Integrand size = 24, antiderivative size = 226 \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}} \] Output:
3/8*a^2*x*(a^2-x^2)^(1/2)*arccos(x/a)^(1/2)+1/4*x*(a^2-x^2)^(3/2)*arccos(x /a)^(1/2)+1/4*a^3*(a^2-x^2)^(1/2)*arccos(x/a)^(3/2)/(1-x^2/a^2)^(1/2)-1/12 8*a^3*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2)*FresnelS(2*2^(1/2)/Pi^(1/2)*arccos( x/a)^(1/2))/(1-x^2/a^2)^(1/2)-1/8*a^3*Pi^(1/2)*(a^2-x^2)^(1/2)*FresnelS(2* arccos(x/a)^(1/2)/Pi^(1/2))/(1-x^2/a^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73 \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=-\frac {a^3 \sqrt {a^2-x^2} \left (32 \arccos \left (\frac {x}{a}\right )^2+16 \sqrt {\pi } \sqrt {\arccos \left (\frac {x}{a}\right )} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+\sqrt {-i \arccos \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-4 i \arccos \left (\frac {x}{a}\right )\right )+\sqrt {i \arccos \left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},4 i \arccos \left (\frac {x}{a}\right )\right )-32 \arccos \left (\frac {x}{a}\right ) \sin \left (2 \arccos \left (\frac {x}{a}\right )\right )\right )}{128 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\arccos \left (\frac {x}{a}\right )}} \] Input:
Integrate[(a^2 - x^2)^(3/2)*Sqrt[ArcCos[x/a]],x]
Output:
-1/128*(a^3*Sqrt[a^2 - x^2]*(32*ArcCos[x/a]^2 + 16*Sqrt[Pi]*Sqrt[ArcCos[x/ a]]*FresnelS[(2*Sqrt[ArcCos[x/a]])/Sqrt[Pi]] + Sqrt[(-I)*ArcCos[x/a]]*Gamm a[3/2, (-4*I)*ArcCos[x/a]] + Sqrt[I*ArcCos[x/a]]*Gamma[3/2, (4*I)*ArcCos[x /a]] - 32*ArcCos[x/a]*Sin[2*ArcCos[x/a]]))/(Sqrt[1 - x^2/a^2]*Sqrt[ArcCos[ x/a]])
Time = 1.66 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5159, 27, 5157, 5147, 4906, 27, 3042, 3786, 3832, 5153, 5225, 4906, 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.
\(\displaystyle \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx\) |
\(\Big \downarrow \) 5159 |
\(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}dx+\frac {a \sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{a^2 \sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}dx+\frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 5157 |
\(\displaystyle \frac {3}{4} a^2 \left (\frac {\sqrt {a^2-x^2} \int \frac {x}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle \frac {3}{4} a^2 \left (-\frac {a \sqrt {a^2-x^2} \int \frac {x \sqrt {1-\frac {x^2}{a^2}}}{a \sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{2 \sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 3786 |
\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \sin \left (2 \arccos \left (\frac {x}{a}\right )\right )d\sqrt {\arccos \left (\frac {x}{a}\right )}}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {3}{4} a^2 \left (\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {x \left (a^2-x^2\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (-\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle -\frac {a^3 \sqrt {a^2-x^2} \int \frac {x \left (1-\frac {x^2}{a^2}\right )^{3/2}}{a \sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (-\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {a^3 \sqrt {a^2-x^2} \int \left (\frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{4 \sqrt {\arccos \left (\frac {x}{a}\right )}}-\frac {\sin \left (4 \arccos \left (\frac {x}{a}\right )\right )}{8 \sqrt {\arccos \left (\frac {x}{a}\right )}}\right )d\arccos \left (\frac {x}{a}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{4} a^2 \left (-\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{4} a^2 \left (-\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \left (\frac {1}{4} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos \left (\frac {x}{a}\right )}\right )\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}\) |
Input:
Int[(a^2 - x^2)^(3/2)*Sqrt[ArcCos[x/a]],x]
Output:
(x*(a^2 - x^2)^(3/2)*Sqrt[ArcCos[x/a]])/4 - (a^3*Sqrt[a^2 - x^2]*(-1/8*(Sq rt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[x/a]]]) + (Sqrt[Pi]*FresnelS[(2 *Sqrt[ArcCos[x/a]])/Sqrt[Pi]])/4))/(8*Sqrt[1 - x^2/a^2]) + (3*a^2*((x*Sqrt [a^2 - x^2]*Sqrt[ArcCos[x/a]])/2 - (a*Sqrt[a^2 - x^2]*ArcCos[x/a]^(3/2))/( 3*Sqrt[1 - x^2/a^2]) - (a*Sqrt[Pi]*Sqrt[a^2 - x^2]*FresnelS[(2*Sqrt[ArcCos [x/a]])/Sqrt[Pi]])/(8*Sqrt[1 - x^2/a^2])))/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (b*c^(m + 1))^(-1) Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x , a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] ]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCos[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcCos[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*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S imp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[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*ArcCos[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_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[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]
\[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \sqrt {\arccos \left (\frac {x}{a}\right )}d x\]
Input:
int((a^2-x^2)^(3/2)*arccos(x/a)^(1/2),x)
Output:
int((a^2-x^2)^(3/2)*arccos(x/a)^(1/2),x)
Exception generated. \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2-x^2)^(3/2)*arccos(x/a)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int \left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {acos}{\left (\frac {x}{a} \right )}}\, dx \] Input:
integrate((a**2-x**2)**(3/2)*acos(x/a)**(1/2),x)
Output:
Integral((-(-a + x)*(a + x))**(3/2)*sqrt(acos(x/a)), x)
Exception generated. \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a^2-x^2)^(3/2)*arccos(x/a)^(1/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int { {\left (a^{2} - x^{2}\right )}^{\frac {3}{2}} \sqrt {\arccos \left (\frac {x}{a}\right )} \,d x } \] Input:
integrate((a^2-x^2)^(3/2)*arccos(x/a)^(1/2),x, algorithm="giac")
Output:
integrate((a^2 - x^2)^(3/2)*sqrt(arccos(x/a)), x)
Timed out. \[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int \sqrt {\mathrm {acos}\left (\frac {x}{a}\right )}\,{\left (a^2-x^2\right )}^{3/2} \,d x \] Input:
int(acos(x/a)^(1/2)*(a^2 - x^2)^(3/2),x)
Output:
int(acos(x/a)^(1/2)*(a^2 - x^2)^(3/2), x)
\[ \int \left (a^2-x^2\right )^{3/2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=-\left (\int \sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acos} \left (\frac {x}{a}\right )}\, x^{2}d x \right )+\left (\int \sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acos} \left (\frac {x}{a}\right )}d x \right ) a^{2} \] Input:
int((a^2-x^2)^(3/2)*acos(x/a)^(1/2),x)
Output:
- int(sqrt(a**2 - x**2)*sqrt(acos(x/a))*x**2,x) + int(sqrt(a**2 - x**2)*s qrt(acos(x/a)),x)*a**2