\(\int (d+e x)^3 \text {arccosh}(c x)^2 \, dx\) [8]

Optimal result

Integrand size = 14, antiderivative size = 334 \[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=2 d^3 x+\frac {4 d e^2 x}{3 c^2}+\frac {3}{4} d^2 e x^2+\frac {3 e^3 x^2}{32 c^2}+\frac {2}{9} d e^2 x^3+\frac {e^3 x^4}{32}-\frac {2 d^3 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{c}-\frac {4 d e^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^3}-\frac {3 d^2 e x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{2 c}-\frac {3 e^3 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{16 c^3}-\frac {2 d e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c}-\frac {e^3 x^3 \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{8 c}-\frac {d^4 \text {arccosh}(c x)^2}{4 e}-\frac {3 d^2 e \text {arccosh}(c x)^2}{4 c^2}-\frac {3 e^3 \text {arccosh}(c x)^2}{32 c^4}+\frac {(d+e x)^4 \text {arccosh}(c x)^2}{4 e} \] Output:

2*d^3*x+4/3*d*e^2*x/c^2+3/4*d^2*e*x^2+3/32*e^3*x^2/c^2+2/9*d*e^2*x^3+1/32* 
e^3*x^4-2*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c-4/3*d*e^2*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c^3-3/2*d^2*e*x*(c*x-1)^(1/2)*(c*x+1)^(1 
/2)*arccosh(c*x)/c-3/16*e^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c^3 
-2/3*d*e^2*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c-1/8*e^3*x^3*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)/c-1/4*d^4*arccosh(c*x)^2/e-3/4*d^2*e* 
arccosh(c*x)^2/c^2-3/32*e^3*arccosh(c*x)^2/c^4+1/4*(e*x+d)^4*arccosh(c*x)^ 
2/e
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.57 \[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\frac {c^2 x \left (3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )-6 c \sqrt {-1+c x} \sqrt {1+c x} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right ) \text {arccosh}(c x)+9 \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arccosh}(c x)^2}{288 c^4} \] Input:

Integrate[(d + e*x)^3*ArcCosh[c*x]^2,x]
 

Output:

(c^2*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 
+ 9*e^3*x^3)) - 6*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2*(64*d + 9*e*x) + c^2 
*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3))*ArcCosh[c*x] + 9*(-24*c 
^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*Ar 
cCosh[c*x]^2)/(288*c^4)
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.08, 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.

\(\displaystyle \int \text {arccosh}(c x)^2 (d+e x)^3 \, dx\)

\(\Big \downarrow \) 6378

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

\(\Big \downarrow \) 6390

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arccosh}(c x)^2 (d+e x)^4}{4 e}-\frac {c \left (\frac {3 e^4 \text {arccosh}(c x)^2}{16 c^5}+\frac {8 d e^3 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^4}+\frac {3 e^4 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{8 c^4}+\frac {3 d^2 e^2 \text {arccosh}(c x)^2}{2 c^3}+\frac {4 d^3 e \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{c^2}+\frac {3 d^2 e^2 x \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{c^2}+\frac {4 d e^3 x^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{3 c^2}+\frac {e^4 x^3 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{4 c^2}+\frac {d^4 \text {arccosh}(c x)^2}{2 c}-\frac {8 d e^3 x}{3 c^3}-\frac {3 e^4 x^2}{16 c^3}-\frac {4 d^3 e x}{c}-\frac {3 d^2 e^2 x^2}{2 c}-\frac {4 d e^3 x^3}{9 c}-\frac {e^4 x^4}{16 c}\right )}{2 e}\)

Input:

Int[(d + e*x)^3*ArcCosh[c*x]^2,x]
 

Output:

