Integrand size = 18, antiderivative size = 267 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\frac {3 b \sqrt {2 d x^2-d^2 x^4} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}+\frac {3 (-b)^{3/2} \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}+\frac {3 (-b)^{3/2} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )} \]
x*(a+b*arcsin(d*x^2-1))^(3/2)+3*(-b)^(3/2)*x*FresnelS((a+b*arcsin(d*x^2-1) )^(1/2)/(-b)^(1/2)/Pi^(1/2))*(cos(1/2*a/b)-sin(1/2*a/b))*Pi^(1/2)/(cos(1/2 *arcsin(d*x^2-1))+sin(1/2*arcsin(d*x^2-1)))+3*(-b)^(3/2)*x*FresnelC((a+b*a rcsin(d*x^2-1))^(1/2)/(-b)^(1/2)/Pi^(1/2))*(cos(1/2*a/b)+sin(1/2*a/b))*Pi^ (1/2)/(cos(1/2*arcsin(d*x^2-1))+sin(1/2*arcsin(d*x^2-1)))+3*b*(-d^2*x^4+2* d*x^2)^(1/2)*(a+b*arcsin(d*x^2-1))^(1/2)/d/x
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.99 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\frac {3 b \sqrt {-d x^2 \left (-2+d x^2\right )} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}+\frac {3 (-b)^{3/2} \sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )}+\frac {3 (-b)^{3/2} \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )} \]
(3*b*Sqrt[-(d*x^2*(-2 + d*x^2))]*Sqrt[a - b*ArcSin[1 - d*x^2]])/(d*x) + x* (a - b*ArcSin[1 - d*x^2])^(3/2) + (3*(-b)^(3/2)*Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]) )/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]) + (3*(-b)^(3/2)*Sq rt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[ a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2 ]/2])
Time = 0.38 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5313, 5318}
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.
\(\displaystyle \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 5313 |
\(\displaystyle -3 b^2 \int \frac {1}{\sqrt {a-b \arcsin \left (1-d x^2\right )}}dx+\frac {3 b \sqrt {2 d x^2-d^2 x^4} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}\) |
\(\Big \downarrow \) 5318 |
\(\displaystyle -3 b^2 \left (-\frac {\sqrt {\pi } x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right )}{\sqrt {-b} \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}-\frac {\sqrt {\pi } x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {a-b \arcsin \left (1-d x^2\right )}}{\sqrt {-b} \sqrt {\pi }}\right )}{\sqrt {-b} \left (\cos \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-d x^2\right )\right )\right )}\right )+\frac {3 b \sqrt {2 d x^2-d^2 x^4} \sqrt {a-b \arcsin \left (1-d x^2\right )}}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2}\) |
(3*b*Sqrt[2*d*x^2 - d^2*x^4]*Sqrt[a - b*ArcSin[1 - d*x^2]])/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^(3/2) - 3*b^2*(-((Sqrt[Pi]*x*FresnelS[Sqrt[a - b*Arc Sin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[ -b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))) - (Sqrt[Pi]*x* FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^ 2]/2])))
3.5.25.3.1 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[1/Sqrt[(a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[(- Sqrt[Pi])*x*(Cos[a/(2*b)] - c*Sin[a/(2*b)])*(FresnelC[(1/(Sqrt[b*c]*Sqrt[Pi ]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/(Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c *Sin[ArcSin[c + d*x^2]/2]))), x] - Simp[Sqrt[Pi]*x*(Cos[a/(2*b)] + c*Sin[a/ (2*b)])*(FresnelS[(1/(Sqrt[b*c]*Sqrt[Pi]))*Sqrt[a + b*ArcSin[c + d*x^2]]]/( Sqrt[b*c]*(Cos[ArcSin[c + d*x^2]/2] - c*Sin[ArcSin[c + d*x^2]/2]))), x] /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1]
\[\int {\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )}^{\frac {3}{2}}d x\]
Exception generated. \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\int { {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\int { {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^{3/2} \, dx=\int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^{3/2} \,d x \]