Integrand size = 22, antiderivative size = 196 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=-\frac {40 b^3 d \sqrt {1-c^2 x^2}}{9 c}-\frac {2 b^3 d \left (1-c^2 x^2\right )^{3/2}}{27 c}-\frac {14}{3} b^2 d x (a+b \arcsin (c x))+\frac {2}{9} b^2 c^2 d x^3 (a+b \arcsin (c x))+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c}+\frac {b d \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c}+\frac {2}{3} d x (a+b \arcsin (c x))^3+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3 \] Output:
-40/9*b^3*d*(-c^2*x^2+1)^(1/2)/c-2/27*b^3*d*(-c^2*x^2+1)^(3/2)/c-14/3*b^2* d*x*(a+b*arcsin(c*x))+2/9*b^2*c^2*d*x^3*(a+b*arcsin(c*x))+2*b*d*(-c^2*x^2+ 1)^(1/2)*(a+b*arcsin(c*x))^2/c+1/3*b*d*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x) )^2/c+2/3*d*x*(a+b*arcsin(c*x))^3+1/3*d*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^3
Time = 0.22 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.22 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=-\frac {d \left (-2 b^3 \sqrt {1-c^2 x^2} \left (-61+c^2 x^2\right )-6 a b^2 c x \left (-21+c^2 x^2\right )+9 a^2 b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+9 a^3 c x \left (-3+c^2 x^2\right )+3 b \left (-2 b^2 c x \left (-21+c^2 x^2\right )+6 a b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+9 a^2 c x \left (-3+c^2 x^2\right )\right ) \arcsin (c x)+9 b^2 \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )\right ) \arcsin (c x)^2+9 b^3 c x \left (-3+c^2 x^2\right ) \arcsin (c x)^3\right )}{27 c} \] Input:
Integrate[(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^3,x]
Output:
-1/27*(d*(-2*b^3*Sqrt[1 - c^2*x^2]*(-61 + c^2*x^2) - 6*a*b^2*c*x*(-21 + c^ 2*x^2) + 9*a^2*b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 9*a^3*c*x*(-3 + c^2*x^ 2) + 3*b*(-2*b^2*c*x*(-21 + c^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x ^2) + 9*a^2*c*x*(-3 + c^2*x^2))*ArcSin[c*x] + 9*b^2*(b*Sqrt[1 - c^2*x^2]*( -7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2))*ArcSin[c*x]^2 + 9*b^3*c*x*(-3 + c^ 2*x^2)*ArcSin[c*x]^3))/c
Time = 0.94 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5158, 5130, 5182, 2009, 5154, 27, 353, 53, 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.
| \(\displaystyle \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx\) |
| \(\Big \downarrow \) 5158 |
| \(\displaystyle -b c d \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx+\frac {2}{3} d \int (a+b \arcsin (c x))^3dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3\) |
| \(\Big \downarrow \) 5130 |
| \(\displaystyle \frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx\right )-b c d \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )-b c d \left (\frac {2 b \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b c d \left (\frac {2 b \int \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+\frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 5154 |
| \(\displaystyle -b c d \left (\frac {2 b \left (-b c \int \frac {x \left (3-c^2 x^2\right )}{3 \sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+\frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -b c d \left (\frac {2 b \left (-\frac {1}{3} b c \int \frac {x \left (3-c^2 x^2\right )}{\sqrt {1-c^2 x^2}}dx-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+\frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle -b c d \left (\frac {2 b \left (-\frac {1}{6} b c \int \frac {3-c^2 x^2}{\sqrt {1-c^2 x^2}}dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+\frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -b c d \left (\frac {2 b \left (-\frac {1}{6} b c \int \left (\sqrt {1-c^2 x^2}+\frac {2}{\sqrt {1-c^2 x^2}}\right )dx^2-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+\frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^3+\frac {2}{3} d \left (x (a+b \arcsin (c x))^3-3 b c \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )\right )-b c d \left (\frac {2 b \left (-\frac {1}{3} c^2 x^3 (a+b \arcsin (c x))+x (a+b \arcsin (c x))-\frac {1}{6} b c \left (-\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^2}-\frac {4 \sqrt {1-c^2 x^2}}{c^2}\right )\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{3 c^2}\right )\) |
Input:
Int[(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^3,x]
Output:
