Integrand size = 14, antiderivative size = 170 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {14 b^3 \sqrt {1-c^2 x^2}}{9 c^3}-\frac {2 b^3 \left (1-c^2 x^2\right )^{3/2}}{27 c^3}-\frac {4 b^2 x (a+b \arccos (c x))}{3 c^2}-\frac {2}{9} b^2 x^3 (a+b \arccos (c x))-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^3}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^3 \] Output:
14/9*b^3*(-c^2*x^2+1)^(1/2)/c^3-2/27*b^3*(-c^2*x^2+1)^(3/2)/c^3-4/3*b^2*x* (a+b*arccos(c*x))/c^2-2/9*b^2*x^3*(a+b*arccos(c*x))-2/3*b*(-c^2*x^2+1)^(1/ 2)*(a+b*arccos(c*x))^2/c^3-1/3*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^ 2/c+1/3*x^3*(a+b*arccos(c*x))^3
Time = 0.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.28 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {9 a^3 c^3 x^3-9 a^2 b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )-6 a b^2 c x \left (6+c^2 x^2\right )+2 b^3 \sqrt {1-c^2 x^2} \left (20+c^2 x^2\right )-3 b \left (-9 a^2 c^3 x^3+6 a b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )+2 b^2 c x \left (6+c^2 x^2\right )\right ) \arccos (c x)-9 b^2 \left (-3 a c^3 x^3+b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )\right ) \arccos (c x)^2+9 b^3 c^3 x^3 \arccos (c x)^3}{27 c^3} \] Input:
Integrate[x^2*(a + b*ArcCos[c*x])^3,x]
Output:
(9*a^3*c^3*x^3 - 9*a^2*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) - 6*a*b^2*c*x*(6 + c^2*x^2) + 2*b^3*Sqrt[1 - c^2*x^2]*(20 + c^2*x^2) - 3*b*(-9*a^2*c^3*x^3 + 6*a*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 2*b^2*c*x*(6 + c^2*x^2))*ArcCos[ c*x] - 9*b^2*(-3*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2))*ArcCos[c*x ]^2 + 9*b^3*c^3*x^3*ArcCos[c*x]^3)/(27*c^3)
Time = 0.85 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5139, 5211, 5139, 243, 53, 2009, 5183, 2009}
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.
| \(\displaystyle \int x^2 (a+b \arccos (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle b c \int \frac {x^3 (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5211 |
| \(\displaystyle b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 b \int x^2 (a+b \arccos (c x))dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5139 |
| \(\displaystyle b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2+\frac {1}{3} x^3 (a+b \arccos (c x))\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2+\frac {1}{3} x^3 (a+b \arccos (c x))\right )}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \arccos (c x))+\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle b c \left (\frac {2 \left (-\frac {2 b \int (a+b \arccos (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \arccos (c x))+\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b c \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^2}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^2}-\frac {2 b \left (a x+b x \arccos (c x)-\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}\right )}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \arccos (c x))+\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^3\) |
Input:
Int[x^2*(a + b*ArcCos[c*x])^3,x]
Output:
(x^3*(a + b*ArcCos[c*x])^3)/3 + b*c*(-1/3*(x^2*Sqrt[1 - c^2*x^2]*(a + b*Ar cCos[c*x])^2)/c^2 - (2*b*((b*c*((-2*Sqrt[1 - c^2*x^2])/c^4 + (2*(1 - c^2*x ^2)^(3/2))/(3*c^4)))/6 + (x^3*(a + b*ArcCos[c*x]))/3))/(3*c) + (2*(-((Sqrt [1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/c^2) - (2*b*(a*x - (b*Sqrt[1 - c^2*x^ 2])/c + b*x*ArcCos[c*x]))/c))/(3*c^2))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -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 = 0.38 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{2}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(235\) |
| default | \(\frac {\frac {a^{3} c^{3} x^{3}}{3}+b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )+3 a \,b^{2} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{2}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(235\) |
| parts | \(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{3}}{3}-\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {4 \sqrt {-c^{2} x^{2}+1}}{3}-\frac {4 c x \arccos \left (c x \right )}{3}-\frac {2 c^{3} x^{3} \arccos \left (c x \right )}{9}+\frac {2 \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )}{c^{3}}+\frac {3 a \,b^{2} \left (\frac {c^{3} x^{3} \arccos \left (c x \right )^{2}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )}{c^{3}}+\frac {3 a^{2} b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(237\) |