((d + e*x)^4*ArcCosh[c*x]^2)/(4*e) - (c*((-4*d^3*e*x)/c - (8*d*e^3*x)/(3*c 
^3) - (3*d^2*e^2*x^2)/(2*c) - (3*e^4*x^2)/(16*c^3) - (4*d*e^3*x^3)/(9*c) - 
 (e^4*x^4)/(16*c) + (4*d^3*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c^ 
2 + (8*d*e^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^4) + (3*d^2*e 
^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c^2 + (3*e^4*x*Sqrt[-1 + c 
*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(8*c^4) + (4*d*e^3*x^2*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x]*ArcCosh[c*x])/(3*c^2) + (e^4*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Ar 
cCosh[c*x])/(4*c^2) + (d^4*ArcCosh[c*x]^2)/(2*c) + (3*d^2*e^2*ArcCosh[c*x] 
^2)/(2*c^3) + (3*e^4*ArcCosh[c*x]^2)/(16*c^5)))/(2*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 = 329, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{3} x +432 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{2} e \,x^{2}+288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d \,e^{2} x^{3}+72 \operatorname {arccosh}\left (c x \right )^{2} e^{3} c^{4} x^{4}-576 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d^{3}-432 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d^{2} e x -192 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d \,e^{2} x^{2}-36 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{3} c^{3} x^{3}-216 \operatorname {arccosh}\left (c x \right )^{2} c^{2} d^{2} e -384 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c d \,e^{2}-54 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{3} c x +576 c^{4} d^{3} x +216 c^{4} d^{2} e \,x^{2}+64 c^{4} d \,e^{2} x^{3}+9 c^{4} e^{3} x^{4}-27 \operatorname {arccosh}\left (c x \right )^{2} e^{3}+384 c^{2} d \,e^{2} x +27 c^{2} e^{3} x^{2}}{288 c^{4}}\)	\(329\)
default	\(\frac {288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{3} x +432 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d^{2} e \,x^{2}+288 \operatorname {arccosh}\left (c x \right )^{2} c^{4} d \,e^{2} x^{3}+72 \operatorname {arccosh}\left (c x \right )^{2} e^{3} c^{4} x^{4}-576 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d^{3}-432 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d^{2} e x -192 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{3} d \,e^{2} x^{2}-36 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{3} c^{3} x^{3}-216 \operatorname {arccosh}\left (c x \right )^{2} c^{2} d^{2} e -384 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c d \,e^{2}-54 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) e^{3} c x +576 c^{4} d^{3} x +216 c^{4} d^{2} e \,x^{2}+64 c^{4} d \,e^{2} x^{3}+9 c^{4} e^{3} x^{4}-27 \operatorname {arccosh}\left (c x \right )^{2} e^{3}+384 c^{2} d \,e^{2} x +27 c^{2} e^{3} x^{2}}{288 c^{4}}\)	\(329\)
orering	\(\frac {\left (111 e^{5} c^{4} x^{6}+699 e^{4} c^{4} x^{5} d +1928 e^{3} c^{4} x^{4} d^{2}+3480 e^{2} c^{4} x^{3} d^{3}+672 c^{4} d^{4} e \,x^{2}+63 e^{5} c^{2} x^{4}+192 c^{4} d^{5} x +1079 e^{4} c^{2} x^{3} d -1632 e^{3} c^{2} x^{2} d^{2}-3600 e^{2} c^{2} d^{3} x -720 c^{2} d^{4} e -180 e^{5} x^{2}-2010 e^{4} d x -402 e^{3} d^{2}\right ) \operatorname {arccosh}\left (c x \right )^{2}}{192 \left (e x +d \right )^{2} c^{4}}-\frac {\left (81 e^{4} c^{4} x^{6}+539 e^{3} c^{4} x^{5} d +1640 e^{2} c^{4} x^{4} d^{2}+3672 e \,c^{4} x^{3} d^{3}+99 e^{4} c^{2} x^{4}+1719 e^{3} c^{2} x^{3} d -1920 e^{2} c^{2} x^{2} d^{2}-4464 e \,c^{2} d^{3} x -576 c^{2} d^{4}-216 e^{4} x^{2}-2742 e^{3} d x -384 e^{2} d^{2}\right ) \left (3 \left (e x +d \right )^{2} \operatorname {arccosh}\left (c x \right )^{2} e +\frac {2 \left (e x +d \right )^{3} \operatorname {arccosh}\left (c x \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{576 \left (e x +d \right )^{4} c^{4}}+\frac {x \left (9 e^{3} c^{2} x^{3}+64 e^{2} c^{2} x^{2} d +216 e \,c^{2} d^{2} x +576 c^{2} d^{3}+27 e^{3} x +384 d \,e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (6 \left (e x +d \right ) \operatorname {arccosh}\left (c x \right )^{2} e^{2}+\frac {12 \left (e x +d \right )^{2} \operatorname {arccosh}\left (c x \right ) e c}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {2 c^{2} \left (e x +d \right )^{3}}{\left (c x -1\right ) \left (c x +1\right )}-\frac {\left (e x +d \right )^{3} \operatorname {arccosh}\left (c x \right ) c^{2}}{\left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {\left (e x +d \right )^{3} \operatorname {arccosh}\left (c x \right ) c^{2}}{\sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}\right )}{576 c^{4} \left (e x +d \right )^{3}}\)	\(568\)