(d*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^3)/3 - b*c*d*(-1/3*((1 - c^2*x^2)^( 3/2)*(a + b*ArcSin[c*x])^2)/c^2 + (2*b*(-1/6*(b*c*((-4*Sqrt[1 - c^2*x^2])/ c^2 - (2*(1 - c^2*x^2)^(3/2))/(3*c^2))) + x*(a + b*ArcSin[c*x]) - (c^2*x^3 *(a + b*ArcSin[c*x]))/3))/(3*c)) + (2*d*(x*(a + b*ArcSin[c*x])^3 - 3*b*c*( -((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2) + (2*b*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/c)))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Ar cSin[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbo l] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin[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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[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*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[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*ArcSin[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 = 0.21 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(\frac {-d \,a^{3} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{3} \left (\frac {\arcsin \left (c x \right )^{3} \left (c^{2} x^{2}-3\right ) c x}{3}-2 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+\frac {40 \sqrt {-c^{2} x^{2}+1}}{9}+4 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{9}-\frac {2 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )-3 d a \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )-3 d \,a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(313\) |
| default | \(\frac {-d \,a^{3} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{3} \left (\frac {\arcsin \left (c x \right )^{3} \left (c^{2} x^{2}-3\right ) c x}{3}-2 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+\frac {40 \sqrt {-c^{2} x^{2}+1}}{9}+4 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{9}-\frac {2 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )-3 d a \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )-3 d \,a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(313\) |
| parts | \(-d \,a^{3} \left (\frac {1}{3} c^{2} x^{3}-x \right )-\frac {d \,b^{3} \left (\frac {\arcsin \left (c x \right )^{3} \left (c^{2} x^{2}-3\right ) c x}{3}-2 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+\frac {40 \sqrt {-c^{2} x^{2}+1}}{9}+4 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-3\right ) c x}{9}-\frac {2 \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{27}\right )}{c}-\frac {3 d a \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )}{c}-\frac {3 d \,a^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(317\) |
| orering | \(\frac {5 x \left (13 c^{4} x^{4}-194 c^{2} x^{2}-179\right ) \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{3}}{81 \left (c^{2} x^{2}-1\right )^{2}}-\frac {\left (25 c^{4} x^{4}-683 c^{2} x^{2}-242\right ) \left (-2 c^{2} d x \left (a +b \arcsin \left (c x \right )\right )^{3}+\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2} b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{81 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 x \left (c^{2} x^{2}-41\right ) \left (-2 c^{2} d \left (a +b \arcsin \left (c x \right )\right )^{3}-\frac {12 c^{3} d x \left (a +b \arcsin \left (c x \right )\right )^{2} b}{\sqrt {-c^{2} x^{2}+1}}+\frac {6 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{2}}{-c^{2} x^{2}+1}+\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3} x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 c^{2}}-\frac {\left (c^{2} x^{2}-61\right ) \left (c x -1\right ) \left (c x +1\right ) \left (-\frac {18 c^{3} d \left (a +b \arcsin \left (c x \right )\right )^{2} b}{\sqrt {-c^{2} x^{2}+1}}-\frac {36 c^{4} d x \left (a +b \arcsin \left (c x \right )\right ) b^{2}}{-c^{2} x^{2}+1}-\frac {18 c^{5} d \,x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} b}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 \left (-c^{2} d \,x^{2}+d \right ) b^{3} c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {18 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right ) b^{2} c^{4} x}{\left (-c^{2} x^{2}+1\right )^{2}}+\frac {9 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{5} x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2} b \,c^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{81 c^{4}}\) | \(553\) |
Input:
int((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^3,x,method=_RETURNVERBOSE)
Output:
1/c*(-d*a^3*(1/3*c^3*x^3-c*x)-d*b^3*(1/3*arcsin(c*x)^3*(c^2*x^2-3)*c*x-2*a rcsin(c*x)^2*(-c^2*x^2+1)^(1/2)+40/9*(-c^2*x^2+1)^(1/2)+4*c*x*arcsin(c*x)+ 1/3*arcsin(c*x)^2*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-2/9*arcsin(c*x)*(c^2*x^2- 3)*c*x-2/27*(c^2*x^2-1)*(-c^2*x^2+1)^(1/2))-3*d*a*b^2*(1/3*arcsin(c*x)^2*( c^2*x^2-3)*c*x+4/3*c*x-4/3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+2/9*arcsin(c*x)* (c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-2/27*(c^2*x^2-3)*c*x)-3*d*a^2*b*(1/3*c^3*x^ 3*arcsin(c*x)-c*x*arcsin(c*x)+1/9*c^2*x^2*(-c^2*x^2+1)^(1/2)-7/9*(-c^2*x^2 +1)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.26 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=-\frac {3 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} c^{3} d x^{3} - 9 \, {\left (3 \, a^{3} - 14 \, a b^{2}\right )} c d x + 9 \, {\left (b^{3} c^{3} d x^{3} - 3 \, b^{3} c d x\right )} \arcsin \left (c x\right )^{3} + 27 \, {\left (a b^{2} c^{3} d x^{3} - 3 \, a b^{2} c d x\right )} \arcsin \left (c x\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{3} d x^{3} - 3 \, {\left (9 \, a^{2} b - 14 \, b^{3}\right )} c d x\right )} \arcsin \left (c x\right ) + {\left ({\left (9 \, a^{2} b - 2 \, b^{3}\right )} c^{2} d x^{2} + 9 \, {\left (b^{3} c^{2} d x^{2} - 7 \, b^{3} d\right )} \arcsin \left (c x\right )^{2} - {\left (63 \, a^{2} b - 122 \, b^{3}\right )} d + 18 \, {\left (a b^{2} c^{2} d x^{2} - 7 \, a b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^3,x, algorithm="fricas")
Output:
-1/27*(3*(3*a^3 - 2*a*b^2)*c^3*d*x^3 - 9*(3*a^3 - 14*a*b^2)*c*d*x + 9*(b^3 *c^3*d*x^3 - 3*b^3*c*d*x)*arcsin(c*x)^3 + 27*(a*b^2*c^3*d*x^3 - 3*a*b^2*c* d*x)*arcsin(c*x)^2 + 3*((9*a^2*b - 2*b^3)*c^3*d*x^3 - 3*(9*a^2*b - 14*b^3) *c*d*x)*arcsin(c*x) + ((9*a^2*b - 2*b^3)*c^2*d*x^2 + 9*(b^3*c^2*d*x^2 - 7* b^3*d)*arcsin(c*x)^2 - (63*a^2*b - 122*b^3)*d + 18*(a*b^2*c^2*d*x^2 - 7*a* b^2*d)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c
Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (184) = 368\).
Time = 0.41 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.11 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=\begin {cases} - \frac {a^{3} c^{2} d x^{3}}{3} + a^{3} d x - a^{2} b c^{2} d x^{3} \operatorname {asin}{\left (c x \right )} - \frac {a^{2} b c d x^{2} \sqrt {- c^{2} x^{2} + 1}}{3} + 3 a^{2} b d x \operatorname {asin}{\left (c x \right )} + \frac {7 a^{2} b d \sqrt {- c^{2} x^{2} + 1}}{3 c} - a b^{2} c^{2} d x^{3} \operatorname {asin}^{2}{\left (c x \right )} + \frac {2 a b^{2} c^{2} d x^{3}}{9} - \frac {2 a b^{2} c d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3} + 3 a b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - \frac {14 a b^{2} d x}{3} + \frac {14 a b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c} - \frac {b^{3} c^{2} d x^{3} \operatorname {asin}^{3}{\left (c x \right )}}{3} + \frac {2 b^{3} c^{2} d x^{3} \operatorname {asin}{\left (c x \right )}}{9} - \frac {b^{3} c d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{3} + \frac {2 b^{3} c d x^{2} \sqrt {- c^{2} x^{2} + 1}}{27} + b^{3} d x \operatorname {asin}^{3}{\left (c x \right )} - \frac {14 b^{3} d x \operatorname {asin}{\left (c x \right )}}{3} + \frac {7 b^{3} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c x \right )}}{3 c} - \frac {122 b^{3} d \sqrt {- c^{2} x^{2} + 1}}{27 c} & \text {for}\: c \neq 0 \\a^{3} d x & \text {otherwise} \end {cases} \] Input:
integrate((-c**2*d*x**2+d)*(a+b*asin(c*x))**3,x)
Output:
Piecewise((-a**3*c**2*d*x**3/3 + a**3*d*x - a**2*b*c**2*d*x**3*asin(c*x) - a**2*b*c*d*x**2*sqrt(-c**2*x**2 + 1)/3 + 3*a**2*b*d*x*asin(c*x) + 7*a**2* b*d*sqrt(-c**2*x**2 + 1)/(3*c) - a*b**2*c**2*d*x**3*asin(c*x)**2 + 2*a*b** 2*c**2*d*x**3/9 - 2*a*b**2*c*d*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/3 + 3*a *b**2*d*x*asin(c*x)**2 - 14*a*b**2*d*x/3 + 14*a*b**2*d*sqrt(-c**2*x**2 + 1 )*asin(c*x)/(3*c) - b**3*c**2*d*x**3*asin(c*x)**3/3 + 2*b**3*c**2*d*x**3*a sin(c*x)/9 - b**3*c*d*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/3 + 2*b**3*c* d*x**2*sqrt(-c**2*x**2 + 1)/27 + b**3*d*x*asin(c*x)**3 - 14*b**3*d*x*asin( c*x)/3 + 7*b**3*d*sqrt(-c**2*x**2 + 1)*asin(c*x)**2/(3*c) - 122*b**3*d*sqr t(-c**2*x**2 + 1)/(27*c), Ne(c, 0)), (a**3*d*x, True))
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (173) = 346\).
