Integrand size = 18, antiderivative size = 156 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=-2 b^2 d x-\frac {4 b^2 e x}{9 c^2}-\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {4 b e \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c}+d x (a+b \arccos (c x))^2+\frac {1}{3} e x^3 (a+b \arccos (c x))^2 \] Output:
-2*b^2*d*x-4/9*b^2*e*x/c^2-2/27*b^2*e*x^3-2*b*d*(-c^2*x^2+1)^(1/2)*(a+b*ar ccos(c*x))/c-4/9*b*e*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c^3-2/9*b*e*x^2* (-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))/c+d*x*(a+b*arccos(c*x))^2+1/3*e*x^3*( a+b*arccos(c*x))^2
Time = 0.12 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {1-c^2 x^2} \left (2 e+c^2 \left (9 d+e x^2\right )\right )-2 b^2 c x \left (6 e+c^2 \left (27 d+e x^2\right )\right )-6 b \left (-3 a c^3 x \left (3 d+e x^2\right )+b \sqrt {1-c^2 x^2} \left (2 e+c^2 \left (9 d+e x^2\right )\right )\right ) \arccos (c x)+9 b^2 c^3 x \left (3 d+e x^2\right ) \arccos (c x)^2}{27 c^3} \] Input:
Integrate[(d + e*x^2)*(a + b*ArcCos[c*x])^2,x]
Output:
(9*a^2*c^3*x*(3*d + e*x^2) - 6*a*b*Sqrt[1 - c^2*x^2]*(2*e + c^2*(9*d + e*x ^2)) - 2*b^2*c*x*(6*e + c^2*(27*d + e*x^2)) - 6*b*(-3*a*c^3*x*(3*d + e*x^2 ) + b*Sqrt[1 - c^2*x^2]*(2*e + c^2*(9*d + e*x^2)))*ArcCos[c*x] + 9*b^2*c^3 *x*(3*d + e*x^2)*ArcCos[c*x]^2)/(27*c^3)
Time = 0.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5173, 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.
| \(\displaystyle \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5173 |
| \(\displaystyle \int \left (d (a+b \arccos (c x))^2+e x^2 (a+b \arccos (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c}-\frac {2 b e x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c}-\frac {4 b e \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^3}+d x (a+b \arccos (c x))^2+\frac {1}{3} e x^3 (a+b \arccos (c x))^2-\frac {4 b^2 e x}{9 c^2}-2 b^2 d x-\frac {2}{27} b^2 e x^3\) |
Input:
Int[(d + e*x^2)*(a + b*ArcCos[c*x])^2,x]
Output:
-2*b^2*d*x - (4*b^2*e*x)/(9*c^2) - (2*b^2*e*x^3)/27 - (2*b*d*Sqrt[1 - c^2* x^2]*(a + b*ArcCos[c*x]))/c - (4*b*e*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]) )/(9*c^3) - (2*b*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(9*c) + d*x* (a + b*ArcCos[c*x])^2 + (e*x^3*(a + b*ArcCos[c*x])^2)/3
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcCos[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.45 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.42
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b^{2} \left (\frac {e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27 c^{2}}+d \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c}+\frac {2 a b \left (\frac {c \arccos \left (c x \right ) x^{3} e}{3}+\arccos \left (c x \right ) d c x +\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-3 d \,c^{2} \sqrt {-c^{2} x^{2}+1}}{3 c^{2}}\right )}{c}\) | \(221\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\arccos \left (c x \right ) d \,c^{3} x +\frac {\arccos \left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(233\) |
| default | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\arccos \left (c x \right )^{2} c x -2 c x -2 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (9 \arccos \left (c x \right )^{2} c^{3} x^{3}-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}-12 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\arccos \left (c x \right ) d \,c^{3} x +\frac {\arccos \left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(233\) |
