Integrand size = 22, antiderivative size = 128 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {14}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3+\frac {4 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{9 c}+\frac {2}{3} d x (a+b \arcsin (c x))^2+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 \]
-14/9*b^2*d*x+2/27*b^2*c^2*d*x^3+2/9*b*d*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c* x))/c+2/3*d*x*(a+b*arcsin(c*x))^2+1/3*d*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2 +4/3*b*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {d \left (-2 b^2 c x \left (-21+c^2 x^2\right )+6 a b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+9 a^2 c x \left (-3+c^2 x^2\right )+6 b \left (b \sqrt {1-c^2 x^2} \left (-7+c^2 x^2\right )+3 a c x \left (-3+c^2 x^2\right )\right ) \arcsin (c x)+9 b^2 c x \left (-3+c^2 x^2\right ) \arcsin (c x)^2\right )}{27 c} \]
-1/27*(d*(-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) + 6*b*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2))*ArcSin[c*x] + 9*b^2*c*x*(-3 + c^2*x^2)*ArcSin[c*x ]^2))/c
Time = 0.51 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5158, 5130, 5182, 24, 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.
\(\displaystyle \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx\) |
\(\Big \downarrow \) 5158 |
\(\displaystyle -\frac {2}{3} b c d \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{3} d \int (a+b \arcsin (c x))^2dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
\(\Big \downarrow \) 5130 |
\(\displaystyle \frac {2}{3} d \left (x (a+b \arcsin (c x))^2-2 b c \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{3} b c d \int x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
\(\Big \downarrow \) 5182 |
\(\displaystyle \frac {2}{3} d \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c d \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {2}{3} b c d \left (\frac {b \int \left (1-c^2 x^2\right )dx}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )+\frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} d \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{3} d \left (x (a+b \arcsin (c x))^2-2 b c \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )\right )-\frac {2}{3} b c d \left (\frac {b \left (x-\frac {c^2 x^3}{3}\right )}{3 c}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^2}\right )\) |
(d*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/3 - (2*b*c*d*((b*(x - (c^2*x^3)/ 3))/(3*c) - ((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^2)))/3 + (2*d*( x*(a + b*ArcSin[c*x])^2 - 2*b*c*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSi n[c*x]))/c^2)))/3
3.2.60.3.1 Defintions of rubi rules used
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_.))^(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 = 0.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {-d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )-2 d a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(173\) |
default | \(\frac {-d \,a^{2} \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )-2 d a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(173\) |
parts | \(-d \,a^{2} \left (\frac {1}{3} c^{2} x^{3}-x \right )-\frac {d \,b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (c^{2} x^{2}-3\right ) c x}{3}+\frac {4 c x}{3}-\frac {4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{3}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{9}-\frac {2 \left (c^{2} x^{2}-3\right ) c x}{27}\right )}{c}-\frac {2 d a b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(174\) |
1/c*(-d*a^2*(1/3*c^3*x^3-c*x)-d*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)-2*d*a*b*(1/3*c^3*x^3*arcsin(c*x)-c*x*ar csin(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 = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.14 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} d x^{3} - 3 \, {\left (9 \, a^{2} - 14 \, b^{2}\right )} c d x + 9 \, {\left (b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (a b c^{3} d x^{3} - 3 \, a b c d x\right )} \arcsin \left (c x\right ) + 6 \, {\left (a b c^{2} d x^{2} - 7 \, a b d + {\left (b^{2} c^{2} d x^{2} - 7 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{27 \, c} \]
-1/27*((9*a^2 - 2*b^2)*c^3*d*x^3 - 3*(9*a^2 - 14*b^2)*c*d*x + 9*(b^2*c^3*d *x^3 - 3*b^2*c*d*x)*arcsin(c*x)^2 + 18*(a*b*c^3*d*x^3 - 3*a*b*c*d*x)*arcsi n(c*x) + 6*(a*b*c^2*d*x^2 - 7*a*b*d + (b^2*c^2*d*x^2 - 7*b^2*d)*arcsin(c*x ))*sqrt(-c^2*x^2 + 1))/c
Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.75 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{2} d x^{3}}{3} + a^{2} d x - \frac {2 a b c^{2} d x^{3} \operatorname {asin}{\left (c x \right )}}{3} - \frac {2 a b c d x^{2} \sqrt {- c^{2} x^{2} + 1}}{9} + 2 a b d x \operatorname {asin}{\left (c x \right )} + \frac {14 a b d \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b^{2} c^{2} d x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} c^{2} d x^{3}}{27} - \frac {2 b^{2} c d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - \frac {14 b^{2} d x}{9} + \frac {14 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} & \text {for}\: c \neq 0 \\a^{2} d x & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**2*d*x**3/3 + a**2*d*x - 2*a*b*c**2*d*x**3*asin(c*x)/3 - 2*a*b*c*d*x**2*sqrt(-c**2*x**2 + 1)/9 + 2*a*b*d*x*asin(c*x) + 14*a*b*d*s qrt(-c**2*x**2 + 1)/(9*c) - b**2*c**2*d*x**3*asin(c*x)**2/3 + 2*b**2*c**2* d*x**3/27 - 2*b**2*c*d*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/9 + b**2*d*x*as in(c*x)**2 - 14*b**2*d*x/9 + 14*b**2*d*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c ), Ne(c, 0)), (a**2*d*x, True))
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (111) = 222\).
Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.82 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {1}{3} \, b^{2} c^{2} d x^{3} \arcsin \left (c x\right )^{2} - \frac {1}{3} \, a^{2} c^{2} d x^{3} - \frac {2}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \arcsin \left (c x\right )^{2} - 2 \, b^{2} d {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d}{c} \]
-1/3*b^2*c^2*d*x^3*arcsin(c*x)^2 - 1/3*a^2*c^2*d*x^3 - 2/9*(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*b*c^2*d - 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4)*arcsi n(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*c^2*d + b^2*d*x*arcsin(c*x)^2 - 2*b^2*d* (x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d*x + 2*(c*x*arcsin(c*x) + sq rt(-c^2*x^2 + 1))*a*b*d/c
Time = 0.33 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.53 \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {1}{3} \, a^{2} c^{2} d x^{3} - \frac {1}{3} \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2} - \frac {2}{3} \, {\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right ) + \frac {2}{3} \, b^{2} d x \arcsin \left (c x\right )^{2} + \frac {2}{27} \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x + \frac {4}{3} \, a b d x \arcsin \left (c x\right ) + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d \arcsin \left (c x\right )}{9 \, c} + a^{2} d x - \frac {40}{27} \, b^{2} d x + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d}{9 \, c} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{3 \, c} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d}{3 \, c} \]
-1/3*a^2*c^2*d*x^3 - 1/3*(c^2*x^2 - 1)*b^2*d*x*arcsin(c*x)^2 - 2/3*(c^2*x^ 2 - 1)*a*b*d*x*arcsin(c*x) + 2/3*b^2*d*x*arcsin(c*x)^2 + 2/27*(c^2*x^2 - 1 )*b^2*d*x + 4/3*a*b*d*x*arcsin(c*x) + 2/9*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsi n(c*x)/c + a^2*d*x - 40/27*b^2*d*x + 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*d/c + 4/ 3*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)/c + 4/3*sqrt(-c^2*x^2 + 1)*a*b*d/c
Timed out. \[ \int \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right ) \,d x \]