Integrand size = 25, antiderivative size = 277 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-\frac {35 b^2 d^3 x^2}{1024}+\frac {35 b^2 d^3 \left (1-c^2 x^2\right )^2}{3072 c^2}+\frac {7 b^2 d^3 \left (1-c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1-c^2 x^2\right )^4}{256 c^2}+\frac {35 b d^3 x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{512 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{768 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))}{192 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))}{32 c}+\frac {35 d^3 (a+b \arccos (c x))^2}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2} \] Output:
-35/1024*b^2*d^3*x^2+35/3072*b^2*d^3*(-c^2*x^2+1)^2/c^2+7/1152*b^2*d^3*(-c ^2*x^2+1)^3/c^2+1/256*b^2*d^3*(-c^2*x^2+1)^4/c^2+35/512*b*d^3*x*(-c^2*x^2+ 1)^(1/2)*(a+b*arccos(c*x))/c+35/768*b*d^3*x*(-c^2*x^2+1)^(3/2)*(a+b*arccos (c*x))/c+7/192*b*d^3*x*(-c^2*x^2+1)^(5/2)*(a+b*arccos(c*x))/c+1/32*b*d^3*x *(-c^2*x^2+1)^(7/2)*(a+b*arccos(c*x))/c+35/1024*d^3*(a+b*arccos(c*x))^2/c^ 2-1/8*d^3*(-c^2*x^2+1)^4*(a+b*arccos(c*x))^2/c^2
Time = 1.48 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.93 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {d^3 \left (c x \left (-1152 a^2 c x \left (-4+6 c^2 x^2-4 c^4 x^4+c^6 x^6\right )+b^2 c x \left (-837+489 c^2 x^2-200 c^4 x^4+36 c^6 x^6\right )+6 a b \sqrt {1-c^2 x^2} \left (-279+326 c^2 x^2-200 c^4 x^4+48 c^6 x^6\right )\right )+6 b c x \left (-384 a c x \left (-4+6 c^2 x^2-4 c^4 x^4+c^6 x^6\right )+b \sqrt {1-c^2 x^2} \left (-279+326 c^2 x^2-200 c^4 x^4+48 c^6 x^6\right )\right ) \arccos (c x)-9 b^2 \left (93-512 c^2 x^2+768 c^4 x^4-512 c^6 x^6+128 c^8 x^8\right ) \arccos (c x)^2+1674 a b \arcsin (c x)\right )}{9216 c^2} \] Input:
Integrate[x*(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2,x]
Output:
(d^3*(c*x*(-1152*a^2*c*x*(-4 + 6*c^2*x^2 - 4*c^4*x^4 + c^6*x^6) + b^2*c*x* (-837 + 489*c^2*x^2 - 200*c^4*x^4 + 36*c^6*x^6) + 6*a*b*Sqrt[1 - c^2*x^2]* (-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*c*x*(-384*a*c*x*(-4 + 6*c^2*x^2 - 4*c^4*x^4 + c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2* x^2 - 200*c^4*x^4 + 48*c^6*x^6))*ArcCos[c*x] - 9*b^2*(93 - 512*c^2*x^2 + 7 68*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8)*ArcCos[c*x]^2 + 1674*a*b*ArcSin[c* x]))/(9216*c^2)
Time = 0.98 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5183, 5159, 241, 5159, 241, 5159, 244, 2009, 5157, 15, 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.
