Integrand size = 14, antiderivative size = 102 \[ \int x^2 (a+b \arccos (c x))^2 \, dx=-\frac {4 b^2 x}{9 c^2}-\frac {2 b^2 x^3}{27}-\frac {4 b \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c^3}-\frac {2 b x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{9 c}+\frac {1}{3} x^3 (a+b \arccos (c x))^2 \]
-4/9*b^2*x/c^2-2/27*b^2*x^3+1/3*x^3*(a+b*arccos(c*x))^2-4/9*b*(a+b*arccos( c*x))*(-c^2*x^2+1)^(1/2)/c^3-2/9*b*x^2*(a+b*arccos(c*x))*(-c^2*x^2+1)^(1/2 )/c
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.19 \[ \int x^2 (a+b \arccos (c x))^2 \, dx=\frac {9 a^2 c^3 x^3-6 a b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )-2 b^2 c x \left (6+c^2 x^2\right )-6 b \left (-3 a c^3 x^3+b \sqrt {1-c^2 x^2} \left (2+c^2 x^2\right )\right ) \arccos (c x)+9 b^2 c^3 x^3 \arccos (c x)^2}{27 c^3} \]
(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) - 6*b*(-3*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2))*ArcCos[c*x ] + 9*b^2*c^3*x^3*ArcCos[c*x]^2)/(27*c^3)
Time = 0.41 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5139, 5211, 15, 5183, 24}
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.
\(\displaystyle \int x^2 (a+b \arccos (c x))^2 \, dx\) |
\(\Big \downarrow \) 5139 |
\(\displaystyle \frac {2}{3} b c \int \frac {x^3 (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx+\frac {1}{3} x^3 (a+b \arccos (c x))^2\) |
\(\Big \downarrow \) 5211 |
\(\displaystyle \frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2}{3} b c \left (\frac {2 \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\) |
\(\Big \downarrow \) 5183 |
\(\displaystyle \frac {2}{3} b c \left (\frac {2 \left (-\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {2}{3} b c \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{3 c^2}+\frac {2 \left (-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c^2}-\frac {b x}{c}\right )}{3 c^2}-\frac {b x^3}{9 c}\right )+\frac {1}{3} x^3 (a+b \arccos (c x))^2\) |
(x^3*(a + b*ArcCos[c*x])^2)/3 + (2*b*c*(-1/9*(b*x^3)/c - (x^2*Sqrt[1 - c^2 *x^2]*(a + b*ArcCos[c*x]))/(3*c^2) + (2*(-((b*x)/c) - (Sqrt[1 - c^2*x^2]*( a + b*ArcCos[c*x]))/c^2))/(3*c^2)))/3
3.2.48.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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 = 1.71 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.23
method | result | size |
parts | \(\frac {a^{2} x^{3}}{3}+\frac {b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )}{c^{3}}+\frac {2 a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(125\) |
derivativedivides | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+2 a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(126\) |
default | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+b^{2} \left (\frac {\arccos \left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {2 \arccos \left (c x \right ) \left (c^{2} x^{2}+2\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 c^{3} x^{3}}{27}-\frac {4 c x}{9}\right )+2 a b \left (\frac {c^{3} x^{3} \arccos \left (c x \right )}{3}-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c^{3}}\) | \(126\) |
1/3*a^2*x^3+b^2/c^3*(1/3*arccos(c*x)^2*c^3*x^3-2/9*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)+2*a*b/c^3*(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 = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int x^2 (a+b \arccos (c x))^2 \, dx=\frac {9 \, b^{2} c^{3} x^{3} \arccos \left (c x\right )^{2} + 18 \, a b c^{3} x^{3} \arccos \left (c x\right ) + {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x - 6 \, {\left (a b c^{2} x^{2} + 2 \, a b + {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c^{3}} \]
1/27*(9*b^2*c^3*x^3*arccos(c*x)^2 + 18*a*b*c^3*x^3*arccos(c*x) + (9*a^2 - 2*b^2)*c^3*x^3 - 12*b^2*c*x - 6*(a*b*c^2*x^2 + 2*a*b + (b^2*c^2*x^2 + 2*b^ 2)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^3
Time = 0.31 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.72 \[ \int x^2 (a+b \arccos (c x))^2 \, dx=\begin {cases} \frac {a^{2} x^{3}}{3} + \frac {2 a b x^{3} \operatorname {acos}{\left (c x \right )}}{3} - \frac {2 a b x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {4 a b \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {b^{2} x^{3} \operatorname {acos}^{2}{\left (c x \right )}}{3} - \frac {2 b^{2} x^{3}}{27} - \frac {2 b^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c} - \frac {4 b^{2} x}{9 c^{2}} - \frac {4 b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {x^{3} \left (a + \frac {\pi b}{2}\right )^{2}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a**2*x**3/3 + 2*a*b*x**3*acos(c*x)/3 - 2*a*b*x**2*sqrt(-c**2*x* *2 + 1)/(9*c) - 4*a*b*sqrt(-c**2*x**2 + 1)/(9*c**3) + b**2*x**3*acos(c*x)* *2/3 - 2*b**2*x**3/27 - 2*b**2*x**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(9*c) - 4*b**2*x/(9*c**2) - 4*b**2*sqrt(-c**2*x**2 + 1)*acos(c*x)/(9*c**3), Ne(c, 0)), (x**3*(a + pi*b/2)**2/3, True))
Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.39 \[ \int x^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {1}{3} \, a^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \arccos \left (c x\right ) - c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arccos \left (c x\right ) + \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} \]
1/3*b^2*x^3*arccos(c*x)^2 + 1/3*a^2*x^3 + 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 - 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
Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.40 \[ \int x^2 (a+b \arccos (c x))^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \arccos \left (c x\right )^{2} + \frac {2}{3} \, a b x^{3} \arccos \left (c x\right ) + \frac {1}{3} \, a^{2} x^{3} - \frac {2}{27} \, b^{2} x^{3} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x^{2} \arccos \left (c x\right )}{9 \, c} - \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b x^{2}}{9 \, c} - \frac {4 \, b^{2} x}{9 \, c^{2}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} \arccos \left (c x\right )}{9 \, c^{3}} - \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b}{9 \, c^{3}} \]
1/3*b^2*x^3*arccos(c*x)^2 + 2/3*a*b*x^3*arccos(c*x) + 1/3*a^2*x^3 - 2/27*b ^2*x^3 - 2/9*sqrt(-c^2*x^2 + 1)*b^2*x^2*arccos(c*x)/c - 2/9*sqrt(-c^2*x^2 + 1)*a*b*x^2/c - 4/9*b^2*x/c^2 - 4/9*sqrt(-c^2*x^2 + 1)*b^2*arccos(c*x)/c^ 3 - 4/9*sqrt(-c^2*x^2 + 1)*a*b/c^3
Timed out. \[ \int x^2 (a+b \arccos (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2 \,d x \]