Integrand size = 14, antiderivative size = 135 \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=-\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \sqrt {1-a^2 x^2}}{15 a^5}+\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^{3/2}}{45 a^5}-\frac {d^2 \left (1-a^2 x^2\right )^{5/2}}{25 a^5}+c^2 x \arccos (a x)+\frac {2}{3} c d x^3 \arccos (a x)+\frac {1}{5} d^2 x^5 \arccos (a x) \]
2/45*d*(5*a^2*c+3*d)*(-a^2*x^2+1)^(3/2)/a^5-1/25*d^2*(-a^2*x^2+1)^(5/2)/a^ 5+c^2*x*arccos(a*x)+2/3*c*d*x^3*arccos(a*x)+1/5*d^2*x^5*arccos(a*x)-1/15*( 15*a^4*c^2+10*a^2*c*d+3*d^2)*(-a^2*x^2+1)^(1/2)/a^5
Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.73 \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=-\frac {\sqrt {1-a^2 x^2} \left (24 d^2+4 a^2 d \left (25 c+3 d x^2\right )+a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )\right )}{225 a^5}+\left (c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}\right ) \arccos (a x) \]
-1/225*(Sqrt[1 - a^2*x^2]*(24*d^2 + 4*a^2*d*(25*c + 3*d*x^2) + a^4*(225*c^ 2 + 50*c*d*x^2 + 9*d^2*x^4)))/a^5 + (c^2*x + (2*c*d*x^3)/3 + (d^2*x^5)/5)* ArcCos[a*x]
Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5171, 27, 1576, 1140, 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.
\(\displaystyle \int \arccos (a x) \left (c+d x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 5171 |
\(\displaystyle a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{15 \sqrt {1-a^2 x^2}}dx+c^2 x \arccos (a x)+\frac {2}{3} c d x^3 \arccos (a x)+\frac {1}{5} d^2 x^5 \arccos (a x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{15} a \int \frac {x \left (3 d^2 x^4+10 c d x^2+15 c^2\right )}{\sqrt {1-a^2 x^2}}dx+c^2 x \arccos (a x)+\frac {2}{3} c d x^3 \arccos (a x)+\frac {1}{5} d^2 x^5 \arccos (a x)\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {1}{30} a \int \frac {3 d^2 x^4+10 c d x^2+15 c^2}{\sqrt {1-a^2 x^2}}dx^2+c^2 x \arccos (a x)+\frac {2}{3} c d x^3 \arccos (a x)+\frac {1}{5} d^2 x^5 \arccos (a x)\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \frac {1}{30} a \int \left (\frac {3 \left (1-a^2 x^2\right )^{3/2} d^2}{a^4}-\frac {2 \left (5 c a^2+3 d\right ) \sqrt {1-a^2 x^2} d}{a^4}+\frac {15 c^2 a^4+10 c d a^2+3 d^2}{a^4 \sqrt {1-a^2 x^2}}\right )dx^2+c^2 x \arccos (a x)+\frac {2}{3} c d x^3 \arccos (a x)+\frac {1}{5} d^2 x^5 \arccos (a x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{30} a \left (\frac {4 d \left (1-a^2 x^2\right )^{3/2} \left (5 a^2 c+3 d\right )}{3 a^6}-\frac {6 d^2 \left (1-a^2 x^2\right )^{5/2}}{5 a^6}-\frac {2 \sqrt {1-a^2 x^2} \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{a^6}\right )+c^2 x \arccos (a x)+\frac {2}{3} c d x^3 \arccos (a x)+\frac {1}{5} d^2 x^5 \arccos (a x)\) |
(a*((-2*(15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Sqrt[1 - a^2*x^2])/a^6 + (4*d*(5 *a^2*c + 3*d)*(1 - a^2*x^2)^(3/2))/(3*a^6) - (6*d^2*(1 - a^2*x^2)^(5/2))/( 5*a^6)))/30 + c^2*x*ArcCos[a*x] + (2*c*d*x^3*ArcCos[a*x])/3 + (d^2*x^5*Arc Cos[a*x])/5
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[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCos[c*x]) u, x ] + Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2 , 0])
Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {\arccos \left (a x \right ) c^{2} a x +\frac {2 a \arccos \left (a x \right ) c d \,x^{3}}{3}+\frac {a \arccos \left (a x \right ) d^{2} x^{5}}{5}+\frac {3 d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )-15 c^{2} a^{4} \sqrt {-a^{2} x^{2}+1}+10 c \,a^{2} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )}{15 a^{4}}}{a}\) | \(169\) |
default | \(\frac {\arccos \left (a x \right ) c^{2} a x +\frac {2 a \arccos \left (a x \right ) c d \,x^{3}}{3}+\frac {a \arccos \left (a x \right ) d^{2} x^{5}}{5}+\frac {3 d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )-15 c^{2} a^{4} \sqrt {-a^{2} x^{2}+1}+10 c \,a^{2} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )}{15 a^{4}}}{a}\) | \(169\) |
