\(\int x^3 (d+e x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [370]

Optimal result

Integrand size = 21, antiderivative size = 301 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{3072 c^7}-\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{4608 c^5}-\frac {b e \left (64 c^2 d+21 e\right ) x^5 \sqrt {-1+c x} \sqrt {1+c x}}{1152 c^3}-\frac {b e^2 x^7 \sqrt {-1+c x} \sqrt {1+c x}}{64 c}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))-\frac {b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{3072 c^8 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/3072*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/ 
c^7-1/4608*b*(288*c^4*d^2+320*c^2*d*e+105*e^2)*x^3*(c*x-1)^(1/2)*(c*x+1)^( 
1/2)/c^5-1/1152*b*e*(64*c^2*d+21*e)*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/ 
64*b*e^2*x^7*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+1/4*d^2*x^4*(a+b*arccosh(c*x))+ 
1/3*d*e*x^6*(a+b*arccosh(c*x))+1/8*e^2*x^8*(a+b*arccosh(c*x))-1/3072*b*(28 
8*c^4*d^2+320*c^2*d*e+105*e^2)*(c^2*x^2-1)^(1/2)*arctanh(c*x/(c^2*x^2-1)^( 
1/2))/c^8/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.71 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {384 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \left (315 e^2+30 c^2 e \left (32 d+7 e x^2\right )+8 c^4 \left (108 d^2+80 d e x^2+21 e^2 x^4\right )+16 c^6 \left (36 d^2 x^2+32 d e x^4+9 e^2 x^6\right )\right )+384 b c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \text {arccosh}(c x)-6 b \left (288 c^4 d^2+320 c^2 d e+105 e^2\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{9216 c^8} \] Input:

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

Output:

(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) - b*c*x*Sqrt[-1 + c*x]*Sqrt 
[1 + c*x]*(315*e^2 + 30*c^2*e*(32*d + 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x 
^2 + 21*e^2*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 384*b*c 
^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x] - 6*b*(288*c^4*d^2 + 3 
20*c^2*d*e + 105*e^2)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(9216*c^8)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6373, 27, 1905, 1590, 363, 262, 262, 224, 219}

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.

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

\(\Big \downarrow \) 6373

\(\displaystyle -b c \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{24} b c \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1905

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {x^4 \left (3 e^2 x^4+8 d e x^2+6 d^2\right )}{\sqrt {c^2 x^2-1}}dx}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {x^4 \left (48 c^2 d^2+e \left (64 d c^2+21 e\right ) x^2\right )}{\sqrt {c^2 x^2-1}}dx}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \int \frac {x^4}{\sqrt {c^2 x^2-1}}dx}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \left (\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right ) \left (\frac {3 \left (\frac {\int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{4} d^2 x^4 (a+b \text {arccosh}(c x))+\frac {1}{3} d e x^6 (a+b \text {arccosh}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{2 c^3}+\frac {x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2-1}}{4 c^2}\right ) \left (288 c^4 d^2+5 e \left (64 c^2 d+21 e\right )\right )}{6 c^2}+\frac {e x^5 \sqrt {c^2 x^2-1} \left (64 c^2 d+21 e\right )}{6 c^2}}{8 c^2}+\frac {3 e^2 x^7 \sqrt {c^2 x^2-1}}{8 c^2}\right )}{24 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

(d^2*x^4*(a + b*ArcCosh[c*x]))/4 + (d*e*x^6*(a + b*ArcCosh[c*x]))/3 + (e^2 
*x^8*(a + b*ArcCosh[c*x]))/8 - (b*c*Sqrt[-1 + c^2*x^2]*((3*e^2*x^7*Sqrt[-1 
 + c^2*x^2])/(8*c^2) + ((e*(64*c^2*d + 21*e)*x^5*Sqrt[-1 + c^2*x^2])/(6*c^ 
2) + ((288*c^4*d^2 + 5*e*(64*c^2*d + 21*e))*((x^3*Sqrt[-1 + c^2*x^2])/(4*c 
^2) + (3*((x*Sqrt[-1 + c^2*x^2])/(2*c^2) + ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2 
]]/(2*c^3)))/(4*c^2)))/(6*c^2))/(8*c^2)))/(24*Sqrt[-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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