| orering | \(\frac {x \left (19 c^{4} e^{3} x^{6}+209 c^{4} d \,e^{2} x^{4}+9 c^{4} d^{2} e \,x^{2}+24 c^{2} e^{3} x^{4}+27 c^{4} d^{3}-232 c^{2} d \,e^{2} x^{2}-48 e^{3} x^{2}\right ) \left (a +b \arccos \left (c x \right )\right )^{2}}{27 \left (e \,x^{2}+d \right )^{2} c^{4}}-\frac {\left (6 c^{4} e^{2} x^{6}+110 c^{4} d e \,x^{4}+17 c^{2} e^{2} x^{4}-138 c^{2} d e \,x^{2}-27 c^{2} d^{2}-30 e^{2} x^{2}-6 d e \right ) \left (2 e x \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {2 \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{27 \left (e \,x^{2}+d \right )^{2} c^{4}}+\frac {x \left (c^{2} e \,x^{2}+27 c^{2} d +6 e \right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 e \left (a +b \arccos \left (c x \right )\right )^{2}-\frac {8 e x \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 \left (e \,x^{2}+d \right ) b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 \left (e \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right ) b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{4} \left (e \,x^{2}+d \right )}\) | \(369\) |
Input:
int((e*x^2+d)*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
a^2*(1/3*x^3*e+d*x)+b^2/c*(1/27*e*(9*arccos(c*x)^2*c^3*x^3-6*(-c^2*x^2+1)^ (1/2)*arccos(c*x)*c^2*x^2-2*c^3*x^3-12*arccos(c*x)*(-c^2*x^2+1)^(1/2)-12*c *x)/c^2+d*(arccos(c*x)^2*c*x-2*c*x-2*arccos(c*x)*(-c^2*x^2+1)^(1/2)))+2*a* b/c*(1/3*c*arccos(c*x)*x^3*e+arccos(c*x)*d*c*x+1/3/c^2*(e*(-1/3*c^2*x^2*(- c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-3*d*c^2*(-c^2*x^2+1)^(1/2)))
Time = 0.13 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.13 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \arccos \left (c x\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{3} d - 4 \, b^{2} c e\right )} x + 18 \, {\left (a b c^{3} e x^{3} + 3 \, a b c^{3} d x\right )} \arccos \left (c x\right ) - 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d + 2 \, a b e + {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \] Input:
integrate((e*x^2+d)*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
1/27*((9*a^2 - 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*arccos (c*x)^2 + 3*(9*(a^2 - 2*b^2)*c^3*d - 4*b^2*c*e)*x + 18*(a*b*c^3*e*x^3 + 3* a*b*c^3*d*x)*arccos(c*x) - 6*(a*b*c^2*e*x^2 + 9*a*b*c^2*d + 2*a*b*e + (b^2 *c^2*e*x^2 + 9*b^2*c^2*d + 2*b^2*e)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^3
Time = 0.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.82 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{3}}{3} + 2 a b d x \operatorname {acos}{\left (c x \right )} + \frac {2 a b e x^{3} \operatorname {acos}{\left (c x \right )}}{3} - \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} - \frac {2 a b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {4 a b e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname {acos}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} e x^{3}}{27} - \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{c} - \frac {2 b^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c} - \frac {4 b^{2} e x}{9 c^{2}} - \frac {4 b^{2} e \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right )^{2} \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)*(a+b*acos(c*x))**2,x)
Output:
Piecewise((a**2*d*x + a**2*e*x**3/3 + 2*a*b*d*x*acos(c*x) + 2*a*b*e*x**3*a cos(c*x)/3 - 2*a*b*d*sqrt(-c**2*x**2 + 1)/c - 2*a*b*e*x**2*sqrt(-c**2*x**2 + 1)/(9*c) - 4*a*b*e*sqrt(-c**2*x**2 + 1)/(9*c**3) + b**2*d*x*acos(c*x)** 2 - 2*b**2*d*x + b**2*e*x**3*acos(c*x)**2/3 - 2*b**2*e*x**3/27 - 2*b**2*d* sqrt(-c**2*x**2 + 1)*acos(c*x)/c - 2*b**2*e*x**2*sqrt(-c**2*x**2 + 1)*acos (c*x)/(9*c) - 4*b**2*e*x/(9*c**2) - 4*b**2*e*sqrt(-c**2*x**2 + 1)*acos(c*x )/(9*c**3), Ne(c, 0)), ((a + pi*b/2)**2*(d*x + e*x**3/3), True))