| \(\displaystyle \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5183 |
| \(\displaystyle -\frac {b d^3 \int \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))dx}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \int \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {1}{8} b c \int x \left (1-c^2 x^2\right )^3dx+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \int \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))dx+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{6} b c \int x \left (1-c^2 x^2\right )^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5159 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int x \left (1-c^2 x^2\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} b c \int \left (x-c^2 x^3\right )dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arccos (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5157 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {b d^3 \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))+\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )+\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}\) |
| \(\Big \downarrow \) 5153 |
| \(\displaystyle -\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \arccos (c x))^2}{8 c^2}-\frac {b d^3 \left (\frac {1}{8} x \left (1-c^2 x^2\right )^{7/2} (a+b \arccos (c x))+\frac {7}{8} \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arccos (c x))+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {(a+b \arccos (c x))^2}{4 b c}+\frac {1}{4} b c x^2\right )+\frac {1}{4} b c \left (\frac {x^2}{2}-\frac {c^2 x^4}{4}\right )\right )-\frac {b \left (1-c^2 x^2\right )^3}{36 c}\right )-\frac {b \left (1-c^2 x^2\right )^4}{64 c}\right )}{4 c}\) |
Input:
Int[x*(d - c^2*d*x^2)^3*(a + b*ArcCos[c*x])^2,x]
Output:
-1/8*(d^3*(1 - c^2*x^2)^4*(a + b*ArcCos[c*x])^2)/c^2 - (b*d^3*(-1/64*(b*(1 - c^2*x^2)^4)/c + (x*(1 - c^2*x^2)^(7/2)*(a + b*ArcCos[c*x]))/8 + (7*(-1/ 36*(b*(1 - c^2*x^2)^3)/c + (x*(1 - c^2*x^2)^(5/2)*(a + b*ArcCos[c*x]))/6 + (5*((b*c*(x^2/2 - (c^2*x^4)/4))/4 + (x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCos[ c*x]))/4 + (3*((b*c*x^2)/4 + (x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/2 - (a + b*ArcCos[c*x])^2/(4*b*c)))/4))/6))/8))/(4*c)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcCos[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] + Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[c*x])^n, x], x] + Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c , d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
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]
Time = 0.28 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {-\frac {d^{3} a^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arccos \left (c x \right ) \left (48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}-200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-279 c x \sqrt {-c^{2} x^{2}+1}+105 \arccos \left (c x \right )\right )}{1536}+\frac {35 \arccos \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )-2 d^{3} a b \left (\frac {\arccos \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arccos \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}+\frac {\arccos \left (c x \right )}{8}+\frac {35 \arcsin \left (c x \right )}{1024}-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}+\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}-\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}+\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(342\) |
| default | \(\frac {-\frac {d^{3} a^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-d^{3} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arccos \left (c x \right ) \left (48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}-200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-279 c x \sqrt {-c^{2} x^{2}+1}+105 \arccos \left (c x \right )\right )}{1536}+\frac {35 \arccos \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )-2 d^{3} a b \left (\frac {\arccos \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arccos \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}+\frac {\arccos \left (c x \right )}{8}+\frac {35 \arcsin \left (c x \right )}{1024}-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}+\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}-\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}+\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(342\) |
