Integrand size = 16, antiderivative size = 135 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=384 b^4 x-\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}-48 b^2 x \left (a-b \arcsin \left (1-d x^2\right )\right )^2+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \]
384*b^4*x-48*b^2*x*(a+b*arcsin(d*x^2-1))^2+x*(a+b*arcsin(d*x^2-1))^4-192*b ^3*(a+b*arcsin(d*x^2-1))*(-d^2*x^4+2*d*x^2)^(1/2)/d/x+8*b*(a+b*arcsin(d*x^ 2-1))^3*(-d^2*x^4+2*d*x^2)^(1/2)/d/x
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.97 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=\frac {8 b \sqrt {-d x^2 \left (-2+d x^2\right )} \left (a-b \arcsin \left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^4-48 b^2 \left (-8 b^2 x+\frac {4 b \sqrt {-d x^2 \left (-2+d x^2\right )} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2\right ) \]
(8*b*Sqrt[-(d*x^2*(-2 + d*x^2))]*(a - b*ArcSin[1 - d*x^2])^3)/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^4 - 48*b^2*(-8*b^2*x + (4*b*Sqrt[-(d*x^2*(-2 + d*x ^2))]*(a - b*ArcSin[1 - d*x^2]))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^2)
Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5313, 5313, 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.
| \(\displaystyle \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx\) |
| \(\Big \downarrow \) 5313 |
| \(\displaystyle -48 b^2 \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2dx+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^4\) |
| \(\Big \downarrow \) 5313 |
| \(\displaystyle -48 b^2 \left (-8 b^2 \int 1dx+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2\right )+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^4\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -48 b^2 \left (\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2-8 b^2 x\right )+\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )^3}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^4\) |
(8*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2])^3)/(d*x) + x*(a - b *ArcSin[1 - d*x^2])^4 - 48*b^2*(-8*b^2*x + (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2]))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^2)
3.5.8.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*( a + b*ArcSin[c + d*x^2])^n, x] + (Simp[2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*((a + b*ArcSin[c + d*x^2])^(n - 1)/(d*x)), x] - Simp[4*b^2*n*(n - 1) Int[(a + b*ArcSin[c + d*x^2])^(n - 2), x], x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^ 2, 1] && GtQ[n, 1]
\[\int {\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )}^{4}d x\]
Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.53 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \arcsin \left (d x^{2} - 1\right )^{4} + 4 \, a b^{3} d x^{2} \arcsin \left (d x^{2} - 1\right )^{3} + 6 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 4 \, {\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arcsin \left (d x^{2} - 1\right ) + {\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 8 \, {\left (b^{4} \arcsin \left (d x^{2} - 1\right )^{3} + 3 \, a b^{3} \arcsin \left (d x^{2} - 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arcsin \left (d x^{2} - 1\right )\right )} \sqrt {-d^{2} x^{4} + 2 \, d x^{2}}}{d x} \]
(b^4*d*x^2*arcsin(d*x^2 - 1)^4 + 4*a*b^3*d*x^2*arcsin(d*x^2 - 1)^3 + 6*(a^ 2*b^2 - 8*b^4)*d*x^2*arcsin(d*x^2 - 1)^2 + 4*(a^3*b - 24*a*b^3)*d*x^2*arcs in(d*x^2 - 1) + (a^4 - 48*a^2*b^2 + 384*b^4)*d*x^2 + 8*(b^4*arcsin(d*x^2 - 1)^3 + 3*a*b^3*arcsin(d*x^2 - 1)^2 + a^3*b - 24*a*b^3 + 3*(a^2*b^2 - 8*b^ 4)*arcsin(d*x^2 - 1))*sqrt(-d^2*x^4 + 2*d*x^2))/(d*x)
\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \]
\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=\int { {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{4} \,d x } \]
b^4*x*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)^4 + 4*(x*arcsin(d*x^2 - 1) - 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(-d*x^2 + 2)*d))*a^3*b + a^4*x + integrate(2*(4*sqrt(-d*x^2 + 2)*b^4*sqrt(d)*x*arctan2(d*x^2 - 1, sqrt(-d*x ^2 + 2)*sqrt(d)*x)^3 + 2*(a*b^3*d*x^2 - 2*a*b^3)*arctan2(d*x^2 - 1, sqrt(- d*x^2 + 2)*sqrt(d)*x)^3 + 3*(a^2*b^2*d*x^2 - 2*a^2*b^2)*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)^2)/(d*x^2 - 2), x)
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (123) = 246\).
