Integrand size = 12, antiderivative size = 125 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {3 b^3 x \sqrt {1-c^2 x^2}}{8 c}-\frac {3}{4} b^2 x^2 (a+b \arccos (c x))-\frac {3 b x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{4 c}-\frac {(a+b \arccos (c x))^3}{4 c^2}+\frac {1}{2} x^2 (a+b \arccos (c x))^3-\frac {3 b^3 \arcsin (c x)}{8 c^2} \] Output:
3/8*b^3*x*(-c^2*x^2+1)^(1/2)/c-3/4*b^2*x^2*(a+b*arccos(c*x))-3/4*b*x*(-c^2 *x^2+1)^(1/2)*(a+b*arccos(c*x))^2/c-1/4*(a+b*arccos(c*x))^3/c^2+1/2*x^2*(a +b*arccos(c*x))^3-3/8*b^3*arcsin(c*x)/c^2
Time = 0.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.48 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {c x \left (4 a^3 c x-6 a b^2 c x-6 a^2 b \sqrt {1-c^2 x^2}+3 b^3 \sqrt {1-c^2 x^2}\right )-6 b c x \left (-2 a^2 c x+b^2 c x+2 a b \sqrt {1-c^2 x^2}\right ) \arccos (c x)-6 b^2 \left (a-2 a c^2 x^2+b c x \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+2 b^3 \left (-1+2 c^2 x^2\right ) \arccos (c x)^3+\left (6 a^2 b-3 b^3\right ) \arcsin (c x)}{8 c^2} \] Input:
Integrate[x*(a + b*ArcCos[c*x])^3,x]
Output:
(c*x*(4*a^3*c*x - 6*a*b^2*c*x - 6*a^2*b*Sqrt[1 - c^2*x^2] + 3*b^3*Sqrt[1 - c^2*x^2]) - 6*b*c*x*(-2*a^2*c*x + b^2*c*x + 2*a*b*Sqrt[1 - c^2*x^2])*ArcC os[c*x] - 6*b^2*(a - 2*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + 2*b^3*(-1 + 2*c^2*x^2)*ArcCos[c*x]^3 + (6*a^2*b - 3*b^3)*ArcSin[c*x])/(8 *c^2)
Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5139, 5211, 5139, 262, 223, 5153}
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.
| \(\displaystyle \int x (a+b \arccos (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {3}{2} b c \int \frac {x^2 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \int x (a+b \arccos (c x))dx}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle \frac {3}{2} b c \left (-\frac {b \left (\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (a+b \arccos (c x))\right )}{c}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {3}{2} b c \left (-\frac {b \left (\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))\right )}{c}+\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {\int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle \frac {3}{2} b c \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c}-\frac {(a+b \arccos (c x))^3}{6 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 c^2}\right )+\frac {1}{2} x^2 (a+b \arccos (c x))^3\) |
Input:
Int[x*(a + b*ArcCos[c*x])^3,x]
Output:
(x^2*(a + b*ArcCos[c*x])^3)/2 + (3*b*c*(-1/2*(x*Sqrt[1 - c^2*x^2]*(a + b*A rcCos[c*x])^2)/c^2 - (a + b*ArcCos[c*x])^3/(6*b*c^3) - (b*((x^2*(a + b*Arc Cos[c*x]))/2 + (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)) )/2))/c))/2
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_.)*(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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {\cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{3}}{4}-\frac {3 \sin \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{8}+\frac {3 \sin \left (2 \arccos \left (c x \right )\right )}{16}-\frac {3 \cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )}{8}\right )+3 a \,b^{2} \left (\frac {\cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{4}-\frac {\cos \left (2 \arccos \left (c x \right )\right )}{8}-\frac {\sin \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(159\) |