parts | \(\frac {d^{2} x^{5} \arccos \left (a x \right )}{5}+\frac {2 c d \,x^{3} \arccos \left (a x \right )}{3}+c^{2} x \arccos \left (a x \right )+\frac {a \left (3 d^{2} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-\frac {15 c^{2} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+10 c d \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )\right )}{15}\) | \(169\) |
1/a*(arccos(a*x)*c^2*a*x+2/3*a*arccos(a*x)*c*d*x^3+1/5*a*arccos(a*x)*d^2*x ^5+1/15/a^4*(3*d^2*(-1/5*a^4*x^4*(-a^2*x^2+1)^(1/2)-4/15*a^2*x^2*(-a^2*x^2 +1)^(1/2)-8/15*(-a^2*x^2+1)^(1/2))-15*c^2*a^4*(-a^2*x^2+1)^(1/2)+10*c*a^2* d*(-1/3*a^2*x^2*(-a^2*x^2+1)^(1/2)-2/3*(-a^2*x^2+1)^(1/2))))
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81 \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=\frac {15 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \arccos \left (a x\right ) - {\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \, {\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{225 \, a^{5}} \]
1/225*(15*(3*a^5*d^2*x^5 + 10*a^5*c*d*x^3 + 15*a^5*c^2*x)*arccos(a*x) - (9 *a^4*d^2*x^4 + 225*a^4*c^2 + 100*a^2*c*d + 2*(25*a^4*c*d + 6*a^2*d^2)*x^2 + 24*d^2)*sqrt(-a^2*x^2 + 1))/a^5
Time = 0.42 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.46 \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=\begin {cases} c^{2} x \operatorname {acos}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acos}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acos}{\left (a x \right )}}{5} - \frac {c^{2} \sqrt {- a^{2} x^{2} + 1}}{a} - \frac {2 c d x^{2} \sqrt {- a^{2} x^{2} + 1}}{9 a} - \frac {d^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} - \frac {4 c d \sqrt {- a^{2} x^{2} + 1}}{9 a^{3}} - \frac {4 d^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{3}} - \frac {8 d^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \]
Piecewise((c**2*x*acos(a*x) + 2*c*d*x**3*acos(a*x)/3 + d**2*x**5*acos(a*x) /5 - c**2*sqrt(-a**2*x**2 + 1)/a - 2*c*d*x**2*sqrt(-a**2*x**2 + 1)/(9*a) - d**2*x**4*sqrt(-a**2*x**2 + 1)/(25*a) - 4*c*d*sqrt(-a**2*x**2 + 1)/(9*a** 3) - 4*d**2*x**2*sqrt(-a**2*x**2 + 1)/(75*a**3) - 8*d**2*sqrt(-a**2*x**2 + 1)/(75*a**5), Ne(a, 0)), (pi*(c**2*x + 2*c*d*x**3/3 + d**2*x**5/5)/2, Tru e))
Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.19 \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=-\frac {1}{225} \, {\left (\frac {9 \, \sqrt {-a^{2} x^{2} + 1} d^{2} x^{4}}{a^{2}} + \frac {50 \, \sqrt {-a^{2} x^{2} + 1} c d x^{2}}{a^{2}} + \frac {225 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2}} + \frac {12 \, \sqrt {-a^{2} x^{2} + 1} d^{2} x^{2}}{a^{4}} + \frac {100 \, \sqrt {-a^{2} x^{2} + 1} c d}{a^{4}} + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} d^{2}}{a^{6}}\right )} a + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \arccos \left (a x\right ) \]
-1/225*(9*sqrt(-a^2*x^2 + 1)*d^2*x^4/a^2 + 50*sqrt(-a^2*x^2 + 1)*c*d*x^2/a ^2 + 225*sqrt(-a^2*x^2 + 1)*c^2/a^2 + 12*sqrt(-a^2*x^2 + 1)*d^2*x^2/a^4 + 100*sqrt(-a^2*x^2 + 1)*c*d/a^4 + 24*sqrt(-a^2*x^2 + 1)*d^2/a^6)*a + 1/15*( 3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*arccos(a*x)
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.19 \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=\frac {1}{5} \, d^{2} x^{5} \arccos \left (a x\right ) + \frac {2}{3} \, c d x^{3} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} d^{2} x^{4}}{25 \, a} + c^{2} x \arccos \left (a x\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c d x^{2}}{9 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} d^{2} x^{2}}{75 \, a^{3}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c d}{9 \, a^{3}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} d^{2}}{75 \, a^{5}} \]
1/5*d^2*x^5*arccos(a*x) + 2/3*c*d*x^3*arccos(a*x) - 1/25*sqrt(-a^2*x^2 + 1 )*d^2*x^4/a + c^2*x*arccos(a*x) - 2/9*sqrt(-a^2*x^2 + 1)*c*d*x^2/a - sqrt( -a^2*x^2 + 1)*c^2/a - 4/75*sqrt(-a^2*x^2 + 1)*d^2*x^2/a^3 - 4/9*sqrt(-a^2* x^2 + 1)*c*d/a^3 - 8/75*sqrt(-a^2*x^2 + 1)*d^2/a^5
Timed out. \[ \int \left (c+d x^2\right )^2 \arccos (a x) \, dx=\int \mathrm {acos}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^2 \,d x \]