Integrand size = 10, antiderivative size = 152 \[ \int x^3 \text {arccosh}(a+b x) \, dx=\frac {7 a x^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{48 b^2}-\frac {x^3 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{16 b}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (4 a \left (16+19 a^2\right )-\left (9+26 a^2\right ) (a+b x)\right )}{96 b^4}-\frac {\left (3+24 a^2+8 a^4\right ) \text {arccosh}(a+b x)}{32 b^4}+\frac {1}{4} x^4 \text {arccosh}(a+b x) \]
-1/32*(8*a^4+24*a^2+3)*arccosh(b*x+a)/b^4+1/4*x^4*arccosh(b*x+a)+7/48*a*x^ 2*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/b^2-1/16*x^3*(b*x+a-1)^(1/2)*(b*x+a+1)^( 1/2)/b+1/96*(4*a*(19*a^2+16)-(26*a^2+9)*(b*x+a))*(b*x+a-1)^(1/2)*(b*x+a+1) ^(1/2)/b^4
Time = 0.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int x^3 \text {arccosh}(a+b x) \, dx=\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (55 a+50 a^3-9 b x-26 a^2 b x+14 a b^2 x^2-6 b^3 x^3\right )+24 b^4 x^4 \text {arccosh}(a+b x)-3 \left (3+24 a^2+8 a^4\right ) \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{96 b^4} \]
(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(55*a + 50*a^3 - 9*b*x - 26*a^2*b*x + 14*a*b^2*x^2 - 6*b^3*x^3) + 24*b^4*x^4*ArcCosh[a + b*x] - 3*(3 + 24*a^2 + 8*a^4)*Log[a + b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]])/(96*b^4)
Time = 0.37 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6411, 25, 27, 6378, 111, 170, 164, 43}
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.
| \(\displaystyle \int x^3 \text {arccosh}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int x^3 \text {arccosh}(a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \text {arccosh}(a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \text {arccosh}(a+b x)d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle -\frac {\frac {1}{4} \int \frac {b^4 x^4}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{4} b^4 x^4 \text {arccosh}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \int \frac {b^2 x^2 \left (4 a^2-7 (a+b x) a+3\right )}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)+\frac {1}{4} b^3 x^3 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{4} b^4 x^4 \text {arccosh}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \int -\frac {b x \left (a \left (12 a^2+23\right )-\left (26 a^2+9\right ) (a+b x)\right )}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {7}{3} a b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{4} b^3 x^3 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{4} b^4 x^4 \text {arccosh}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4+24 a^2+3\right ) \int \frac {1}{\sqrt {a+b x-1} \sqrt {a+b x+1}}d(a+b x)-\frac {1}{2} \sqrt {a+b x-1} \sqrt {a+b x+1} \left (4 a \left (19 a^2+16\right )-\left (26 a^2+9\right ) (a+b x)\right )\right )-\frac {7}{3} a b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{4} b^3 x^3 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{4} b^4 x^4 \text {arccosh}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 43 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (8 a^4+24 a^2+3\right ) \text {arccosh}(a+b x)-\frac {1}{2} \sqrt {a+b x-1} \sqrt {a+b x+1} \left (4 a \left (19 a^2+16\right )-\left (26 a^2+9\right ) (a+b x)\right )\right )-\frac {7}{3} a b^2 x^2 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )+\frac {1}{4} b^3 x^3 \sqrt {a+b x-1} \sqrt {a+b x+1}\right )-\frac {1}{4} b^4 x^4 \text {arccosh}(a+b x)}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCosh[a + b*x]) + ((b^3*x^3*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/4 + ((-7*a*b^2*x^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/3 + (- 1/2*(Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(4*a*(16 + 19*a^2) - (9 + 26*a^2 )*(a + b*x))) + (3*(3 + 24*a^2 + 8*a^4)*ArcCosh[a + b*x])/2)/3)/4)/4)/b^4)
3.1.83.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(284\) vs. \(2(130)=260\).