| default | \(\frac {\frac {c^{2} x^{2} a^{3}}{2}+b^{3} \left (\frac {\cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{3}}{4}-\frac {3 \sin \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{8}+\frac {3 \sin \left (2 \arccos \left (c x \right )\right )}{16}-\frac {3 \cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )}{8}\right )+3 a \,b^{2} \left (\frac {\cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{4}-\frac {\cos \left (2 \arccos \left (c x \right )\right )}{8}-\frac {\sin \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )}{4}\right )+3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(159\) |
| parts | \(\frac {x^{2} a^{3}}{2}+\frac {b^{3} \left (\frac {\cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{3}}{4}-\frac {3 \sin \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{8}+\frac {3 \sin \left (2 \arccos \left (c x \right )\right )}{16}-\frac {3 \cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )}{8}\right )}{c^{2}}+\frac {3 a \,b^{2} \left (\frac {\cos \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )^{2}}{4}-\frac {\cos \left (2 \arccos \left (c x \right )\right )}{8}-\frac {\sin \left (2 \arccos \left (c x \right )\right ) \arccos \left (c x \right )}{4}\right )}{c^{2}}+\frac {3 a^{2} b \left (\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}-\frac {c x \sqrt {-c^{2} x^{2}+1}}{4}+\frac {\arcsin \left (c x \right )}{4}\right )}{c^{2}}\) | \(161\) |
| orering | \(\frac {\left (15 c^{4} x^{4}-20 c^{2} x^{2}+8\right ) \left (a +b \arccos \left (c x \right )\right )^{3}}{16 c^{4} x^{2}}-\frac {\left (7 c^{4} x^{4}-16 c^{2} x^{2}+8\right ) \left (\left (a +b \arccos \left (c x \right )\right )^{3}-\frac {3 x \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{16 c^{4} x^{2}}+\frac {\left (c x -1\right ) \left (c x +1\right ) \left (c^{2} x^{2}-2\right ) \left (-\frac {6 \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {6 x \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {3 x^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{8 c^{4} x}-\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {18 \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {12 \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 x \,b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 x^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}-\frac {9 x^{3} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{5}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{16 c^{4}}\) | \(376\) |
Input:
int(x*(a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c^2*(1/2*c^2*x^2*a^3+b^3*(1/4*cos(2*arccos(c*x))*arccos(c*x)^3-3/8*sin(2 *arccos(c*x))*arccos(c*x)^2+3/16*sin(2*arccos(c*x))-3/8*cos(2*arccos(c*x)) *arccos(c*x))+3*a*b^2*(1/4*cos(2*arccos(c*x))*arccos(c*x)^2-1/8*cos(2*arcc os(c*x))-1/4*sin(2*arccos(c*x))*arccos(c*x))+3*a^2*b*(1/2*c^2*x^2*arccos(c *x)-1/4*c*x*(-c^2*x^2+1)^(1/2)+1/4*arcsin(c*x)))
Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.35 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} c^{2} x^{2} + 2 \, {\left (2 \, b^{3} c^{2} x^{2} - b^{3}\right )} \arccos \left (c x\right )^{3} + 6 \, {\left (2 \, a b^{2} c^{2} x^{2} - a b^{2}\right )} \arccos \left (c x\right )^{2} + 3 \, {\left (2 \, {\left (2 \, a^{2} b - b^{3}\right )} c^{2} x^{2} - 2 \, a^{2} b + b^{3}\right )} \arccos \left (c x\right ) - 3 \, {\left (2 \, b^{3} c x \arccos \left (c x\right )^{2} + 4 \, a b^{2} c x \arccos \left (c x\right ) + {\left (2 \, a^{2} b - b^{3}\right )} c x\right )} \sqrt {-c^{2} x^{2} + 1}}{8 \, c^{2}} \] Input:
integrate(x*(a+b*arccos(c*x))^3,x, algorithm="fricas")
Output:
1/8*(2*(2*a^3 - 3*a*b^2)*c^2*x^2 + 2*(2*b^3*c^2*x^2 - b^3)*arccos(c*x)^3 + 6*(2*a*b^2*c^2*x^2 - a*b^2)*arccos(c*x)^2 + 3*(2*(2*a^2*b - b^3)*c^2*x^2 - 2*a^2*b + b^3)*arccos(c*x) - 3*(2*b^3*c*x*arccos(c*x)^2 + 4*a*b^2*c*x*ar ccos(c*x) + (2*a^2*b - b^3)*c*x)*sqrt(-c^2*x^2 + 1))/c^2
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (116) = 232\).
