Integrand size = 24, antiderivative size = 219 \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\frac {3 \sqrt {c-a^2 c x^2} \sqrt {\arccos (a x)}}{16 a \sqrt {1-a^2 x^2}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \sqrt {\arccos (a x)}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2}+\frac {\sqrt {c-a^2 c x^2} \arccos (a x)^{5/2}}{5 a \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {\pi } \sqrt {c-a^2 c x^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}} \] Output:
3/16*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(1/2)/a/(-a^2*x^2+1)^(1/2)-3/8*a*x^2 *(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(1/2)/(-a^2*x^2+1)^(1/2)+1/2*x*(-a^2*c*x ^2+c)^(1/2)*arccos(a*x)^(3/2)+1/5*(-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(5/2)/a /(-a^2*x^2+1)^(1/2)-3/32*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)*FresnelC(2*arccos(a *x)^(1/2)/Pi^(1/2))/a/(-a^2*x^2+1)^(1/2)
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=-\frac {\sqrt {c \left (1-a^2 x^2\right )} \left (15 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )+2 \sqrt {\arccos (a x)} (-15 \cos (2 \arccos (a x))+4 \arccos (a x) (4 \arccos (a x)-5 \sin (2 \arccos (a x))))\right )}{160 a \sqrt {1-a^2 x^2}} \] Input:
Integrate[Sqrt[c - a^2*c*x^2]*ArcCos[a*x]^(3/2),x]
Output:
-1/160*(Sqrt[c*(1 - a^2*x^2)]*(15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/ Sqrt[Pi]] + 2*Sqrt[ArcCos[a*x]]*(-15*Cos[2*ArcCos[a*x]] + 4*ArcCos[a*x]*(4 *ArcCos[a*x] - 5*Sin[2*ArcCos[a*x]]))))/(a*Sqrt[1 - a^2*x^2])
Time = 0.89 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5157, 5141, 5153, 5225, 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.
\(\displaystyle \int \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 5157 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \int x \sqrt {\arccos (a x)}dx}{4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arccos (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 5141 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{4} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}dx+\frac {1}{2} x^2 \sqrt {\arccos (a x)}\right )}{4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\arccos (a x)^{3/2}}{\sqrt {1-a^2 x^2}}dx}{2 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 5153 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{4} a \int \frac {x^2}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}dx+\frac {1}{2} x^2 \sqrt {\arccos (a x)}\right )}{4 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\int \frac {a^2 x^2}{\sqrt {\arccos (a x)}}d\arccos (a x)}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\int \frac {\sin \left (\arccos (a x)+\frac {\pi }{2}\right )^2}{\sqrt {\arccos (a x)}}d\arccos (a x)}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\int \left (\frac {\cos (2 \arccos (a x))}{2 \sqrt {\arccos (a x)}}+\frac {1}{2 \sqrt {\arccos (a x)}}\right )d\arccos (a x)}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 a \sqrt {c-a^2 c x^2} \left (\frac {1}{2} x^2 \sqrt {\arccos (a x)}-\frac {\frac {1}{2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arccos (a x)}}{4 a^2}\right )}{4 \sqrt {1-a^2 x^2}}-\frac {\arccos (a x)^{5/2} \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \arccos (a x)^{3/2} \sqrt {c-a^2 c x^2}\) |
Input:
Int[Sqrt[c - a^2*c*x^2]*ArcCos[a*x]^(3/2),x]
Output:
(x*Sqrt[c - a^2*c*x^2]*ArcCos[a*x]^(3/2))/2 - (Sqrt[c - a^2*c*x^2]*ArcCos[ a*x]^(5/2))/(5*a*Sqrt[1 - a^2*x^2]) + (3*a*Sqrt[c - a^2*c*x^2]*((x^2*Sqrt[ ArcCos[a*x]])/2 - (Sqrt[ArcCos[a*x]] + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a *x]])/Sqrt[Pi]])/2)/(4*a^2)))/(4*Sqrt[1 - a^2*x^2])
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_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcCos[c*x])^n/(m + 1)), x] + Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcCos[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_.) + 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_.)*(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]
\[\int \sqrt {-a^{2} c \,x^{2}+c}\, \arccos \left (a x \right )^{\frac {3}{2}}d x\]
Input:
int((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(3/2),x)
Output:
int((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(3/2),x)
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \] Input:
integrate((-a**2*c*x**2+c)**(1/2)*acos(a*x)**(3/2),x)
Output:
Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*acos(a*x)**(3/2), x)
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a^2*c*x^2+c)^(1/2)*arccos(a*x)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\int {\mathrm {acos}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \] Input:
int(acos(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2),x)
Output:
int(acos(a*x)^(3/2)*(c - a^2*c*x^2)^(1/2), x)
\[ \int \sqrt {c-a^2 c x^2} \arccos (a x)^{3/2} \, dx=\sqrt {c}\, \left (\int \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, \mathit {acos} \left (a x \right )d x \right ) \] Input:
int((-a^2*c*x^2+c)^(1/2)*acos(a*x)^(3/2),x)
Output:
sqrt(c)*int(sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*acos(a*x),x)