Time = 0.81 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.88
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arccosh}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccosh}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccosh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (24 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-96 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+72 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-32 a \sqrt {\left (b x +a \right )^{2}-1}\, \left (b x +a \right )^{2}+6 \sqrt {\left (b x +a \right )^{2}-1}\, \left (b x +a \right )^{3}+72 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-64 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{96 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{4}}\) | \(285\) |
| default | \(\frac {\frac {\operatorname {arccosh}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccosh}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccosh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccosh}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccosh}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (24 a^{4} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-96 a^{3} \sqrt {\left (b x +a \right )^{2}-1}+72 a^{2} \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-32 a \sqrt {\left (b x +a \right )^{2}-1}\, \left (b x +a \right )^{2}+6 \sqrt {\left (b x +a \right )^{2}-1}\, \left (b x +a \right )^{3}+72 a^{2} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-64 a \sqrt {\left (b x +a \right )^{2}-1}+9 \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}+9 \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )\right )}{96 \sqrt {\left (b x +a \right )^{2}-1}}}{b^{4}}\) | \(285\) |
| parts | \(\frac {x^{4} \operatorname {arccosh}\left (b x +a \right )}{4}+\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-6 \,\operatorname {csgn}\left (b \right ) b^{3} x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+14 \,\operatorname {csgn}\left (b \right ) a \,b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-26 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) a^{2} b x +50 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) a^{3}-24 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a^{4}-9 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) b x +55 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right ) a -72 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right ) a^{2}-9 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \operatorname {csgn}\left (b \right )+b x +a \right ) \operatorname {csgn}\left (b \right )\right )\right ) \operatorname {csgn}\left (b \right )}{96 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\) | \(323\) |
1/b^4*(1/4*arccosh(b*x+a)*a^4-arccosh(b*x+a)*a^3*(b*x+a)+3/2*arccosh(b*x+a )*a^2*(b*x+a)^2-arccosh(b*x+a)*a*(b*x+a)^3+1/4*arccosh(b*x+a)*(b*x+a)^4-1/ 96*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(24*a^4*ln(b*x+a+((b*x+a)^2-1)^(1/2))-9 6*a^3*((b*x+a)^2-1)^(1/2)+72*a^2*(b*x+a)*((b*x+a)^2-1)^(1/2)-32*a*((b*x+a) ^2-1)^(1/2)*(b*x+a)^2+6*((b*x+a)^2-1)^(1/2)*(b*x+a)^3+72*a^2*ln(b*x+a+((b* x+a)^2-1)^(1/2))-64*a*((b*x+a)^2-1)^(1/2)+9*(b*x+a)*((b*x+a)^2-1)^(1/2)+9* ln(b*x+a+((b*x+a)^2-1)^(1/2)))/((b*x+a)^2-1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.72 \[ \int x^3 \text {arccosh}(a+b x) \, dx=\frac {3 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} - 24 \, a^{2} - 3\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} + 9\right )} b x - 55 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{96 \, b^{4}} \]
1/96*(3*(8*b^4*x^4 - 8*a^4 - 24*a^2 - 3)*log(b*x + a + sqrt(b^2*x^2 + 2*a* b*x + a^2 - 1)) - (6*b^3*x^3 - 14*a*b^2*x^2 - 50*a^3 + (26*a^2 + 9)*b*x - 55*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^4
\[ \int x^3 \text {arccosh}(a+b x) \, dx=\int x^{3} \operatorname {acosh}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (129) = 258\).
Time = 0.20 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.11 \[ \int x^3 \text {arccosh}(a+b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {arcosh}\left (b x + a\right ) - \frac {1}{96} \, {\left (\frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x^{3}}{b^{2}} - \frac {14 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x^{2}}{b^{3}} + \frac {105 \, a^{4} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{5}} + \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2} x}{b^{4}} - \frac {90 \, {\left (a^{2} - 1\right )} a^{2} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{5}} - \frac {105 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{3}}{b^{5}} - \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} x}{b^{4}} + \frac {9 \, {\left (a^{2} - 1\right )}^{2} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{b^{5}} + \frac {55 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} a}{b^{5}}\right )} b \]
1/4*x^4*arccosh(b*x + a) - 1/96*(6*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*x^3/b ^2 - 14*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*a*x^2/b^3 + 105*a^4*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^5 + 35*sqrt(b^2*x^2 + 2 *a*b*x + a^2 - 1)*a^2*x/b^4 - 90*(a^2 - 1)*a^2*log(2*b^2*x + 2*a*b + 2*sqr t(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^5 - 105*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*a^3/b^5 - 9*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - 1)*x/b^4 + 9*(a^2 - 1)^2*log(2*b^2*x + 2*a*b + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*b)/b^5 + 55*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(a^2 - 1)*a/b^5)*b
Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07 \[ \int x^3 \text {arccosh}(a+b x) \, dx=\frac {1}{4} \, x^{4} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} - 1}\right ) - \frac {1}{96} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{b^{2}} - \frac {7 \, a}{b^{3}}\right )} + \frac {26 \, a^{2} b^{3} + 9 \, b^{3}}{b^{7}}\right )} x - \frac {5 \, {\left (10 \, a^{3} b^{2} + 11 \, a b^{2}\right )}}{b^{7}}\right )} - \frac {3 \, {\left (8 \, a^{4} + 24 \, a^{2} + 3\right )} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )} {\left | b \right |} \right |}\right )}{b^{4} {\left | b \right |}}\right )} b \]
1/4*x^4*log(b*x + a + sqrt((b*x + a)^2 - 1)) - 1/96*(sqrt(b^2*x^2 + 2*a*b* x + a^2 - 1)*((2*x*(3*x/b^2 - 7*a/b^3) + (26*a^2*b^3 + 9*b^3)/b^7)*x - 5*( 10*a^3*b^2 + 11*a*b^2)/b^7) - 3*(8*a^4 + 24*a^2 + 3)*log(abs(-a*b - (x*abs (b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))*abs(b)))/(b^4*abs(b)))*b
Timed out. \[ \int x^3 \text {arccosh}(a+b x) \, dx=\int x^3\,\mathrm {acosh}\left (a+b\,x\right ) \,d x \]