Time = 0.28 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.15 \[ \int x (a+b \arccos (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{2}}{2} + \frac {3 a^{2} b x^{2} \operatorname {acos}{\left (c x \right )}}{2} - \frac {3 a^{2} b x \sqrt {- c^{2} x^{2} + 1}}{4 c} - \frac {3 a^{2} b \operatorname {acos}{\left (c x \right )}}{4 c^{2}} + \frac {3 a b^{2} x^{2} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {3 a b^{2} x^{2}}{4} - \frac {3 a b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{2 c} - \frac {3 a b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{4 c^{2}} + \frac {b^{3} x^{2} \operatorname {acos}^{3}{\left (c x \right )}}{2} - \frac {3 b^{3} x^{2} \operatorname {acos}{\left (c x \right )}}{4} - \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{4 c} + \frac {3 b^{3} x \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {b^{3} \operatorname {acos}^{3}{\left (c x \right )}}{4 c^{2}} + \frac {3 b^{3} \operatorname {acos}{\left (c x \right )}}{8 c^{2}} & \text {for}\: c \neq 0 \\\frac {x^{2} \left (a + \frac {\pi b}{2}\right )^{3}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(a+b*acos(c*x))**3,x)
Output:
Piecewise((a**3*x**2/2 + 3*a**2*b*x**2*acos(c*x)/2 - 3*a**2*b*x*sqrt(-c**2 *x**2 + 1)/(4*c) - 3*a**2*b*acos(c*x)/(4*c**2) + 3*a*b**2*x**2*acos(c*x)** 2/2 - 3*a*b**2*x**2/4 - 3*a*b**2*x*sqrt(-c**2*x**2 + 1)*acos(c*x)/(2*c) - 3*a*b**2*acos(c*x)**2/(4*c**2) + b**3*x**2*acos(c*x)**3/2 - 3*b**3*x**2*ac os(c*x)/4 - 3*b**3*x*sqrt(-c**2*x**2 + 1)*acos(c*x)**2/(4*c) + 3*b**3*x*sq rt(-c**2*x**2 + 1)/(8*c) - b**3*acos(c*x)**3/(4*c**2) + 3*b**3*acos(c*x)/( 8*c**2), Ne(c, 0)), (x**2*(a + pi*b/2)**3/2, True))
\[ \int x (a+b \arccos (c x))^3 \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{3} x \,d x } \] Input:
integrate(x*(a+b*arccos(c*x))^3,x, algorithm="maxima")
Output:
1/2*b^3*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 + 1/2*a^3*x^2 + 3 /4*(2*x^2*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a^ 2*b - integrate(3/2*(sqrt(c*x + 1)*sqrt(-c*x + 1)*b^3*c*x^2*arctan2(sqrt(c *x + 1)*sqrt(-c*x + 1), c*x)^2 - 2*(a*b^2*c^2*x^3 - a*b^2*x)*arctan2(sqrt( c*x + 1)*sqrt(-c*x + 1), c*x)^2)/(c^2*x^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (109) = 218\).
