\(\int x (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [381]

Optimal result

Integrand size = 19, antiderivative size = 311 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (256 c^6 d^3+288 c^4 d^2 e+160 c^2 d e^2+35 e^3\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{1024 c^7}-\frac {b e \left (288 c^4 d^2+160 c^2 d e+35 e^2\right ) x^3 \sqrt {-1+c x} \sqrt {1+c x}}{1536 c^5}-\frac {b e^2 \left (32 c^2 d+7 e\right ) x^5 \sqrt {-1+c x} \sqrt {1+c x}}{384 c^3}-\frac {b e^3 x^7 \sqrt {-1+c x} \sqrt {1+c x}}{64 c}+\frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{1024 c^8 e \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/1024*b*(256*c^6*d^3+288*c^4*d^2*e+160*c^2*d*e^2+35*e^3)*x*(c*x-1)^(1/2) 
*(c*x+1)^(1/2)/c^7-1/1536*b*e*(288*c^4*d^2+160*c^2*d*e+35*e^2)*x^3*(c*x-1) 
^(1/2)*(c*x+1)^(1/2)/c^5-1/384*b*e^2*(32*c^2*d+7*e)*x^5*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)/c^3-1/64*b*e^3*x^7*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c+1/8*(e*x^2+d)^4 
*(a+b*arccosh(c*x))/e-1/1024*b*(128*c^8*d^4+256*c^6*d^3*e+288*c^4*d^2*e^2+ 
160*c^2*d*e^3+35*e^4)*(c^2*x^2-1)^(1/2)*arctanh(c*x/(c^2*x^2-1)^(1/2))/c^8 
/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.26 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.82 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {c x \left (384 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )-b \sqrt {-1+c x} \sqrt {1+c x} \left (105 e^3+10 c^2 e^2 \left (48 d+7 e x^2\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+16 c^6 \left (48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right )\right )\right )+384 b c^8 x^2 \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right ) \text {arccosh}(c x)-6 b \left (256 c^6 d^3+288 c^4 d^2 e+160 c^2 d e^2+35 e^3\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{3072 c^8} \] Input:

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

Output:

(c*x*(384*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) - b*Sqrt[- 
1 + c*x]*Sqrt[1 + c*x]*(105*e^3 + 10*c^2*e^2*(48*d + 7*e*x^2) + 8*c^4*e*(1 
08*d^2 + 40*d*e*x^2 + 7*e^2*x^4) + 16*c^6*(48*d^3 + 36*d^2*e*x^2 + 16*d*e^ 
2*x^4 + 3*e^3*x^6))) + 384*b*c^8*x^2*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + 
e^3*x^6)*ArcCosh[c*x] - 6*b*(256*c^6*d^3 + 288*c^4*d^2*e + 160*c^2*d*e^2 + 
 35*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(3072*c^8)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6372, 648, 318, 403, 403, 299, 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.

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

\(\Big \downarrow \) 6372

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\int \frac {\left (e x^2+d\right ) \left (e \left (104 d^2 c^4+104 d e c^2+35 e^2\right ) x^2+d \left (48 d^2 c^4+20 d e c^2+7 e^2\right )\right )}{\sqrt {c^2 x^2-1}}dx}{6 c^2}+\frac {7 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\int \frac {5 e \left (2 d c^2+e\right ) \left (40 d^2 c^4+40 d e c^2+21 e^2\right ) x^2+d \left (192 d^3 c^6+184 d^2 e c^4+132 d e^2 c^2+35 e^3\right )}{\sqrt {c^2 x^2-1}}dx}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {7 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {3 \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {5 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{2 c^2}}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {7 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {3 \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {5 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{2 c^2}}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {7 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\frac {3 \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right )}{2 c^3}+\frac {5 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{2 c^2}}{4 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{4 c^2}}{6 c^2}+\frac {7 e x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{6 c^2}}{8 c^2}+\frac {e x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^3}{8 c^2}\right )}{8 e \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