| parts | \(-\frac {d^{3} a^{2} \left (c^{2} x^{2}-1\right )^{4}}{8 c^{2}}-\frac {d^{3} b^{2} \left (\frac {\arccos \left (c x \right )^{2} \left (c^{2} x^{2}-1\right )^{4}}{8}-\frac {\arccos \left (c x \right ) \left (48 c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}-200 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}+326 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-279 c x \sqrt {-c^{2} x^{2}+1}+105 \arccos \left (c x \right )\right )}{1536}+\frac {35 \arccos \left (c x \right )^{2}}{1024}-\frac {\left (c^{2} x^{2}-1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}-1\right )^{3}}{1152}-\frac {35 \left (c^{2} x^{2}-1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}-\frac {35}{1024}\right )}{c^{2}}-\frac {2 d^{3} a b \left (\frac {\arccos \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arccos \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arccos \left (c x \right )}{4}-\frac {c^{2} x^{2} \arccos \left (c x \right )}{2}+\frac {\arccos \left (c x \right )}{8}+\frac {35 \arcsin \left (c x \right )}{1024}-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}+\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}-\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}+\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(347\) |
| orering | \(\frac {\left (6084 c^{10} x^{10}-32348 c^{8} x^{8}+72453 c^{6} x^{6}-97420 c^{4} x^{4}+34749 c^{2} x^{2}-5022\right ) \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2}}{18432 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (756 c^{8} x^{8}-4160 c^{6} x^{6}+9913 c^{4} x^{4}-15489 c^{2} x^{2}+3348\right ) \left (\left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right )^{2}-6 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2}-\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{18432 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (c^{2} x^{2}-1\right )}+\frac {x \left (36 c^{6} x^{6}-200 c^{4} x^{4}+489 c^{2} x^{2}-837\right ) \left (-18 \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right )^{2} d \,c^{2} x -\frac {4 \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b c}{\sqrt {-c^{2} x^{2}+1}}+24 x^{3} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arccos \left (c x \right )\right )^{2} d^{2} c^{4}+\frac {24 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arccos \left (c x \right )\right ) d \,c^{3} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {2 x \left (-c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{-c^{2} x^{2}+1}-\frac {2 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arccos \left (c x \right )\right ) b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{18432 c^{2} \left (c x -1\right )^{2} \left (c x +1\right )^{2}}\) | \(521\) |
Input:
int(x*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/c^2*(-1/8*d^3*a^2*(c^2*x^2-1)^4-d^3*b^2*(1/8*arccos(c*x)^2*(c^2*x^2-1)^4 -1/1536*arccos(c*x)*(48*c^7*x^7*(-c^2*x^2+1)^(1/2)-200*c^5*x^5*(-c^2*x^2+1 )^(1/2)+326*c^3*x^3*(-c^2*x^2+1)^(1/2)-279*c*x*(-c^2*x^2+1)^(1/2)+105*arcc os(c*x))+35/1024*arccos(c*x)^2-1/256*(c^2*x^2-1)^4+7/1152*(c^2*x^2-1)^3-35 /3072*(c^2*x^2-1)^2+35/1024*c^2*x^2-35/1024)-2*d^3*a*b*(1/8*arccos(c*x)*c^ 8*x^8-1/2*arccos(c*x)*c^6*x^6+3/4*c^4*x^4*arccos(c*x)-1/2*c^2*x^2*arccos(c *x)+1/8*arccos(c*x)+35/1024*arcsin(c*x)-1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)+25 /384*c^5*x^5*(-c^2*x^2+1)^(1/2)-163/1536*c^3*x^3*(-c^2*x^2+1)^(1/2)+93/102 4*c*x*(-c^2*x^2+1)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.28 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-\frac {36 \, {\left (32 \, a^{2} - b^{2}\right )} c^{8} d^{3} x^{8} - 8 \, {\left (576 \, a^{2} - 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \, {\left (2304 \, a^{2} - 163 \, b^{2}\right )} c^{4} d^{3} x^{4} - 9 \, {\left (512 \, a^{2} - 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \, {\left (128 \, b^{2} c^{8} d^{3} x^{8} - 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} - 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \arccos \left (c x\right )^{2} + 18 \, {\left (128 \, a b c^{8} d^{3} x^{8} - 512 \, a b c^{6} d^{3} x^{6} + 768 \, a b c^{4} d^{3} x^{4} - 512 \, a b c^{2} d^{3} x^{2} + 93 \, a b d^{3}\right )} \arccos \left (c x\right ) - 6 \, {\left (48 \, a b c^{7} d^{3} x^{7} - 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} - 279 \, a b c d^{3} x + {\left (48 \, b^{2} c^{7} d^{3} x^{7} - 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} - 279 \, b^{2} c d^{3} x\right )} \arccos \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{9216 \, c^{2}} \] Input:
integrate(x*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="fricas")
Output:
-1/9216*(36*(32*a^2 - b^2)*c^8*d^3*x^8 - 8*(576*a^2 - 25*b^2)*c^6*d^3*x^6 + 3*(2304*a^2 - 163*b^2)*c^4*d^3*x^4 - 9*(512*a^2 - 93*b^2)*c^2*d^3*x^2 + 9*(128*b^2*c^8*d^3*x^8 - 512*b^2*c^6*d^3*x^6 + 768*b^2*c^4*d^3*x^4 - 512*b ^2*c^2*d^3*x^2 + 93*b^2*d^3)*arccos(c*x)^2 + 18*(128*a*b*c^8*d^3*x^8 - 512 *a*b*c^6*d^3*x^6 + 768*a*b*c^4*d^3*x^4 - 512*a*b*c^2*d^3*x^2 + 93*a*b*d^3) *arccos(c*x) - 6*(48*a*b*c^7*d^3*x^7 - 200*a*b*c^5*d^3*x^5 + 326*a*b*c^3*d ^3*x^3 - 279*a*b*c*d^3*x + (48*b^2*c^7*d^3*x^7 - 200*b^2*c^5*d^3*x^5 + 326 *b^2*c^3*d^3*x^3 - 279*b^2*c*d^3*x)*arccos(c*x))*sqrt(-c^2*x^2 + 1))/c^2
Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (258) = 516\).
