Integrand size = 10, antiderivative size = 137 \[ \int x^3 \arccos (a+b x) \, dx=\frac {7 a x^2 \sqrt {1-(a+b x)^2}}{48 b^2}-\frac {x^3 \sqrt {1-(a+b x)^2}}{16 b}+\frac {\left (4 a \left (16+19 a^2\right )-\left (9+26 a^2\right ) (a+b x)\right ) \sqrt {1-(a+b x)^2}}{96 b^4}+\frac {1}{4} x^4 \arccos (a+b x)+\frac {\left (3+24 a^2+8 a^4\right ) \arcsin (a+b x)}{32 b^4} \]
1/4*x^4*arccos(b*x+a)+1/32*(8*a^4+24*a^2+3)*arcsin(b*x+a)/b^4+7/48*a*x^2*( 1-(b*x+a)^2)^(1/2)/b^2-1/16*x^3*(1-(b*x+a)^2)^(1/2)/b+1/96*(4*a*(19*a^2+16 )-(26*a^2+9)*(b*x+a))*(1-(b*x+a)^2)^(1/2)/b^4
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int x^3 \arccos (a+b x) \, dx=\frac {\sqrt {1-a^2-2 a b x-b^2 x^2} \left (55 a+50 a^3-9 b x-26 a^2 b x+14 a b^2 x^2-6 b^3 x^3\right )+24 b^4 x^4 \arccos (a+b x)+3 \left (3+24 a^2+8 a^4\right ) \arcsin (a+b x)}{96 b^4} \]
(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(55*a + 50*a^3 - 9*b*x - 26*a^2*b*x + 1 4*a*b^2*x^2 - 6*b^3*x^3) + 24*b^4*x^4*ArcCos[a + b*x] + 3*(3 + 24*a^2 + 8* a^4)*ArcSin[a + b*x])/(96*b^4)
Time = 0.35 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5305, 25, 27, 5243, 497, 25, 687, 25, 676, 223}
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.
| \(\displaystyle \int x^3 \arccos (a+b x) \, dx\) |
| \(\Big \downarrow \) 5305 |
| \(\displaystyle \frac {\int x^3 \arccos (a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -x^3 \arccos (a+b x)d(a+b x)}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -b^3 x^3 \arccos (a+b x)d(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 5243 |
| \(\displaystyle -\frac {-\frac {1}{4} \int \frac {b^4 x^4}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \int -\frac {b^2 x^2 \left (4 a^2-7 (a+b x) a+3\right )}{\sqrt {1-(a+b x)^2}}d(a+b x)+\frac {1}{4} b^3 x^3 \sqrt {1-(a+b x)^2}\right )-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} b^3 x^3 \sqrt {1-(a+b x)^2}-\frac {1}{4} \int \frac {b^2 x^2 \left (4 a^2-7 (a+b x) a+3\right )}{\sqrt {1-(a+b x)^2}}d(a+b x)\right )-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {b x \left (a \left (12 a^2+23\right )-\left (26 a^2+9\right ) (a+b x)\right )}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {7}{3} a b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{4} b^3 x^3 \sqrt {1-(a+b x)^2}\right )-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (-\frac {1}{3} \int -\frac {b x \left (a \left (12 a^2+23\right )-\left (26 a^2+9\right ) (a+b x)\right )}{\sqrt {1-(a+b x)^2}}d(a+b x)-\frac {7}{3} a b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{4} b^3 x^3 \sqrt {1-(a+b x)^2}\right )-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \left (-\frac {3}{2} \left (8 a^4+24 a^2+3\right ) \int \frac {1}{\sqrt {1-(a+b x)^2}}d(a+b x)-2 a \left (19 a^2+16\right ) \sqrt {1-(a+b x)^2}+\frac {1}{2} \left (26 a^2+9\right ) (a+b x) \sqrt {1-(a+b x)^2}\right )-\frac {7}{3} a b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{4} b^3 x^3 \sqrt {1-(a+b x)^2}\right )-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{4} \left (\frac {1}{3} \left (-2 a \left (19 a^2+16\right ) \sqrt {1-(a+b x)^2}+\frac {1}{2} \left (26 a^2+9\right ) (a+b x) \sqrt {1-(a+b x)^2}-\frac {3}{2} \left (8 a^4+24 a^2+3\right ) \arcsin (a+b x)\right )-\frac {7}{3} a b^2 x^2 \sqrt {1-(a+b x)^2}\right )+\frac {1}{4} b^3 x^3 \sqrt {1-(a+b x)^2}\right )-\frac {1}{4} b^4 x^4 \arccos (a+b x)}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCos[a + b*x]) + ((b^3*x^3*Sqrt[1 - (a + b*x)^2])/4 + ( (-7*a*b^2*x^2*Sqrt[1 - (a + b*x)^2])/3 + (-2*a*(16 + 19*a^2)*Sqrt[1 - (a + b*x)^2] + ((9 + 26*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^2])/2 - (3*(3 + 24*a ^2 + 8*a^4)*ArcSin[a + b*x])/2)/3)/4)/4)/b^4)