((d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e) - (b*c*Sqrt[-1 + c^2*x^2]*((e*x 
*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^3)/(8*c^2) + ((7*e*(2*c^2*d + e)*x*Sqrt[-1 
 + c^2*x^2]*(d + e*x^2)^2)/(6*c^2) + ((e*(104*c^4*d^2 + 104*c^2*d*e + 35*e 
^2)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2))/(4*c^2) + ((5*e*(2*c^2*d + e)*(40*c^ 
4*d^2 + 40*c^2*d*e + 21*e^2)*x*Sqrt[-1 + c^2*x^2])/(2*c^2) + (3*(128*c^8*d 
^4 + 256*c^6*d^3*e + 288*c^4*d^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*ArcTanh[(c* 
x)/Sqrt[-1 + c^2*x^2]])/(2*c^3))/(4*c^2))/(6*c^2))/(8*c^2)))/(8*e*Sqrt[-1 
+ c*x]*Sqrt[1 + c*x])
 

Defintions of rubi rules used

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 299

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

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 403

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

rule 648

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

rule 6372

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.38

method	result	size
orering	\(\frac {\left (720 c^{8} e^{4} x^{10}+3760 c^{8} d \,e^{3} x^{8}+8128 c^{8} d^{2} e^{2} x^{6}+56 c^{6} e^{4} x^{8}+9792 c^{8} d^{3} e \,x^{4}+456 c^{6} d \,e^{3} x^{6}+2304 c^{8} d^{4} x^{2}+2080 c^{6} d^{2} e^{2} x^{4}+98 c^{4} e^{4} x^{6}-5856 c^{6} d^{3} e \,x^{2}+1134 c^{4} d \,e^{3} x^{4}-1536 c^{6} d^{4}-6752 c^{4} d^{2} e^{2} x^{2}+245 c^{2} e^{4} x^{4}-1728 c^{4} d^{3} e -3805 c^{2} d \,e^{3} x^{2}-960 c^{2} d^{2} e^{2}-840 e^{4} x^{2}-210 d \,e^{3}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{3072 \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (48 c^{6} e^{3} x^{6}+256 c^{6} d \,e^{2} x^{4}+576 c^{6} d^{2} e \,x^{2}+56 c^{4} e^{3} x^{4}+768 c^{6} d^{3}+320 c^{4} d \,e^{2} x^{2}+864 c^{4} d^{2} e +70 c^{2} e^{3} x^{2}+480 c^{2} d \,e^{2}+105 e^{3}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (\left (e \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+6 x^{2} \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e +\frac {x \left (e \,x^{2}+d \right )^{3} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{3072 c^{8} \left (e \,x^{2}+d \right )^{3}}\)	\(430\)
parts	\(\frac {a \left (e \,x^{2}+d \right )^{4}}{8 e}+\frac {b \left (\frac {c^{2} e^{3} \operatorname {arccosh}\left (c x \right ) x^{8}}{8}+\frac {c^{2} e^{2} \operatorname {arccosh}\left (c x \right ) x^{6} d}{2}+\frac {3 c^{2} e \,\operatorname {arccosh}\left (c x \right ) x^{4} d^{2}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{2} x^{2} d^{3}}{2}+\frac {c^{2} \operatorname {arccosh}\left (c x \right ) d^{4}}{8 e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (384 d^{4} c^{8} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+768 d^{3} c^{7} e x \sqrt {c^{2} x^{2}-1}+576 c^{7} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+256 c^{7} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{5}+48 e^{4} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+768 d^{3} c^{6} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+864 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{5} d^{2} x +320 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} d \,x^{3}+56 e^{4} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) c^{4} d^{2}+480 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{3} d x +70 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+480 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) c^{2} d +105 e^{4} c x \sqrt {c^{2} x^{2}-1}+105 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 c^{6} e \sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\)	\(476\)
derivativedivides	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{4} c^{8}}{8 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \operatorname {arccosh}\left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (384 d^{4} c^{8} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+768 d^{3} c^{7} e x \sqrt {c^{2} x^{2}-1}+576 c^{7} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+256 c^{7} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{5}+48 e^{4} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+768 d^{3} c^{6} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+864 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{5} d^{2} x +320 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} d \,x^{3}+56 e^{4} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) c^{4} d^{2}+480 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{3} d x +70 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+480 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) c^{2} d +105 e^{4} c x \sqrt {c^{2} x^{2}-1}+105 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{2}}\)	\(487\)
default	\(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{8 c^{6} e}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{4} c^{8}}{8 e}+\frac {\operatorname {arccosh}\left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \,\operatorname {arccosh}\left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \operatorname {arccosh}\left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \operatorname {arccosh}\left (c x \right ) c^{8} x^{8}}{8}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (384 d^{4} c^{8} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+768 d^{3} c^{7} e x \sqrt {c^{2} x^{2}-1}+576 c^{7} d^{2} e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+256 c^{7} d \,e^{3} \sqrt {c^{2} x^{2}-1}\, x^{5}+48 e^{4} \sqrt {c^{2} x^{2}-1}\, c^{7} x^{7}+768 d^{3} c^{6} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+864 e^{2} \sqrt {c^{2} x^{2}-1}\, c^{5} d^{2} x +320 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} d \,x^{3}+56 e^{4} c^{5} x^{5} \sqrt {c^{2} x^{2}-1}+864 e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) c^{4} d^{2}+480 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{3} d x +70 e^{4} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+480 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) c^{2} d +105 e^{4} c x \sqrt {c^{2} x^{2}-1}+105 e^{4} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{3072 e \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{2}}\)	\(487\)