Time = 1.23 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.09 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{6} d^{3} x^{8}}{8} + \frac {a^{2} c^{4} d^{3} x^{6}}{2} - \frac {3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{2}}{2} - \frac {a b c^{6} d^{3} x^{8} \operatorname {acos}{\left (c x \right )}}{4} + \frac {a b c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname {acos}{\left (c x \right )} - \frac {25 a b c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{192} - \frac {3 a b c^{2} d^{3} x^{4} \operatorname {acos}{\left (c x \right )}}{2} + \frac {163 a b c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname {acos}{\left (c x \right )} - \frac {93 a b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{512 c} - \frac {93 a b d^{3} \operatorname {acos}{\left (c x \right )}}{512 c^{2}} - \frac {b^{2} c^{6} d^{3} x^{8} \operatorname {acos}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{6} d^{3} x^{8}}{256} + \frac {b^{2} c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{32} + \frac {b^{2} c^{4} d^{3} x^{6} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {25 b^{2} c^{4} d^{3} x^{6}}{1152} - \frac {25 b^{2} c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{192} - \frac {3 b^{2} c^{2} d^{3} x^{4} \operatorname {acos}^{2}{\left (c x \right )}}{4} + \frac {163 b^{2} c^{2} d^{3} x^{4}}{3072} + \frac {163 b^{2} c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{768} + \frac {b^{2} d^{3} x^{2} \operatorname {acos}^{2}{\left (c x \right )}}{2} - \frac {93 b^{2} d^{3} x^{2}}{1024} - \frac {93 b^{2} d^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{512 c} - \frac {93 b^{2} d^{3} \operatorname {acos}^{2}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {d^{3} x^{2} \left (a + \frac {\pi b}{2}\right )^{2}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(-c**2*d*x**2+d)**3*(a+b*acos(c*x))**2,x)
Output:
Piecewise((-a**2*c**6*d**3*x**8/8 + a**2*c**4*d**3*x**6/2 - 3*a**2*c**2*d* *3*x**4/4 + a**2*d**3*x**2/2 - a*b*c**6*d**3*x**8*acos(c*x)/4 + a*b*c**5*d **3*x**7*sqrt(-c**2*x**2 + 1)/32 + a*b*c**4*d**3*x**6*acos(c*x) - 25*a*b*c **3*d**3*x**5*sqrt(-c**2*x**2 + 1)/192 - 3*a*b*c**2*d**3*x**4*acos(c*x)/2 + 163*a*b*c*d**3*x**3*sqrt(-c**2*x**2 + 1)/768 + a*b*d**3*x**2*acos(c*x) - 93*a*b*d**3*x*sqrt(-c**2*x**2 + 1)/(512*c) - 93*a*b*d**3*acos(c*x)/(512*c **2) - b**2*c**6*d**3*x**8*acos(c*x)**2/8 + b**2*c**6*d**3*x**8/256 + b**2 *c**5*d**3*x**7*sqrt(-c**2*x**2 + 1)*acos(c*x)/32 + b**2*c**4*d**3*x**6*ac os(c*x)**2/2 - 25*b**2*c**4*d**3*x**6/1152 - 25*b**2*c**3*d**3*x**5*sqrt(- c**2*x**2 + 1)*acos(c*x)/192 - 3*b**2*c**2*d**3*x**4*acos(c*x)**2/4 + 163* b**2*c**2*d**3*x**4/3072 + 163*b**2*c*d**3*x**3*sqrt(-c**2*x**2 + 1)*acos( c*x)/768 + b**2*d**3*x**2*acos(c*x)**2/2 - 93*b**2*d**3*x**2/1024 - 93*b** 2*d**3*x*sqrt(-c**2*x**2 + 1)*acos(c*x)/(512*c) - 93*b**2*d**3*acos(c*x)** 2/(1024*c**2), Ne(c, 0)), (d**3*x**2*(a + pi*b/2)**2/2, True))
\[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int { -{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arccos \left (c x\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="maxima")
Output:
-1/8*a^2*c^6*d^3*x^8 + 1/2*a^2*c^4*d^3*x^6 - 1/1536*(384*x^8*arccos(c*x) - (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt( -c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9 )*c)*a*b*c^6*d^3 - 3/4*a^2*c^2*d^3*x^4 + 1/48*(48*x^6*arccos(c*x) - (8*sqr t(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*a*b*c^4*d^3 - 3/16*(8*x^4*arccos(c*x) - (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c *x)/c^5)*c)*a*b*c^2*d^3 + 1/2*a^2*d^3*x^2 + 1/2*(2*x^2*arccos(c*x) - c*(sq rt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*a*b*d^3 - 1/8*(b^2*c^6*d^3*x^8 - 4*b^2*c^4*d^3*x^6 + 6*b^2*c^2*d^3*x^4 - 4*b^2*d^3*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + integrate(1/4*(b^2*c^7*d^3*x^8 - 4*b^2*c^5*d ^3*x^6 + 6*b^2*c^3*d^3*x^4 - 4*b^2*c*d^3*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1) *arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^2*x^2 - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (245) = 490\).
