Integrand size = 24, antiderivative size = 215 \[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\frac {3 a \sqrt {a^2-x^2} \sqrt {\arcsin \left (\frac {x}{a}\right )}}{16 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\arcsin \left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a \sqrt {\pi } \sqrt {a^2-x^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}} \]
1/2*x*arcsin(x/a)^(3/2)*(a^2-x^2)^(1/2)+1/5*a*arcsin(x/a)^(5/2)*(a^2-x^2)^ (1/2)/(1-x^2/a^2)^(1/2)-3/32*a*FresnelC(2*arcsin(x/a)^(1/2)/Pi^(1/2))*Pi^( 1/2)*(a^2-x^2)^(1/2)/(1-x^2/a^2)^(1/2)+3/16*a*(a^2-x^2)^(1/2)*arcsin(x/a)^ (1/2)/(1-x^2/a^2)^(1/2)-3/8*x^2*(a^2-x^2)^(1/2)*arcsin(x/a)^(1/2)/a/(1-x^2 /a^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72 \[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\frac {\sqrt {a^2-x^2} \left (160 x \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^2+64 a \arcsin \left (\frac {x}{a}\right )^3+15 i \sqrt {2} a \sqrt {-i \arcsin \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 i \arcsin \left (\frac {x}{a}\right )\right )-15 i \sqrt {2} a \sqrt {i \arcsin \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 i \arcsin \left (\frac {x}{a}\right )\right )\right )}{320 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}} \]
(Sqrt[a^2 - x^2]*(160*x*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^2 + 64*a*ArcSin[x/a] ^3 + (15*I)*Sqrt[2]*a*Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (-2*I)*ArcSin[x/a] ] - (15*I)*Sqrt[2]*a*Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (2*I)*ArcSin[x/a]]))/( 320*Sqrt[1 - x^2/a^2]*Sqrt[ArcSin[x/a]])
Time = 0.89 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5156, 5140, 5152, 5224, 3042, 3793, 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.
\(\displaystyle \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 5156 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \int x \sqrt {\arcsin \left (\frac {x}{a}\right )}dx}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {\sqrt {a^2-x^2} \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
\(\Big \downarrow \) 5140 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {\sqrt {a^2-x^2} \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {x^2}{a^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}}d\arcsin \left (\frac {x}{a}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {\sin \left (\arcsin \left (\frac {x}{a}\right )\right )^2}{\sqrt {\arcsin \left (\frac {x}{a}\right )}}d\arcsin \left (\frac {x}{a}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \left (\frac {1}{2 \sqrt {\arcsin \left (\frac {x}{a}\right )}}-\frac {\cos \left (2 \arcsin \left (\frac {x}{a}\right )\right )}{2 \sqrt {\arcsin \left (\frac {x}{a}\right )}}\right )d\arcsin \left (\frac {x}{a}\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \left (\sqrt {\arcsin \left (\frac {x}{a}\right )}-\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arcsin \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )\right )\right )}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2}\) |
(x*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2))/2 + (a*Sqrt[a^2 - x^2]*ArcSin[x/a]^( 5/2))/(5*Sqrt[1 - x^2/a^2]) - (3*Sqrt[a^2 - x^2]*((x^2*Sqrt[ArcSin[x/a]])/ 2 - (a^2*(Sqrt[ArcSin[x/a]] - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[x/a]])/Sqr t[Pi]])/2))/4))/(4*a*Sqrt[1 - x^2/a^2])
3.5.61.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \sqrt {a^{2}-x^{2}}\, \arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}\, dx \]
Exception generated. \[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\int { \sqrt {a^{2} - x^{2}} \arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sqrt {a^2-x^2} \arcsin \left (\frac {x}{a}\right )^{3/2} \, dx=\int {\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}\,\sqrt {a^2-x^2} \,d x \]