Input:

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

Output:

1/3072*(720*c^8*e^4*x^10+3760*c^8*d*e^3*x^8+8128*c^8*d^2*e^2*x^6+56*c^6*e^ 
4*x^8+9792*c^8*d^3*e*x^4+456*c^6*d*e^3*x^6+2304*c^8*d^4*x^2+2080*c^6*d^2*e 
^2*x^4+98*c^4*e^4*x^6-5856*c^6*d^3*e*x^2+1134*c^4*d*e^3*x^4-1536*c^6*d^4-6 
752*c^4*d^2*e^2*x^2+245*c^2*e^4*x^4-1728*c^4*d^3*e-3805*c^2*d*e^3*x^2-960* 
c^2*d^2*e^2-840*e^4*x^2-210*d*e^3)/(e*x^2+d)/c^8*(a+b*arccosh(c*x))-1/3072 
*(48*c^6*e^3*x^6+256*c^6*d*e^2*x^4+576*c^6*d^2*e*x^2+56*c^4*e^3*x^4+768*c^ 
6*d^3+320*c^4*d*e^2*x^2+864*c^4*d^2*e+70*c^2*e^3*x^2+480*c^2*d*e^2+105*e^3 
)/c^8/(e*x^2+d)^3*(c*x-1)*(c*x+1)*((e*x^2+d)^3*(a+b*arccosh(c*x))+6*x^2*(e 
*x^2+d)^2*(a+b*arccosh(c*x))*e+x*(e*x^2+d)^3*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/ 
2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.92 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {384 \, a c^{8} e^{3} x^{8} + 1536 \, a c^{8} d e^{2} x^{6} + 2304 \, a c^{8} d^{2} e x^{4} + 1536 \, a c^{8} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} e^{3} x^{8} + 512 \, b c^{8} d e^{2} x^{6} + 768 \, b c^{8} d^{2} e x^{4} + 512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} - 288 \, b c^{4} d^{2} e - 160 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} e^{3} x^{7} + 8 \, {\left (32 \, b c^{7} d e^{2} + 7 \, b c^{5} e^{3}\right )} x^{5} + 2 \, {\left (288 \, b c^{7} d^{2} e + 160 \, b c^{5} d e^{2} + 35 \, b c^{3} e^{3}\right )} x^{3} + 3 \, {\left (256 \, b c^{7} d^{3} + 288 \, b c^{5} d^{2} e + 160 \, b c^{3} d e^{2} + 35 \, b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{3072 \, c^{8}} \] Input:

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

Output:

1/3072*(384*a*c^8*e^3*x^8 + 1536*a*c^8*d*e^2*x^6 + 2304*a*c^8*d^2*e*x^4 + 
1536*a*c^8*d^3*x^2 + 3*(128*b*c^8*e^3*x^8 + 512*b*c^8*d*e^2*x^6 + 768*b*c^ 
8*d^2*e*x^4 + 512*b*c^8*d^3*x^2 - 256*b*c^6*d^3 - 288*b*c^4*d^2*e - 160*b* 
c^2*d*e^2 - 35*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (48*b*c^7*e^3*x^7 + 8 
*(32*b*c^7*d*e^2 + 7*b*c^5*e^3)*x^5 + 2*(288*b*c^7*d^2*e + 160*b*c^5*d*e^2 
 + 35*b*c^3*e^3)*x^3 + 3*(256*b*c^7*d^3 + 288*b*c^5*d^2*e + 160*b*c^3*d*e^ 
2 + 35*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^8
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.32 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{2} e + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d e^{2} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b e^{3} \] Input:

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

Output:

1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/2*a*d^3*x^2 + 1/4*(2 
*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2* 
x^2 - 1)*c)/c^3))*b*d^3 + 3/32*(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*e + 1/96*(48*x^6*arccosh(c*x) - (8*sqrt(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^2 + 1/3072*(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*s 
qrt(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^3
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.39 \[ \int x \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1536 \mathit {acosh} \left (c x \right ) b \,c^{8} d^{3} x^{2}+2304 \mathit {acosh} \left (c x \right ) b \,c^{8} d^{2} e \,x^{4}+1536 \mathit {acosh} \left (c x \right ) b \,c^{8} d \,e^{2} x^{6}+384 \mathit {acosh} \left (c x \right ) b \,c^{8} e^{3} x^{8}-768 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} d^{3} x -576 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} d^{2} e \,x^{3}-256 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} d \,e^{2} x^{5}-48 \sqrt {c^{2} x^{2}-1}\, b \,c^{7} e^{3} x^{7}-864 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} d^{2} e x -320 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} d \,e^{2} x^{3}-56 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} e^{3} x^{5}-480 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} d \,e^{2} x -70 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} e^{3} x^{3}-105 \sqrt {c^{2} x^{2}-1}\, b c \,e^{3} x -768 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{6} d^{3}-864 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{4} d^{2} e -480 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,c^{2} d \,e^{2}-105 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b \,e^{3}+1536 a \,c^{8} d^{3} x^{2}+2304 a \,c^{8} d^{2} e \,x^{4}+1536 a \,c^{8} d \,e^{2} x^{6}+384 a \,c^{8} e^{3} x^{8}}{3072 c^{8}} \] Input:

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

Output:

(1536*acosh(c*x)*b*c**8*d**3*x**2 + 2304*acosh(c*x)*b*c**8*d**2*e*x**4 + 1 
536*acosh(c*x)*b*c**8*d*e**2*x**6 + 384*acosh(c*x)*b*c**8*e**3*x**8 - 768* 
sqrt(c**2*x**2 - 1)*b*c**7*d**3*x - 576*sqrt(c**2*x**2 - 1)*b*c**7*d**2*e* 
x**3 - 256*sqrt(c**2*x**2 - 1)*b*c**7*d*e**2*x**5 - 48*sqrt(c**2*x**2 - 1) 
*b*c**7*e**3*x**7 - 864*sqrt(c**2*x**2 - 1)*b*c**5*d**2*e*x - 320*sqrt(c** 
2*x**2 - 1)*b*c**5*d*e**2*x**3 - 56*sqrt(c**2*x**2 - 1)*b*c**5*e**3*x**5 - 
 480*sqrt(c**2*x**2 - 1)*b*c**3*d*e**2*x - 70*sqrt(c**2*x**2 - 1)*b*c**3*e 
**3*x**3 - 105*sqrt(c**2*x**2 - 1)*b*c*e**3*x - 768*log(sqrt(c**2*x**2 - 1 
) + c*x)*b*c**6*d**3 - 864*log(sqrt(c**2*x**2 - 1) + c*x)*b*c**4*d**2*e - 
480*log(sqrt(c**2*x**2 - 1) + c*x)*b*c**2*d*e**2 - 105*log(sqrt(c**2*x**2 
- 1) + c*x)*b*e**3 + 1536*a*c**8*d**3*x**2 + 2304*a*c**8*d**2*e*x**4 + 153 
6*a*c**8*d*e**2*x**6 + 384*a*c**8*e**3*x**8)/(3072*c**8)