| orering | \(\frac {5 \left (13 c^{6} x^{6}+40 c^{4} x^{4}-152 c^{2} x^{2}+96\right ) \left (a +b \arccos \left (c x \right )\right )^{3}}{81 c^{6} x^{3}}-\frac {\left (25 c^{6} x^{6}+166 c^{4} x^{4}-572 c^{2} x^{2}+360\right ) \left (2 x \left (a +b \arccos \left (c x \right )\right )^{3}-\frac {3 x^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{81 c^{6} x^{4}}+\frac {2 \left (c x -1\right ) \left (c x +1\right ) \left (c^{4} x^{4}+12 c^{2} x^{2}-20\right ) \left (2 \left (a +b \arccos \left (c x \right )\right )^{3}-\frac {12 x \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {6 x^{2} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {3 x^{3} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{6} x^{3}}-\frac {\left (c^{2} x^{2}+20\right ) \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (-\frac {18 \left (a +b \arccos \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {36 x \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {21 x^{2} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {6 x^{2} b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 x^{3} \left (a +b \arccos \left (c x \right )\right ) b^{2} c^{4}}{\left (-c^{2} x^{2}+1\right )^{2}}-\frac {9 x^{4} \left (a +b \arccos \left (c x \right )\right )^{2} b \,c^{5}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{81 c^{6} x^{2}}\) | \(463\) |
Input:
int(x^2*(a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c^3*(1/3*a^3*c^3*x^3+b^3*(1/3*c^3*x^3*arccos(c*x)^3-1/3*arccos(c*x)^2*(c ^2*x^2+2)*(-c^2*x^2+1)^(1/2)+4/3*(-c^2*x^2+1)^(1/2)-4/3*c*x*arccos(c*x)-2/ 9*c^3*x^3*arccos(c*x)+2/27*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2))+3*a*b^2*(1/3*c^ 3*x^3*arccos(c*x)^2-2/9*arccos(c*x)*(c^2*x^2+2)*(-c^2*x^2+1)^(1/2)-2/27*c^ 3*x^3-4/9*c*x)+3*a^2*b*(1/3*c^3*x^3*arccos(c*x)-1/9*c^2*x^2*(-c^2*x^2+1)^( 1/2)-2/9*(-c^2*x^2+1)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.15 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {9 \, b^{3} c^{3} x^{3} \arccos \left (c x\right )^{3} + 27 \, a b^{2} c^{3} x^{3} \arccos \left (c x\right )^{2} + 3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} x^{3} - 36 \, a b^{2} c x + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} x^{3} - 12 \, b^{3} c x\right )} \arccos \left (c x\right ) - {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} x^{2} + 18 \, a^{2} b - 40 \, b^{3} + 9 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (a b^{2} c^{2} x^{2} + 2 \, a b^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \] Input:
integrate(x^2*(a+b*arccos(c*x))^3,x, algorithm="fricas")
Output:
1/27*(9*b^3*c^3*x^3*arccos(c*x)^3 + 27*a*b^2*c^3*x^3*arccos(c*x)^2 + 3*(3* a^3 - 2*a*b^2)*c^3*x^3 - 36*a*b^2*c*x + 3*((9*a^2*b - 2*b^3)*c^3*x^3 - 12* b^3*c*x)*arccos(c*x) - ((9*a^2*b - 2*b^3)*c^2*x^2 + 18*a^2*b - 40*b^3 + 9* (b^3*c^2*x^2 + 2*b^3)*arccos(c*x)^2 + 18*(a*b^2*c^2*x^2 + 2*a*b^2)*arccos( c*x))*sqrt(-c^2*x^2 + 1))/c^3
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (158) = 316\).
Time = 0.37 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.96 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\begin {cases} \frac {a^{3} x^{3}}{3} + a^{2} b x^{3} \operatorname {acos}{\left (c x \right )} - \frac {a^{2} b x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} - \frac {2 a^{2} b \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + a b^{2} x^{3} \operatorname {acos}^{2}{\left (c x \right )} - \frac {2 a b^{2} x^{3}}{9} - \frac {2 a b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3 c} - \frac {4 a b^{2} x}{3 c^{2}} - \frac {4 a b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{3 c^{3}} + \frac {b^{3} x^{3} \operatorname {acos}^{3}{\left (c x \right )}}{3} - \frac {2 b^{3} x^{3} \operatorname {acos}{\left (c x \right )}}{9} - \frac {b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{3 c} + \frac {2 b^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{27 c} - \frac {4 b^{3} x \operatorname {acos}{\left (c x \right )}}{3 c^{2}} - \frac {2 b^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{3 c^{3}} + \frac {40 b^{3} \sqrt {- c^{2} x^{2} + 1}}{27 c^{3}} & \text {for}\: c \neq 0 \\\frac {x^{3} \left (a + \frac {\pi b}{2}\right )^{3}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(a+b*acos(c*x))**3,x)