Time = 0.13 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.42 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \arccos \left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} e - 2 \, b^{2} d {\left (x + \frac {\sqrt {-c^{2} x^{2} + 1} \arccos \left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} a b d}{c} \] Input:
integrate((e*x^2+d)*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
1/3*b^2*e*x^3*arccos(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arccos(c*x)^2 + 2/9* (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*b*e - 2/27*(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)*b^2*e - 2*b^2*d*(x + sqrt(-c^2*x^ 2 + 1)*arccos(c*x)/c) + a^2*d*x + 2*(c*x*arccos(c*x) - sqrt(-c^2*x^2 + 1)) *a*b*d/c
Time = 0.15 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.49 \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \arccos \left (c x\right )^{2} + \frac {2}{3} \, a b e x^{3} \arccos \left (c x\right ) + \frac {1}{3} \, a^{2} e x^{3} - \frac {2}{27} \, b^{2} e x^{3} + b^{2} d x \arccos \left (c x\right )^{2} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e x^{2} \arccos \left (c x\right )}{9 \, c} + 2 \, a b d x \arccos \left (c x\right ) - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e x^{2}}{9 \, c} + a^{2} d x - 2 \, b^{2} d x - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arccos \left (c x\right )}{c} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} - \frac {4 \, b^{2} e x}{9 \, c^{2}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e \arccos \left (c x\right )}{9 \, c^{3}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b e}{9 \, c^{3}} \] Input:
integrate((e*x^2+d)*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
1/3*b^2*e*x^3*arccos(c*x)^2 + 2/3*a*b*e*x^3*arccos(c*x) + 1/3*a^2*e*x^3 - 2/27*b^2*e*x^3 + b^2*d*x*arccos(c*x)^2 - 2/9*sqrt(-c^2*x^2 + 1)*b^2*e*x^2* arccos(c*x)/c + 2*a*b*d*x*arccos(c*x) - 2/9*sqrt(-c^2*x^2 + 1)*a*b*e*x^2/c + a^2*d*x - 2*b^2*d*x - 2*sqrt(-c^2*x^2 + 1)*b^2*d*arccos(c*x)/c - 2*sqrt (-c^2*x^2 + 1)*a*b*d/c - 4/9*b^2*e*x/c^2 - 4/9*sqrt(-c^2*x^2 + 1)*b^2*e*ar ccos(c*x)/c^3 - 4/9*sqrt(-c^2*x^2 + 1)*a*b*e/c^3
Timed out. \[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a + b*acos(c*x))^2*(d + e*x^2),x)
Output:
int((a + b*acos(c*x))^2*(d + e*x^2), x)
\[ \int \left (d+e x^2\right ) (a+b \arccos (c x))^2 \, dx=\frac {9 \mathit {acos} \left (c x \right )^{2} b^{2} c^{3} d x -18 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c^{2} d +18 \mathit {acos} \left (c x \right ) a b \,c^{3} d x +6 \mathit {acos} \left (c x \right ) a b \,c^{3} e \,x^{3}-18 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} d -2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} e \,x^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a b e +9 \left (\int \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3} e +9 a^{2} c^{3} d x +3 a^{2} c^{3} e \,x^{3}-18 b^{2} c^{3} d x}{9 c^{3}} \] Input:
int((e*x^2+d)*(a+b*acos(c*x))^2,x)
Output:
(9*acos(c*x)**2*b**2*c**3*d*x - 18*sqrt( - c**2*x**2 + 1)*acos(c*x)*b**2*c **2*d + 18*acos(c*x)*a*b*c**3*d*x + 6*acos(c*x)*a*b*c**3*e*x**3 - 18*sqrt( - c**2*x**2 + 1)*a*b*c**2*d - 2*sqrt( - c**2*x**2 + 1)*a*b*c**2*e*x**2 - 4*sqrt( - c**2*x**2 + 1)*a*b*e + 9*int(acos(c*x)**2*x**2,x)*b**2*c**3*e + 9*a**2*c**3*d*x + 3*a**2*c**3*e*x**3 - 18*b**2*c**3*d*x)/(9*c**3)