Time = 0.17 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.81 \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=-\frac {1}{8} \, b^{2} c^{6} d^{3} x^{8} \arccos \left (c x\right )^{2} - \frac {1}{4} \, a b c^{6} d^{3} x^{8} \arccos \left (c x\right ) - \frac {1}{8} \, a^{2} c^{6} d^{3} x^{8} + \frac {1}{256} \, b^{2} c^{6} d^{3} x^{8} + \frac {1}{32} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{5} d^{3} x^{7} \arccos \left (c x\right ) + \frac {1}{32} \, \sqrt {-c^{2} x^{2} + 1} a b c^{5} d^{3} x^{7} + \frac {1}{2} \, b^{2} c^{4} d^{3} x^{6} \arccos \left (c x\right )^{2} + a b c^{4} d^{3} x^{6} \arccos \left (c x\right ) + \frac {1}{2} \, a^{2} c^{4} d^{3} x^{6} - \frac {25}{1152} \, b^{2} c^{4} d^{3} x^{6} - \frac {25}{192} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c^{3} d^{3} x^{5} \arccos \left (c x\right ) - \frac {25}{192} \, \sqrt {-c^{2} x^{2} + 1} a b c^{3} d^{3} x^{5} - \frac {3}{4} \, b^{2} c^{2} d^{3} x^{4} \arccos \left (c x\right )^{2} - \frac {3}{2} \, a b c^{2} d^{3} x^{4} \arccos \left (c x\right ) - \frac {3}{4} \, a^{2} c^{2} d^{3} x^{4} + \frac {163}{3072} \, b^{2} c^{2} d^{3} x^{4} + \frac {163}{768} \, \sqrt {-c^{2} x^{2} + 1} b^{2} c d^{3} x^{3} \arccos \left (c x\right ) + \frac {163}{768} \, \sqrt {-c^{2} x^{2} + 1} a b c d^{3} x^{3} + \frac {1}{2} \, b^{2} d^{3} x^{2} \arccos \left (c x\right )^{2} + a b d^{3} x^{2} \arccos \left (c x\right ) + \frac {1}{2} \, a^{2} d^{3} x^{2} - \frac {93}{1024} \, b^{2} d^{3} x^{2} - \frac {93 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} x \arccos \left (c x\right )}{512 \, c} - \frac {93 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3} x}{512 \, c} - \frac {93 \, b^{2} d^{3} \arccos \left (c x\right )^{2}}{1024 \, c^{2}} - \frac {93 \, a b d^{3} \arccos \left (c x\right )}{512 \, c^{2}} + \frac {9209 \, b^{2} d^{3}}{294912 \, c^{2}} \] Input:
integrate(x*(-c^2*d*x^2+d)^3*(a+b*arccos(c*x))^2,x, algorithm="giac")
Output:
-1/8*b^2*c^6*d^3*x^8*arccos(c*x)^2 - 1/4*a*b*c^6*d^3*x^8*arccos(c*x) - 1/8 *a^2*c^6*d^3*x^8 + 1/256*b^2*c^6*d^3*x^8 + 1/32*sqrt(-c^2*x^2 + 1)*b^2*c^5 *d^3*x^7*arccos(c*x) + 1/32*sqrt(-c^2*x^2 + 1)*a*b*c^5*d^3*x^7 + 1/2*b^2*c ^4*d^3*x^6*arccos(c*x)^2 + a*b*c^4*d^3*x^6*arccos(c*x) + 1/2*a^2*c^4*d^3*x ^6 - 25/1152*b^2*c^4*d^3*x^6 - 25/192*sqrt(-c^2*x^2 + 1)*b^2*c^3*d^3*x^5*a rccos(c*x) - 25/192*sqrt(-c^2*x^2 + 1)*a*b*c^3*d^3*x^5 - 3/4*b^2*c^2*d^3*x ^4*arccos(c*x)^2 - 3/2*a*b*c^2*d^3*x^4*arccos(c*x) - 3/4*a^2*c^2*d^3*x^4 + 