Output:
Piecewise((a**3*x**3/3 + a**2*b*x**3*acos(c*x) - a**2*b*x**2*sqrt(-c**2*x* *2 + 1)/(3*c) - 2*a**2*b*sqrt(-c**2*x**2 + 1)/(3*c**3) + a*b**2*x**3*acos( c*x)**2 - 2*a*b**2*x**3/9 - 2*a*b**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/( 3*c) - 4*a*b**2*x/(3*c**2) - 4*a*b**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(3*c* *3) + b**3*x**3*acos(c*x)**3/3 - 2*b**3*x**3*acos(c*x)/9 - b**3*x**2*sqrt( -c**2*x**2 + 1)*acos(c*x)**2/(3*c) + 2*b**3*x**2*sqrt(-c**2*x**2 + 1)/(27* c) - 4*b**3*x*acos(c*x)/(3*c**2) - 2*b**3*sqrt(-c**2*x**2 + 1)*acos(c*x)** 2/(3*c**3) + 40*b**3*sqrt(-c**2*x**2 + 1)/(27*c**3), Ne(c, 0)), (x**3*(a + pi*b/2)**3/3, True))
Time = 0.15 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.61 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} - \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right )^{2} - 2 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} - \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arccos \left (c x\right )}{c^{3}}\right )}\right )} b^{3} \] Input:
integrate(x^2*(a+b*arccos(c*x))^3,x, algorithm="maxima")
Output:
1/3*b^3*x^3*arccos(c*x)^3 + a*b^2*x^3*arccos(c*x)^2 + 1/3*a^3*x^3 + 1/3*(3 *x^3*arccos(c*x) - c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^ 4))*a^2*b - 2/9*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^ 4)*arccos(c*x) + (c^2*x^3 + 6*x)/c^2)*a*b^2 - 1/27*(9*c*(sqrt(-c^2*x^2 + 1 )*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arccos(c*x)^2 - 2*c*((sqrt(-c^2*x^2 + 1)*x^2 + 20*sqrt(-c^2*x^2 + 1)/c^2)/c^2 - 3*(c^2*x^3 + 6*x)*arccos(c*x)/ c^3))*b^3
Time = 0.16 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.70 \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \arccos \left (c x\right )^{3} + a b^{2} x^{3} \arccos \left (c x\right )^{2} + a^{2} b x^{3} \arccos \left (c x\right ) - \frac {2}{9} \, b^{3} x^{3} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b^{3} x^{2} \arccos \left (c x\right )^{2}}{3 \, c} + \frac {1}{3} \, a^{3} x^{3} - \frac {2}{9} \, a b^{2} x^{3} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} x^{2} \arccos \left (c x\right )}{3 \, c} - \frac {\sqrt {-c^{2} x^{2} + 1} a^{2} b x^{2}}{3 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} x^{2}}{27 \, c} - \frac {4 \, b^{3} x \arccos \left (c x\right )}{3 \, c^{2}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} \arccos \left (c x\right )^{2}}{3 \, c^{3}} - \frac {4 \, a b^{2} x}{3 \, c^{2}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} \arccos \left (c x\right )}{3 \, c^{3}} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b}{3 \, c^{3}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} b^{3}}{27 \, c^{3}} \] Input:
integrate(x^2*(a+b*arccos(c*x))^3,x, algorithm="giac")
Output:
1/3*b^3*x^3*arccos(c*x)^3 + a*b^2*x^3*arccos(c*x)^2 + a^2*b*x^3*arccos(c*x ) - 2/9*b^3*x^3*arccos(c*x) - 1/3*sqrt(-c^2*x^2 + 1)*b^3*x^2*arccos(c*x)^2 /c + 1/3*a^3*x^3 - 2/9*a*b^2*x^3 - 2/3*sqrt(-c^2*x^2 + 1)*a*b^2*x^2*arccos (c*x)/c - 1/3*sqrt(-c^2*x^2 + 1)*a^2*b*x^2/c + 2/27*sqrt(-c^2*x^2 + 1)*b^3 *x^2/c - 4/3*b^3*x*arccos(c*x)/c^2 - 2/3*sqrt(-c^2*x^2 + 1)*b^3*arccos(c*x )^2/c^3 - 4/3*a*b^2*x/c^2 - 4/3*sqrt(-c^2*x^2 + 1)*a*b^2*arccos(c*x)/c^3 - 2/3*sqrt(-c^2*x^2 + 1)*a^2*b/c^3 + 40/27*sqrt(-c^2*x^2 + 1)*b^3/c^3
Timed out. \[ \int x^2 (a+b \arccos (c x))^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3 \,d x \] Input:
int(x^2*(a + b*acos(c*x))^3,x)
Output:
int(x^2*(a + b*acos(c*x))^3, x)
\[ \int x^2 (a+b \arccos (c x))^3 \, dx=\frac {3 \mathit {acos} \left (c x \right ) a^{2} b \,c^{3} x^{3}-\sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2} b +3 \left (\int \mathit {acos} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}+9 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}+a^{3} c^{3} x^{3}}{3 c^{3}} \] Input:
int(x^2*(a+b*acos(c*x))^3,x)
Output:
(3*acos(c*x)*a**2*b*c**3*x**3 - sqrt( - c**2*x**2 + 1)*a**2*b*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a**2*b + 3*int(acos(c*x)**3*x**2,x)*b**3*c**3 + 9 *int(acos(c*x)**2*x**2,x)*a*b**2*c**3 + a**3*c**3*x**3)/(3*c**3)