3.1.24.3.1 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_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 1))), x] + Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(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*A rcCos[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.11 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.72
| method | result | size |
| derivativedivides | \(\frac {\frac {\arccos \left (b x +a \right ) a^{4}}{4}-\arccos \left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \arccos \left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\arccos \left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {a^{4} \arcsin \left (b x +a \right )}{4}-\frac {\left (b x +a \right )^{3} \sqrt {1-\left (b x +a \right )^{2}}}{16}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{32}+\frac {3 \arcsin \left (b x +a \right )}{32}+a^{3} \sqrt {1-\left (b x +a \right )^{2}}+\frac {3 a^{2} \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )}{2}-a \left (-\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{3}\right )}{b^{4}}\) | \(235\) |
| default | \(\frac {\frac {\arccos \left (b x +a \right ) a^{4}}{4}-\arccos \left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \arccos \left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\arccos \left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\arccos \left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {a^{4} \arcsin \left (b x +a \right )}{4}-\frac {\left (b x +a \right )^{3} \sqrt {1-\left (b x +a \right )^{2}}}{16}-\frac {3 \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{32}+\frac {3 \arcsin \left (b x +a \right )}{32}+a^{3} \sqrt {1-\left (b x +a \right )^{2}}+\frac {3 a^{2} \left (-\frac {\left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{2}+\frac {\arcsin \left (b x +a \right )}{2}\right )}{2}-a \left (-\frac {\left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}}{3}-\frac {2 \sqrt {1-\left (b x +a \right )^{2}}}{3}\right )}{b^{4}}\) | \(235\) |
| parts | \(\frac {x^{4} \arccos \left (b x +a \right )}{4}+\frac {b \left (-\frac {x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {7 a \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{3 b^{2}}\right )}{4 b}+\frac {3 \left (-a^{2}+1\right ) \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{4}\) | \(515\) |
1/b^4*(1/4*arccos(b*x+a)*a^4-arccos(b*x+a)*a^3*(b*x+a)+3/2*arccos(b*x+a)*a ^2*(b*x+a)^2-arccos(b*x+a)*a*(b*x+a)^3+1/4*arccos(b*x+a)*(b*x+a)^4+1/4*a^4 *arcsin(b*x+a)-1/16*(b*x+a)^3*(1-(b*x+a)^2)^(1/2)-3/32*(b*x+a)*(1-(b*x+a)^ 2)^(1/2)+3/32*arcsin(b*x+a)+a^3*(1-(b*x+a)^2)^(1/2)+3/2*a^2*(-1/2*(b*x+a)* (1-(b*x+a)^2)^(1/2)+1/2*arcsin(b*x+a))-a*(-1/3*(b*x+a)^2*(1-(b*x+a)^2)^(1/ 2)-2/3*(1-(b*x+a)^2)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.69 \[ \int x^3 \arccos (a+b x) \, dx=\frac {3 \, {\left (8 \, b^{4} x^{4} - 8 \, a^{4} - 24 \, a^{2} - 3\right )} \arccos \left (b x + a\right ) - {\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} + {\left (26 \, a^{2} + 9\right )} b x - 55 \, a\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{96 \, b^{4}} \]
1/96*(3*(8*b^4*x^4 - 8*a^4 - 24*a^2 - 3)*arccos(b*x + a) - (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
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (117) = 234\).