Input:

int((e*x+d)^3*arccosh(c*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/288/c^4*(288*arccosh(c*x)^2*c^4*d^3*x+432*arccosh(c*x)^2*c^4*d^2*e*x^2+2 
88*arccosh(c*x)^2*c^4*d*e^2*x^3+72*arccosh(c*x)^2*e^3*c^4*x^4-576*(c*x-1)^ 
(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^3*d^3-432*(c*x-1)^(1/2)*(c*x+1)^(1/2)*a 
rccosh(c*x)*c^3*d^2*e*x-192*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c^3*d 
*e^2*x^2-36*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*e^3*c^3*x^3-216*arcco 
sh(c*x)^2*c^2*d^2*e-384*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*c*d*e^2-5 
4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*e^3*c*x+576*c^4*d^3*x+216*c^4*d 
^2*e*x^2+64*c^4*d*e^2*x^3+9*c^4*e^3*x^4-27*arccosh(c*x)^2*e^3+384*c^2*d*e^ 
2*x+27*c^2*e^3*x^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.71 \[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\frac {9 \, c^{4} e^{3} x^{4} + 64 \, c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, c^{4} d^{2} e + c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, c^{4} e^{3} x^{4} + 32 \, c^{4} d e^{2} x^{3} + 48 \, c^{4} d^{2} e x^{2} + 32 \, c^{4} d^{3} x - 24 \, c^{2} d^{2} e - 3 \, e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (6 \, c^{3} e^{3} x^{3} + 32 \, c^{3} d e^{2} x^{2} + 96 \, c^{3} d^{3} + 64 \, c d e^{2} + 9 \, {\left (8 \, c^{3} d^{2} e + c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 192 \, {\left (3 \, c^{4} d^{3} + 2 \, c^{2} d e^{2}\right )} x}{288 \, c^{4}} \] Input:

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="fricas")
 

Output:

1/288*(9*c^4*e^3*x^4 + 64*c^4*d*e^2*x^3 + 27*(8*c^4*d^2*e + c^2*e^3)*x^2 + 
 9*(8*c^4*e^3*x^4 + 32*c^4*d*e^2*x^3 + 48*c^4*d^2*e*x^2 + 32*c^4*d^3*x - 2 
4*c^2*d^2*e - 3*e^3)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(6*c^3*e^3*x^3 + 3 
2*c^3*d*e^2*x^2 + 96*c^3*d^3 + 64*c*d*e^2 + 9*(8*c^3*d^2*e + c*e^3)*x)*sqr 
t(c^2*x^2 - 1)*log(c*x + sqrt(c^2*x^2 - 1)) + 192*(3*c^4*d^3 + 2*c^2*d*e^2 
)*x)/c^4
 

Sympy [F]

\[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\int \left (d + e x\right )^{3} \operatorname {acosh}^{2}{\left (c x \right )}\, dx \] Input:

integrate((e*x+d)**3*acosh(c*x)**2,x)
 

Output:

Integral((d + e*x)**3*acosh(c*x)**2, x)
 

Maxima [F]

\[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\int { {\left (e x + d\right )}^{3} \operatorname {arcosh}\left (c x\right )^{2} \,d x } \] Input:

integrate((e*x+d)^3*arccosh(c*x)^2,x, algorithm="maxima")
 

Output:

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)*log(c*x + sqrt(c*x + 1 
)*sqrt(c*x - 1))^2 - integrate(1/2*(c^3*e^3*x^6 + 4*c^3*d*e^2*x^5 - 6*c*d^ 
2*e*x^2 - 4*c*d^3*x + (6*c^3*d^2*e - c*e^3)*x^4 + 4*(c^3*d^3 - c*d*e^2)*x^ 
3 + (c^2*e^3*x^5 + 4*c^2*d*e^2*x^4 + 6*c^2*d^2*e*x^3 + 4*c^2*d^3*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. \[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^3*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. \[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\int {\mathrm {acosh}\left (c\,x\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \] Input:

int(acosh(c*x)^2*(d + e*x)^3,x)
 

Output:

int(acosh(c*x)^2*(d + e*x)^3, x)
 

Reduce [F]

\[ \int (d+e x)^3 \text {arccosh}(c x)^2 \, dx=\left (\int \mathit {acosh} \left (c x \right )^{2}d x \right ) d^{3}+\left (\int \mathit {acosh} \left (c x \right )^{2} x^{3}d x \right ) e^{3}+3 \left (\int \mathit {acosh} \left (c x \right )^{2} x^{2}d x \right ) d \,e^{2}+3 \left (\int \mathit {acosh} \left (c x \right )^{2} x d x \right ) d^{2} e \] Input:

int((e*x+d)^3*acosh(c*x)^2,x)
                                                                                    
                                                                                    
 

Output:

int(acosh(c*x)**2,x)*d**3 + int(acosh(c*x)**2*x**3,x)*e**3 + 3*int(acosh(c 
*x)**2*x**2,x)*d*e**2 + 3*int(acosh(c*x)**2*x,x)*d**2*e