\(\int \sqrt {a^2-x^2} \sqrt {\arccos (\frac {x}{a})} \, dx\) [458]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [F]
Fricas [F(-2)]
Sympy [F]
Maxima [F(-2)]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}+\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {\pi } \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}} \] Output:

1/2*x*(a^2-x^2)^(1/2)*arccos(x/a)^(1/2)+1/3*a*(a^2-x^2)^(1/2)*arccos(x/a)^ 
(3/2)/(1-x^2/a^2)^(1/2)-1/8*a*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)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\frac {a \sqrt {a^2-x^2} \left (-3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )+2 \sqrt {\arccos \left (\frac {x}{a}\right )} \left (-4 \arccos \left (\frac {x}{a}\right )+3 \sin \left (2 \arccos \left (\frac {x}{a}\right )\right )\right )\right )}{24 \sqrt {1-\frac {x^2}{a^2}}} \] Input:

Integrate[Sqrt[a^2 - x^2]*Sqrt[ArcCos[x/a]],x]
 

Output:

(a*Sqrt[a^2 - x^2]*(-3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[x/a]])/Sqrt[Pi]] + 
 2*Sqrt[ArcCos[x/a]]*(-4*ArcCos[x/a] + 3*Sin[2*ArcCos[x/a]])))/(24*Sqrt[1 
- x^2/a^2])
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5157, 5147, 4906, 27, 3042, 3786, 3832, 5153}

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.

\(\displaystyle \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx\)

\(\Big \downarrow \) 5157

\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {x}{\sqrt {\arccos \left (\frac {x}{a}\right )}}dx}{4 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 5147

\(\displaystyle -\frac {a \sqrt {a^2-x^2} \int \frac {x \sqrt {1-\frac {x^2}{a^2}}}{a \sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{2 \sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \frac {\sin \left (2 \arccos \left (\frac {x}{a}\right )\right )}{\sqrt {\arccos \left (\frac {x}{a}\right )}}d\arccos \left (\frac {x}{a}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \int \sin \left (2 \arccos \left (\frac {x}{a}\right )\right )d\sqrt {\arccos \left (\frac {x}{a}\right )}}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {\sqrt {a^2-x^2} \int \frac {\sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}}dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

\(\Big \downarrow \) 5153

\(\displaystyle -\frac {\sqrt {\pi } a \sqrt {a^2-x^2} \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a \sqrt {a^2-x^2} \arccos \left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )}\)

Input:

Int[Sqrt[a^2 - x^2]*Sqrt[ArcCos[x/a]],x]
 

Output:

(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])
 

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3786
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]
 

rule 3832
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]
 

rule 4906
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]
 

rule 5147
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]
 

rule 5153
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]
 

rule 5157
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]
 
Maple [F]

\[\int \sqrt {a^{2}-x^{2}}\, \sqrt {\arccos \left (\frac {x}{a}\right )}d x\]

Input:

int((a^2-x^2)^(1/2)*arccos(x/a)^(1/2),x)
 

Output:

int((a^2-x^2)^(1/2)*arccos(x/a)^(1/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a^2-x^2)^(1/2)*arccos(x/a)^(1/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \sqrt {\operatorname {acos}{\left (\frac {x}{a} \right )}}\, dx \] Input:

integrate((a**2-x**2)**(1/2)*acos(x/a)**(1/2),x)
 

Output:

Integral(sqrt(-(-a + x)*(a + x))*sqrt(acos(x/a)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a^2-x^2)^(1/2)*arccos(x/a)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int { \sqrt {a^{2} - x^{2}} \sqrt {\arccos \left (\frac {x}{a}\right )} \,d x } \] Input:

integrate((a^2-x^2)^(1/2)*arccos(x/a)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(a^2 - x^2)*sqrt(arccos(x/a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int \sqrt {\mathrm {acos}\left (\frac {x}{a}\right )}\,\sqrt {a^2-x^2} \,d x \] Input:

int(acos(x/a)^(1/2)*(a^2 - x^2)^(1/2),x)
 

Output:

int(acos(x/a)^(1/2)*(a^2 - x^2)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a^2-x^2} \sqrt {\arccos \left (\frac {x}{a}\right )} \, dx=\int \sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acos} \left (\frac {x}{a}\right )}d x \] Input:

int((a^2-x^2)^(1/2)*acos(x/a)^(1/2),x)
 

Output:

int(sqrt(a**2 - x**2)*sqrt(acos(x/a)),x)