Time = 0.15 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.27 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=-\frac {1}{3} \, b^{3} c^{2} d x^{3} \arcsin \left (c x\right )^{3} - a b^{2} c^{2} d x^{3} \arcsin \left (c x\right )^{2} - \frac {1}{3} \, a^{3} c^{2} d x^{3} + b^{3} d x \arcsin \left (c x\right )^{3} - \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a^{2} b c^{2} d - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} a b^{2} c^{2} d - \frac {1}{27} \, {\left (9 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right )^{2} - 2 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-c^{2} x^{2} + 1}}{c^{2}}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{3} + 6 \, x\right )} \arcsin \left (c x\right )}{c^{3}}\right )}\right )} b^{3} c^{2} d + 3 \, a b^{2} d x \arcsin \left (c x\right )^{2} + 3 \, {\left (\frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )^{2}}{c} - \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )}}{c}\right )} b^{3} d - 6 \, a b^{2} d {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{3} d x + \frac {3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a^{2} b d}{c} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^3,x, algorithm="maxima")
Output:
-1/3*b^3*c^2*d*x^3*arcsin(c*x)^3 - a*b^2*c^2*d*x^3*arcsin(c*x)^2 - 1/3*a^3 *c^2*d*x^3 + b^3*d*x*arcsin(c*x)^3 - 1/3*(3*x^3*arcsin(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*c^2*d - 2/9*(3*c*(sqr t(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*a*b^2*c^2*d - 1/27*(9*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt( -c^2*x^2 + 1)/c^4)*arcsin(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)*arcsin(c*x)/c^3))*b^3*c^2*d + 3 *a*b^2*d*x*arcsin(c*x)^2 + 3*(sqrt(-c^2*x^2 + 1)*arcsin(c*x)^2/c - 2*(c*x* arcsin(c*x) + sqrt(-c^2*x^2 + 1))/c)*b^3*d - 6*a*b^2*d*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^3*d*x + 3*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a ^2*b*d/c
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (173) = 346\).