Time = 1.01 (sec) , antiderivative size = 617, normalized size of antiderivative = 4.57 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=4 \, {\left (x \arcsin \left (d x^{2} - 1\right ) - \frac {2 \, \sqrt {2} \mathrm {sgn}\left (x\right )}{\sqrt {d}} + \frac {2 \, \sqrt {-d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\left (x\right )}\right )} a^{3} b + 6 \, {\left (x \arcsin \left (d x^{2} - 1\right )^{2} + \frac {2 \, {\left (\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 4 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d {\left | d \right |}} + \frac {4 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (d x^{2} - 1\right ) + \frac {2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} a^{2} b^{2} + 2 \, {\left (2 \, x \arcsin \left (d x^{2} - 1\right )^{3} - \frac {3 \, {\left (\sqrt {2} \pi ^{2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \pi \sqrt {d} {\left | d \right |} - 32 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d^{2}} + \frac {12 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (d x^{2} - 1\right )^{2} - \frac {2 \, {\left (2 \, \sqrt {d^{2} x^{2}} \arcsin \left (\frac {d^{2} x^{2} - d}{d}\right ) - \frac {4 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {-d^{2} x^{2} + 2 \, d}\right )} d}{{\left | d \right |}} - \frac {\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 4 \, \sqrt {2} d^{\frac {3}{2}}}{{\left | d \right |}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} a b^{3} + {\left (x \arcsin \left (d x^{2} - 1\right )^{4} + \frac {{\left (\sqrt {2} \pi ^{3} \sqrt {d} {\left | d \right |} - 12 \, \sqrt {2} \pi ^{2} d^{\frac {3}{2}} - 96 \, \sqrt {2} \pi \sqrt {d} {\left | d \right |} + 384 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d {\left | d \right |}} + \frac {4 \, {\left (2 \, \sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (d x^{2} - 1\right )^{3} - \frac {3 \, {\left (4 \, \sqrt {d^{2} x^{2}} \arcsin \left (\frac {d^{2} x^{2} - d}{d}\right )^{2} + \frac {8 \, {\left (2 \, \sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (\frac {d^{2} x^{2} - d}{d}\right ) + \frac {4 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}} + \frac {\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 4 \, \sqrt {2} d^{\frac {3}{2}}}{{\left | d \right |}}\right )} d}{{\left | d \right |}} - \frac {\sqrt {2} \pi ^{2} d^{\frac {3}{2}} + 8 \, \sqrt {2} \pi \sqrt {d} {\left | d \right |} - 32 \, \sqrt {2} d^{\frac {3}{2}}}{d}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} b^{4} + a^{4} x \]
4*(x*arcsin(d*x^2 - 1) - 2*sqrt(2)*sgn(x)/sqrt(d) + 2*sqrt(-d^2*x^2 + 2*d) /(d*sgn(x)))*a^3*b + 6*(x*arcsin(d*x^2 - 1)^2 + 2*(sqrt(2)*pi*sqrt(d)*abs( d) - 4*sqrt(2)*d^(3/2))*sgn(x)/(d*abs(d)) + 4*(sqrt(-d^2*x^2 + 2*d)*arcsin (d*x^2 - 1) + 2*(sqrt(2)*sqrt(d) - sqrt(d^2*x^2))*d/abs(d))/(d*sgn(x)))*a^ 2*b^2 + 2*(2*x*arcsin(d*x^2 - 1)^3 - 3*(sqrt(2)*pi^2*d^(3/2) + 8*sqrt(2)*p i*sqrt(d)*abs(d) - 32*sqrt(2)*d^(3/2))*sgn(x)/d^2 + 12*(sqrt(-d^2*x^2 + 2* d)*arcsin(d*x^2 - 1)^2 - 2*(2*sqrt(d^2*x^2)*arcsin((d^2*x^2 - d)/d) - 4*(s qrt(2)*sqrt(d) - sqrt(-d^2*x^2 + 2*d))*d/abs(d) - (sqrt(2)*pi*sqrt(d)*abs( d) - 4*sqrt(2)*d^(3/2))/abs(d))*d/abs(d))/(d*sgn(x)))*a*b^3 + (x*arcsin(d* x^2 - 1)^4 + (sqrt(2)*pi^3*sqrt(d)*abs(d) - 12*sqrt(2)*pi^2*d^(3/2) - 96*s qrt(2)*pi*sqrt(d)*abs(d) + 384*sqrt(2)*d^(3/2))*sgn(x)/(d*abs(d)) + 4*(2*s qrt(-d^2*x^2 + 2*d)*arcsin(d*x^2 - 1)^3 - 3*(4*sqrt(d^2*x^2)*arcsin((d^2*x ^2 - d)/d)^2 + 8*(2*sqrt(-d^2*x^2 + 2*d)*arcsin((d^2*x^2 - d)/d) + 4*(sqrt (2)*sqrt(d) - sqrt(d^2*x^2))*d/abs(d) + (sqrt(2)*pi*sqrt(d)*abs(d) - 4*sqr t(2)*d^(3/2))/abs(d))*d/abs(d) - (sqrt(2)*pi^2*d^(3/2) + 8*sqrt(2)*pi*sqrt (d)*abs(d) - 32*sqrt(2)*d^(3/2))/d)*d/abs(d))/(d*sgn(x)))*b^4 + a^4*x
Timed out. \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^4 \,d x \]