2))*ArcCosh[c*x] + 9* 
(-3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcCosh[c*x]^2)/(54*c^3)
 

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6378, 6390, 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.

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

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arccosh}(c x)^2 (d+e x)^3}{3 e}-\frac {2 c \left (\frac {2 e^3 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^4}+\frac {3 d e^2 \text {arccosh}(c x)^2}{4 c^3}+\frac {3 d^2 e \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{c^2}+\frac {3 d e^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{2 c^2}+\frac {e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^2}+\frac {d^3 \text {arccosh}(c x)^2}{2 c}-\frac {2 e^3 x}{3 c^3}-\frac {3 d^2 e x}{c}-\frac {3 d e^2 x^2}{4 c}-\frac {e^3 x^3}{9 c}\right )}{3 e}\)

Input:

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

Output:

((d + e*x)^3*ArcCosh[c*x]^2)/(3*e) - (2*c*((-3*d^2*e*x)/c - (2*e^3*x)/(3*c 
^3) - (3*d*e^2*x^2)/(4*c) - (e^3*x^3)/(9*c) + (3*d^2*e*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x]*ArcCosh[c*x])/c^2 + (2*e^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[ 
c*x])/(3*c^4) + (3*d*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c 
^2) + (e^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^2) + (d^3*A 
rcCosh[c*x]^2)/(2*c) + (3*d*e^2*ArcCosh[c*x]^2)/(4*c^3)))/(3*e)
 

Defintions of rubi rules used

rule 2009

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

rule 6378

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] 
 - Simp[b*c*(n/(e*(m + 1)))   Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( 
n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x] /; FreeQ[{a, b, c, d, e, m}, 
 x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6390

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand 
Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, 
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && 
 EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ 
d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 
] || (EqQ[m, 2] && LtQ[p, -2]))
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d^{2} x +54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d e \,x^{2}+18 \operatorname {arccosh}\left (c x \right )^{2} e^{2} c^{3} x^{3}-108 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{2} d^{2}-54 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{2} d e x -12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{2} c^{2} x^{2}-27 \operatorname {arccosh}\left (c x \right )^{2} c d e -24 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{2}+108 c^{3} d^{2} x +27 c^{3} d e \,x^{2}+4 e^{2} c^{3} x^{3}+24 e^{2} c x}{54 c^{3}}\)	\(207\)
default	\(\frac {54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d^{2} x +54 \operatorname {arccosh}\left (c x \right )^{2} c^{3} d e \,x^{2}+18 \operatorname {arccosh}\left (c x \right )^{2} e^{2} c^{3} x^{3}-108 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{2} d^{2}-54 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{2} d e x -12 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{2} c^{2} x^{2}-27 \operatorname {arccosh}\left (c x \right )^{2} c d e -24 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{2}+108 c^{3} d^{2} x +27 c^{3} d e \,x^{2}+4 e^{2} c^{3} x^{3}+24 e^{2} c x}{54 c^{3}}\)	\(207\)
orering	\(\frac {\left (38 e^{4} c^{4} x^{5}+206 e^{3} c^{4} x^{4} d +531 c^{4} d^{2} e^{2} x^{3}+162 c^{4} d^{3} e \,x^{2}+54 c^{4} d^{4} x +48 e^{4} c^{2} x^{3}-174 c^{2} d \,e^{3} x^{2}-540 c^{2} d^{2} e^{2} x -135 c^{2} d^{3} e -96 e^{4} x -24 d \,e^{3}\right ) \operatorname {arccosh}\left (c x \right )^{2}}{54 \left (e x +d \right )^{2} c^{4}}-\frac {\left (24 e^{3} c^{4} x^{5}+143 e^{2} c^{4} x^{4} d +459 c^{4} d^{2} e \,x^{3}+68 e^{3} c^{2} x^{3}-174 e^{2} c^{2} x^{2} d -594 e \,c^{2} d^{2} x -108 c^{2} d^{3}-120 e^{3} x -24 d \,e^{2}\right ) \left (2 \left (e x +d \right ) \operatorname {arccosh}\left (c x \right )^{2} e +\frac {2 \left (e x +d \right )^{2} \operatorname {arccosh}\left (c x \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{108 \left (e x +d \right )^{3} c^{4}}+\frac {x \left (4 c^{2} e^{2} x^{2}+27 c^{2} d e x +108 c^{2} d^{2}+24 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 e^{2} \operatorname {arccosh}\left (c x \right )^{2}+\frac {8 \left (e x +d \right ) \operatorname {arccosh}\left (c x \right ) e c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 c^{2} \left (e x +d \right )^{2}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (e x +d \right )^{2} \operatorname {arccosh}\left (c x \right ) c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (e x +d \right )^{2} \operatorname {arccosh}\left (c x \right ) c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{108 c^{4} \left (e x +d \right )^{2}}\)	\(460\)

Input:

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

Output:

1/54/c^3*(54*arccosh(c*x)^2*c^3*d^2*x+54*arccosh(c*x)^2*c^3*d*e*x^2+18*arc 
cosh(c*x)^2*e^2*c^3*x^3-108*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^2*d 
^2-54*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^2*d*e*x-12*(c*x-1)^(1/2)* 
(c*x+1)^(1/2)*arccosh(c*x)*e^2*c^2*x^2-27*arccosh(c*x)^2*c*d*e-24*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*e^2+108*c^3*d^2*x+27*c^3*d*e*x^2+4*e^2*c^ 
3*x^3+24*e^2*c*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.76 \[ \int (d+e x)^2 \text {arccosh}(c x)^2 \, dx=\frac {4 \, c^{3} e^{2} x^{3} + 27 \, c^{3} d e x^{2} + 9 \, {\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, {\left (9 \, c^{3} d^{2} + 2 \, c e^{2}\right )} x}{54 \, c^{3}} \] Input:

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

Output:

1/54*(4*c^3*e^2*x^3 + 27*c^3*d*e*x^2 + 9*(2*c^3*e^2*x^3 + 6*c^3*d*e*x^2 + 
6*c^3*d^2*x - 3*c*d*e)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(2*c^2*e^2*x^2 + 
 9*c^2*d*e*x + 18*c^2*d^2 + 4*e^2)*sqrt(c^2*x^2 - 1)*log(c*x + sqrt(c^2*x^ 
2 - 1)) + 12*(9*c^3*d^2 + 2*c*e^2)*x)/c^3
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) 
^2 - integrate(2/3*(c^3*e^2*x^5 + 3*c^3*d*e*x^4 - 3*c*d*e*x^2 - 3*c*d^2*x 
+ (3*c^3*d^2 - c*e^2)*x^3 + (c^2*e^2*x^4 + 3*c^2*d*e*x^3 + 3*c^2*d^2*x^2)* 
sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*x 
^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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