Time = 0.17 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.90 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=-\frac {1}{3} \, a^{3} c^{2} d x^{3} - \frac {1}{3} \, {\left (c^{2} x^{2} - 1\right )} b^{3} d x \arcsin \left (c x\right )^{3} - {\left (c^{2} x^{2} - 1\right )} a b^{2} d x \arcsin \left (c x\right )^{2} + \frac {2}{3} \, b^{3} d x \arcsin \left (c x\right )^{3} - {\left (c^{2} x^{2} - 1\right )} a^{2} b d x \arcsin \left (c x\right ) + \frac {2}{9} \, {\left (c^{2} x^{2} - 1\right )} b^{3} d x \arcsin \left (c x\right ) + 2 \, a b^{2} d x \arcsin \left (c x\right )^{2} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3} d \arcsin \left (c x\right )^{2}}{3 \, c} + \frac {2}{9} \, {\left (c^{2} x^{2} - 1\right )} a b^{2} d x + 2 \, a^{2} b d x \arcsin \left (c x\right ) - \frac {40}{9} \, b^{3} d x \arcsin \left (c x\right ) + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b^{2} d \arcsin \left (c x\right )}{3 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{3} d \arcsin \left (c x\right )^{2}}{c} + a^{3} d x - \frac {40}{9} \, a b^{2} d x + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} b d}{3 \, c} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{3} d}{27 \, c} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b^{2} d \arcsin \left (c x\right )}{c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a^{2} b d}{c} - \frac {40 \, \sqrt {-c^{2} x^{2} + 1} b^{3} d}{9 \, c} \] Input:
integrate((-c^2*d*x^2+d)*(a+b*arcsin(c*x))^3,x, algorithm="giac")
Output:
-1/3*a^3*c^2*d*x^3 - 1/3*(c^2*x^2 - 1)*b^3*d*x*arcsin(c*x)^3 - (c^2*x^2 - 1)*a*b^2*d*x*arcsin(c*x)^2 + 2/3*b^3*d*x*arcsin(c*x)^3 - (c^2*x^2 - 1)*a^2 *b*d*x*arcsin(c*x) + 2/9*(c^2*x^2 - 1)*b^3*d*x*arcsin(c*x) + 2*a*b^2*d*x*a rcsin(c*x)^2 + 1/3*(-c^2*x^2 + 1)^(3/2)*b^3*d*arcsin(c*x)^2/c + 2/9*(c^2*x ^2 - 1)*a*b^2*d*x + 2*a^2*b*d*x*arcsin(c*x) - 40/9*b^3*d*x*arcsin(c*x) + 2 /3*(-c^2*x^2 + 1)^(3/2)*a*b^2*d*arcsin(c*x)/c + 2*sqrt(-c^2*x^2 + 1)*b^3*d *arcsin(c*x)^2/c + a^3*d*x - 40/9*a*b^2*d*x + 1/3*(-c^2*x^2 + 1)^(3/2)*a^2 *b*d/c - 2/27*(-c^2*x^2 + 1)^(3/2)*b^3*d/c + 4*sqrt(-c^2*x^2 + 1)*a*b^2*d* arcsin(c*x)/c + 2*sqrt(-c^2*x^2 + 1)*a^2*b*d/c - 40/9*sqrt(-c^2*x^2 + 1)*b ^3*d/c
Timed out. \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3\,\left (d-c^2\,d\,x^2\right ) \,d x \] Input:
int((a + b*asin(c*x))^3*(d - c^2*d*x^2),x)
Output:
int((a + b*asin(c*x))^3*(d - c^2*d*x^2), x)
\[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^3 \, dx=\frac {d \left (3 \mathit {asin} \left (c x \right )^{3} b^{3} c x +9 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} b^{3}+9 \mathit {asin} \left (c x \right )^{2} a \,b^{2} c x +18 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a \,b^{2}-3 \mathit {asin} \left (c x \right ) a^{2} b \,c^{3} x^{3}+9 \mathit {asin} \left (c x \right ) a^{2} b c x -18 \mathit {asin} \left (c x \right ) b^{3} c x -\sqrt {-c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}+7 \sqrt {-c^{2} x^{2}+1}\, a^{2} b -18 \sqrt {-c^{2} x^{2}+1}\, b^{3}-3 \left (\int \mathit {asin} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}-9 \left (\int \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}-a^{3} c^{3} x^{3}+3 a^{3} c x -18 a \,b^{2} c x \right )}{3 c} \] Input:
int((-c^2*d*x^2+d)*(a+b*asin(c*x))^3,x)
Output:
(d*(3*asin(c*x)**3*b**3*c*x + 9*sqrt( - c**2*x**2 + 1)*asin(c*x)**2*b**3 + 9*asin(c*x)**2*a*b**2*c*x + 18*sqrt( - c**2*x**2 + 1)*asin(c*x)*a*b**2 - 3*asin(c*x)*a**2*b*c**3*x**3 + 9*asin(c*x)*a**2*b*c*x - 18*asin(c*x)*b**3* c*x - sqrt( - c**2*x**2 + 1)*a**2*b*c**2*x**2 + 7*sqrt( - c**2*x**2 + 1)*a **2*b - 18*sqrt( - c**2*x**2 + 1)*b**3 - 3*int(asin(c*x)**3*x**2,x)*b**3*c **3 - 9*int(asin(c*x)**2*x**2,x)*a*b**2*c**3 - a**3*c**3*x**3 + 3*a**3*c*x - 18*a*b**2*c*x))/(3*c)