Time = 0.16 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.85 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {1}{2} \, b^{3} x^{2} \arccos \left (c x\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \arccos \left (c x\right )^{2} + \frac {3}{2} \, a^{2} b x^{2} \arccos \left (c x\right ) - \frac {3}{4} \, b^{3} x^{2} \arccos \left (c x\right ) - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x \arccos \left (c x\right )^{2}}{4 \, c} + \frac {1}{2} \, a^{3} x^{2} - \frac {3}{4} \, a b^{2} x^{2} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x \arccos \left (c x\right )}{2 \, c} - \frac {b^{3} \arccos \left (c x\right )^{3}}{4 \, c^{2}} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b x}{4 \, c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x}{8 \, c} - \frac {3 \, a b^{2} \arccos \left (c x\right )^{2}}{4 \, c^{2}} - \frac {3 \, a^{2} b \arccos \left (c x\right )}{4 \, c^{2}} + \frac {3 \, b^{3} \arccos \left (c x\right )}{8 \, c^{2}} + \frac {3 \, a b^{2}}{8 \, c^{2}} \] Input:
integrate(x*(a+b*arccos(c*x))^3,x, algorithm="giac")
Output:
1/2*b^3*x^2*arccos(c*x)^3 + 3/2*a*b^2*x^2*arccos(c*x)^2 + 3/2*a^2*b*x^2*ar ccos(c*x) - 3/4*b^3*x^2*arccos(c*x) - 3/4*sqrt(-c^2*x^2 + 1)*b^3*x*arccos( c*x)^2/c + 1/2*a^3*x^2 - 3/4*a*b^2*x^2 - 3/2*sqrt(-c^2*x^2 + 1)*a*b^2*x*ar ccos(c*x)/c - 1/4*b^3*arccos(c*x)^3/c^2 - 3/4*sqrt(-c^2*x^2 + 1)*a^2*b*x/c + 3/8*sqrt(-c^2*x^2 + 1)*b^3*x/c - 3/4*a*b^2*arccos(c*x)^2/c^2 - 3/4*a^2* b*arccos(c*x)/c^2 + 3/8*b^3*arccos(c*x)/c^2 + 3/8*a*b^2/c^2
Timed out. \[ \int x (a+b \arccos (c x))^3 \, dx=\int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \] Input:
int(x*(a + b*acos(c*x))^3,x)
Output:
int(x*(a + b*acos(c*x))^3, x)
Time = 0.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int x (a+b \arccos (c x))^3 \, dx=\frac {4 \mathit {acos} \left (c x \right )^{3} b^{3} c^{2} x^{2}-2 \mathit {acos} \left (c x \right )^{3} b^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} b^{3} c x +12 \mathit {acos} \left (c x \right )^{2} a \,b^{2} c^{2} x^{2}-6 \mathit {acos} \left (c x \right )^{2} a \,b^{2}-12 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) a \,b^{2} c x +12 \mathit {acos} \left (c x \right ) a^{2} b \,c^{2} x^{2}-6 \mathit {acos} \left (c x \right ) b^{3} c^{2} x^{2}+6 \mathit {asin} \left (c x \right ) a^{2} b -3 \mathit {asin} \left (c x \right ) b^{3}-6 \sqrt {-c^{2} x^{2}+1}\, a^{2} b c x +3 \sqrt {-c^{2} x^{2}+1}\, b^{3} c x +4 a^{3} c^{2} x^{2}-6 a \,b^{2} c^{2} x^{2}}{8 c^{2}} \] Input:
int(x*(a+b*acos(c*x))^3,x)
Output:
(4*acos(c*x)**3*b**3*c**2*x**2 - 2*acos(c*x)**3*b**3 - 6*sqrt( - c**2*x**2 + 1)*acos(c*x)**2*b**3*c*x + 12*acos(c*x)**2*a*b**2*c**2*x**2 - 6*acos(c* x)**2*a*b**2 - 12*sqrt( - c**2*x**2 + 1)*acos(c*x)*a*b**2*c*x + 12*acos(c* x)*a**2*b*c**2*x**2 - 6*acos(c*x)*b**3*c**2*x**2 + 6*asin(c*x)*a**2*b - 3* asin(c*x)*b**3 - 6*sqrt( - c**2*x**2 + 1)*a**2*b*c*x + 3*sqrt( - c**2*x**2 + 1)*b**3*c*x + 4*a**3*c**2*x**2 - 6*a*b**2*c**2*x**2)/(8*c**2)