Time = 0.36 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.86 \[ \int x^3 \arccos (a+b x) \, dx=\begin {cases} - \frac {a^{4} \operatorname {acos}{\left (a + b x \right )}}{4 b^{4}} + \frac {25 a^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{48 b^{4}} - \frac {13 a^{2} x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{48 b^{3}} - \frac {3 a^{2} \operatorname {acos}{\left (a + b x \right )}}{4 b^{4}} + \frac {7 a x^{2} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{48 b^{2}} + \frac {55 a \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{96 b^{4}} + \frac {x^{4} \operatorname {acos}{\left (a + b x \right )}}{4} - \frac {x^{3} \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{16 b} - \frac {3 x \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32 b^{3}} - \frac {3 \operatorname {acos}{\left (a + b x \right )}}{32 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acos}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-a**4*acos(a + b*x)/(4*b**4) + 25*a**3*sqrt(-a**2 - 2*a*b*x - b **2*x**2 + 1)/(48*b**4) - 13*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/ (48*b**3) - 3*a**2*acos(a + b*x)/(4*b**4) + 7*a*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(48*b**2) + 55*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(9 6*b**4) + x**4*acos(a + b*x)/4 - x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1 )/(16*b) - 3*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(32*b**3) - 3*acos(a + b*x)/(32*b**4), Ne(b, 0)), (x**4*acos(a)/4, True))
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (120) = 240\).
Time = 0.27 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.43 \[ \int x^3 \arccos (a+b x) \, dx=\frac {1}{4} \, x^{4} \arccos \left (b x + a\right ) - \frac {1}{96} \, {\left (\frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{3}}{b^{2}} - \frac {14 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x^{2}}{b^{3}} + \frac {105 \, a^{4} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{5}} + \frac {35 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} x}{b^{4}} - \frac {90 \, {\left (a^{2} - 1\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{5}} - \frac {105 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{b^{5}} - \frac {9 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} x}{b^{4}} + \frac {9 \, {\left (a^{2} - 1\right )}^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{5}} + \frac {55 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} a}{b^{5}}\right )} b \]
1/4*x^4*arccos(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*arcsin(-(b^ 2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/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*arcsin(-(b^2*x + a*b)/sqrt(a^2*b^ 2 - (a^2 - 1)*b^2))/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*arcs in(-(b^2*x + a*b)/sqrt(a^2*b^2 - (a^2 - 1)*b^2))/b^5 + 55*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^2 - 1)*a/b^5)*b
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (120) = 240\).
Time = 0.28 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.77 \[ \int x^3 \arccos (a+b x) \, dx=\frac {{\left (b x + a\right )}^{4} \arccos \left (b x + a\right )}{4 \, b^{4}} - \frac {{\left (b x + a\right )}^{3} a \arccos \left (b x + a\right )}{b^{4}} + \frac {3 \, {\left (b x + a\right )}^{2} a^{2} \arccos \left (b x + a\right )}{2 \, b^{4}} - \frac {{\left (b x + a\right )} a^{3} \arccos \left (b x + a\right )}{b^{4}} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}^{3}}{16 \, b^{4}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}^{2} a}{3 \, b^{4}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )} a^{2}}{4 \, b^{4}} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} a^{3}}{b^{4}} - \frac {3 \, a^{2} \arccos \left (b x + a\right )}{4 \, b^{4}} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (b x + a\right )}}{32 \, b^{4}} + \frac {2 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} a}{3 \, b^{4}} - \frac {3 \, \arccos \left (b x + a\right )}{32 \, b^{4}} \]
1/4*(b*x + a)^4*arccos(b*x + a)/b^4 - (b*x + a)^3*a*arccos(b*x + a)/b^4 + 3/2*(b*x + a)^2*a^2*arccos(b*x + a)/b^4 - (b*x + a)*a^3*arccos(b*x + a)/b^ 4 - 1/16*sqrt(-(b*x + a)^2 + 1)*(b*x + a)^3/b^4 + 1/3*sqrt(-(b*x + a)^2 + 1)*(b*x + a)^2*a/b^4 - 3/4*sqrt(-(b*x + a)^2 + 1)*(b*x + a)*a^2/b^4 + sqrt (-(b*x + a)^2 + 1)*a^3/b^4 - 3/4*a^2*arccos(b*x + a)/b^4 - 3/32*sqrt(-(b*x + a)^2 + 1)*(b*x + a)/b^4 + 2/3*sqrt(-(b*x + a)^2 + 1)*a/b^4 - 3/32*arcco s(b*x + a)/b^4
Timed out. \[ \int x^3 \arccos (a+b x) \, dx=\int x^3\,\mathrm {acos}\left (a+b\,x\right ) \,d x \]