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 6373

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]}, Sim 
p[(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] && 
NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le 
Q[m + p, 0]))
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.21

method	result	size
parts	\(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} d^{2} x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccosh}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccosh}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4} d^{2}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (576 c^{7} d^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+512 c^{7} d e \sqrt {c^{2} x^{2}-1}\, x^{5}+144 e^{2} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+864 \sqrt {c^{2} x^{2}-1}\, c^{5} d^{2} x +640 e \sqrt {c^{2} x^{2}-1}\, c^{5} d \,x^{3}+168 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 d^{2} c^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+960 c^{3} d e x \sqrt {c^{2} x^{2}-1}+210 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+960 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+315 e^{2} c x \sqrt {c^{2} x^{2}-1}+315 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 c^{4} \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(364\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} e^{2} x^{8} c^{8}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (576 c^{7} d^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+512 c^{7} d e \sqrt {c^{2} x^{2}-1}\, x^{5}+144 e^{2} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+864 \sqrt {c^{2} x^{2}-1}\, c^{5} d^{2} x +640 e \sqrt {c^{2} x^{2}-1}\, c^{5} d \,x^{3}+168 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 d^{2} c^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+960 c^{3} d e x \sqrt {c^{2} x^{2}-1}+210 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+960 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+315 e^{2} c x \sqrt {c^{2} x^{2}-1}+315 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}}{c^{4}}\)	\(377\)
default	\(\frac {\frac {a \left (\frac {1}{4} c^{8} d^{2} x^{4}+\frac {1}{3} c^{8} d e \,x^{6}+\frac {1}{8} e^{2} x^{8} c^{8}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{8} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) d \,c^{8} e \,x^{6}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (576 c^{7} d^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+512 c^{7} d e \sqrt {c^{2} x^{2}-1}\, x^{5}+144 e^{2} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+864 \sqrt {c^{2} x^{2}-1}\, c^{5} d^{2} x +640 e \sqrt {c^{2} x^{2}-1}\, c^{5} d \,x^{3}+168 e^{2} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 d^{2} c^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+960 c^{3} d e x \sqrt {c^{2} x^{2}-1}+210 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{3} x^{3}+960 d \,c^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+315 e^{2} c x \sqrt {c^{2} x^{2}-1}+315 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{9216 \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}}{c^{4}}\)	\(377\)
orering	\(\frac {\left (2160 c^{8} e^{3} x^{10}+8240 c^{8} d \,e^{2} x^{8}+10944 c^{8} d^{2} e \,x^{6}+168 x^{8} e^{3} c^{6}+4032 c^{8} d^{3} x^{4}+968 x^{6} e^{2} c^{6} d +2400 x^{4} e \,c^{6} d^{2}+294 x^{6} e^{3} c^{4}+864 c^{6} d^{3} x^{2}+2366 x^{4} e^{2} c^{4} d -5952 x^{2} e \,c^{4} d^{2}+735 x^{4} e^{3} c^{2}-3456 c^{4} d^{3}-7365 x^{2} e^{2} c^{2} d -3840 c^{2} d^{2} e -2520 x^{2} e^{3}-1260 d \,e^{2}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{9216 \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (144 x^{6} c^{6} e^{2}+512 x^{4} c^{6} d e +576 x^{2} c^{6} d^{2}+168 c^{4} e^{2} x^{4}+640 c^{4} d e \,x^{2}+864 c^{4} d^{2}+210 c^{2} e^{2} x^{2}+960 c^{2} d e +315 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (3 x^{2} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+4 x^{4} \left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e +\frac {x^{3} \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{9216 x^{2} c^{8} \left (e \,x^{2}+d \right )^{2}}\)	\(391\)

Input:

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

Output:

a*(1/8*e^2*x^8+1/3*d*e*x^6+1/4*d^2*x^4)+b/c^4*(1/8*c^4*arccosh(c*x)*e^2*x^ 
8+1/3*c^4*arccosh(c*x)*d*e*x^6+1/4*arccosh(c*x)*c^4*x^4*d^2-1/9216/c^4*(c* 
x-1)^(1/2)*(c*x+1)^(1/2)*(576*c^7*d^2*(c^2*x^2-1)^(1/2)*x^3+512*c^7*d*e*(c 
^2*x^2-1)^(1/2)*x^5+144*e^2*c^7*x^7*(c^2*x^2-1)^(1/2)+864*(c^2*x^2-1)^(1/2 
)*c^5*d^2*x+640*e*(c^2*x^2-1)^(1/2)*c^5*d*x^3+168*e^2*c^5*x^5*(c^2*x^2-1)^ 
(1/2)+864*d^2*c^4*ln(c*x+(c^2*x^2-1)^(1/2))+960*c^3*d*e*x*(c^2*x^2-1)^(1/2 
)+210*e^2*(c^2*x^2-1)^(1/2)*c^3*x^3+960*d*c^2*e*ln(c*x+(c^2*x^2-1)^(1/2))+ 
315*e^2*c*x*(c^2*x^2-1)^(1/2)+315*e^2*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2- 
1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.75 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1152 \, a c^{8} e^{2} x^{8} + 3072 \, a c^{8} d e x^{6} + 2304 \, a c^{8} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} e^{2} x^{8} + 1024 \, b c^{8} d e x^{6} + 768 \, b c^{8} d^{2} x^{4} - 288 \, b c^{4} d^{2} - 320 \, b c^{2} d e - 105 \, b e^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (144 \, b c^{7} e^{2} x^{7} + 8 \, {\left (64 \, b c^{7} d e + 21 \, b c^{5} e^{2}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} + 320 \, b c^{5} d e + 105 \, b c^{3} e^{2}\right )} x^{3} + 3 \, {\left (288 \, b c^{5} d^{2} + 320 \, b c^{3} d e + 105 \, b c e^{2}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{9216 \, c^{8}} \] Input:

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

Output:

1/9216*(1152*a*c^8*e^2*x^8 + 3072*a*c^8*d*e*x^6 + 2304*a*c^8*d^2*x^4 + 3*( 
384*b*c^8*e^2*x^8 + 1024*b*c^8*d*e*x^6 + 768*b*c^8*d^2*x^4 - 288*b*c^4*d^2 
 - 320*b*c^2*d*e - 105*b*e^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (144*b*c^7*e^ 
2*x^7 + 8*(64*b*c^7*d*e + 21*b*c^5*e^2)*x^5 + 2*(288*b*c^7*d^2 + 320*b*c^5 
*d*e + 105*b*c^3*e^2)*x^3 + 3*(288*b*c^5*d^2 + 320*b*c^3*d*e + 105*b*c*e^2 
)*x)*sqrt(c^2*x^2 - 1))/c^8
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.10 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d e + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b e^{2} \] Input:

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

Output:

1/8*a*e^2*x^8 + 1/3*a*d*e*x^6 + 1/4*a*d^2*x^4 + 1/32*(8*x^4*arccosh(c*x) - 
 (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x 
+ 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*d^2 + 1/144*(48*x^6*arccosh(c*x) - (8*s 
qrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 
- 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*d*e + 1/307 
2*(384*x^8*arccosh(c*x) - (48*sqrt(c^2*x^2 - 1)*x^7/c^2 + 56*sqrt(c^2*x^2 
- 1)*x^5/c^4 + 70*sqrt(c^2*x^2 - 1)*x^3/c^6 + 105*sqrt(c^2*x^2 - 1)*x/c^8 
+ 105*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^9)*c)*b*e^2
 

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.14 \[ \int x^3 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {2304 \mathit {acosh} \left (c x \right ) b \,c^{8} d^{2} x^{4}+3072 \mathit {acosh} \left (c x \right ) b \,c^{8} d e \,x^{6}+1152 \mathit {acosh} \left (c x \right ) b \,c^{8} e^{2} x^{8}-576 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} d^{2} x^{3}-512 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} d e \,x^{5}-144 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} e^{2} x^{7}-864 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} d^{2} x -640 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} d e \,x^{3}-168 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} e^{2} x^{5}-960 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} d e x -210 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} e^{2} x^{3}-315 \sqrt {c^{2} x^{2}-1}\, b c \,e^{2} x -864 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{4} d^{2}-960 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{2} d e -315 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,e^{2}+2304 a \,c^{8} d^{2} x^{4}+3072 a \,c^{8} d e \,x^{6}+1152 a \,c^{8} e^{2} x^{8}}{9216 c^{8}} \] Input:

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

Output:

(2304*acosh(c*x)*b*c**8*d**2*x**4 + 3072*acosh(c*x)*b*c**8*d*e*x**6 + 1152 
*acosh(c*x)*b*c**8*e**2*x**8 - 576*sqrt(c**2*x**2 - 1)*b*c**7*d**2*x**3 - 
512*sqrt(c**2*x**2 - 1)*b*c**7*d*e*x**5 - 144*sqrt(c**2*x**2 - 1)*b*c**7*e 
**2*x**7 - 864*sqrt(c**2*x**2 - 1)*b*c**5*d**2*x - 640*sqrt(c**2*x**2 - 1) 
*b*c**5*d*e*x**3 - 168*sqrt(c**2*x**2 - 1)*b*c**5*e**2*x**5 - 960*sqrt(c** 
2*x**2 - 1)*b*c**3*d*e*x - 210*sqrt(c**2*x**2 - 1)*b*c**3*e**2*x**3 - 315* 
sqrt(c**2*x**2 - 1)*b*c*e**2*x - 864*log(sqrt(c**2*x**2 - 1) + c*x)*b*c**4 
*d**2 - 960*log(sqrt(c**2*x**2 - 1) + c*x)*b*c**2*d*e - 315*log(sqrt(c**2* 
x**2 - 1) + c*x)*b*e**2 + 2304*a*c**8*d**2*x**4 + 3072*a*c**8*d*e*x**6 + 1 
152*a*c**8*e**2*x**8)/(9216*c**8)