163/3072*b^2*c^2*d^3*x^4 + 163/768*sqrt(-c^2*x^2 + 1)*b^2*c*d^3*x^3*arcco s(c*x) + 163/768*sqrt(-c^2*x^2 + 1)*a*b*c*d^3*x^3 + 1/2*b^2*d^3*x^2*arccos (c*x)^2 + a*b*d^3*x^2*arccos(c*x) + 1/2*a^2*d^3*x^2 - 93/1024*b^2*d^3*x^2 - 93/512*sqrt(-c^2*x^2 + 1)*b^2*d^3*x*arccos(c*x)/c - 93/512*sqrt(-c^2*x^2 + 1)*a*b*d^3*x/c - 93/1024*b^2*d^3*arccos(c*x)^2/c^2 - 93/512*a*b*d^3*arc cos(c*x)/c^2 + 9209/294912*b^2*d^3/c^2
Timed out. \[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \] Input:
int(x*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^3,x)
Output:
int(x*(a + b*acos(c*x))^2*(d - c^2*d*x^2)^3, x)
\[ \int x \left (d-c^2 d x^2\right )^3 (a+b \arccos (c x))^2 \, dx=\frac {d^{3} \left (768 \mathit {acos} \left (c x \right )^{2} b^{2} c^{2} x^{2}-384 \mathit {acos} \left (c x \right )^{2} b^{2}-768 \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) b^{2} c x -384 \mathit {acos} \left (c x \right ) a b \,c^{8} x^{8}+1536 \mathit {acos} \left (c x \right ) a b \,c^{6} x^{6}-2304 \mathit {acos} \left (c x \right ) a b \,c^{4} x^{4}+1536 \mathit {acos} \left (c x \right ) a b \,c^{2} x^{2}+279 \mathit {asin} \left (c x \right ) a b +48 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{7} x^{7}-200 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{5} x^{5}+326 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-279 \sqrt {-c^{2} x^{2}+1}\, a b c x -1536 \left (\int \mathit {acos} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}+4608 \left (\int \mathit {acos} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}-4608 \left (\int \mathit {acos} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}-192 a^{2} c^{8} x^{8}+768 a^{2} c^{6} x^{6}-1152 a^{2} c^{4} x^{4}+768 a^{2} c^{2} x^{2}-384 b^{2} c^{2} x^{2}\right )}{1536 c^{2}} \] Input:
int(x*(-c^2*d*x^2+d)^3*(a+b*acos(c*x))^2,x)
Output:
(d**3*(768*acos(c*x)**2*b**2*c**2*x**2 - 384*acos(c*x)**2*b**2 - 768*sqrt( - c**2*x**2 + 1)*acos(c*x)*b**2*c*x - 384*acos(c*x)*a*b*c**8*x**8 + 1536* acos(c*x)*a*b*c**6*x**6 - 2304*acos(c*x)*a*b*c**4*x**4 + 1536*acos(c*x)*a* b*c**2*x**2 + 279*asin(c*x)*a*b + 48*sqrt( - c**2*x**2 + 1)*a*b*c**7*x**7 - 200*sqrt( - c**2*x**2 + 1)*a*b*c**5*x**5 + 326*sqrt( - c**2*x**2 + 1)*a* b*c**3*x**3 - 279*sqrt( - c**2*x**2 + 1)*a*b*c*x - 1536*int(acos(c*x)**2*x **7,x)*b**2*c**8 + 4608*int(acos(c*x)**2*x**5,x)*b**2*c**6 - 4608*int(acos (c*x)**2*x**3,x)*b**2*c**4 - 192*a**2*c**8*x**8 + 768*a**2*c**6*x**6 - 115 2*a**2*c**4*x**4 + 768*a**2*c**2*x**2 - 384*b**2*c**2*x**2))/(1536*c**2)