\(\int \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2 \, dx\) [45]

Optimal result

Integrand size = 26, antiderivative size = 124 \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2 \, dx=-\frac {1}{4} b^2 \sqrt {\pi } x \sqrt {1-c^2 x^2}+\frac {1}{2} b c \sqrt {\pi } x^2 (a+b \arccos (c x))+\frac {1}{2} x \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2-\frac {\sqrt {\pi } (a+b \arccos (c x))^3}{6 b c}+\frac {b^2 \sqrt {\pi } \arcsin (c x)}{4 c} \] Output:

-1/4*b^2*Pi^(1/2)*x*(-c^2*x^2+1)^(1/2)+1/2*b*c*Pi^(1/2)*x^2*(a+b*arccos(c* 
x))+1/2*x*(-Pi*c^2*x^2+Pi)^(1/2)*(a+b*arccos(c*x))^2-1/6*Pi^(1/2)*(a+b*arc 
cos(c*x))^3/b/c+1/4*b^2*Pi^(1/2)*arcsin(c*x)/c
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04 \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {\pi } \left (-4 b^2 \arccos (c x)^3+6 b \arccos (c x) (b \cos (2 \arccos (c x))+2 a \sin (2 \arccos (c x)))+6 b \arccos (c x)^2 (-2 a+b \sin (2 \arccos (c x)))+3 \left (4 a^2 c x \sqrt {1-c^2 x^2}+4 a^2 \arcsin (c x)+2 a b \cos (2 \arccos (c x))-b^2 \sin (2 \arccos (c x))\right )\right )}{24 c} \] Input:

Integrate[Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcCos[c*x])^2,x]
 

Output:

(Sqrt[Pi]*(-4*b^2*ArcCos[c*x]^3 + 6*b*ArcCos[c*x]*(b*Cos[2*ArcCos[c*x]] + 
2*a*Sin[2*ArcCos[c*x]]) + 6*b*ArcCos[c*x]^2*(-2*a + b*Sin[2*ArcCos[c*x]]) 
+ 3*(4*a^2*c*x*Sqrt[1 - c^2*x^2] + 4*a^2*ArcSin[c*x] + 2*a*b*Cos[2*ArcCos[ 
c*x]] - b^2*Sin[2*ArcCos[c*x]])))/(24*c)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5157, 5139, 262, 223, 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.

\(\displaystyle \int \sqrt {\pi -\pi c^2 x^2} (a+b \arccos (c x))^2 \, dx\)

\(\Big \downarrow \) 5157

\(\displaystyle \frac {1}{2} \sqrt {\pi } \int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx+\sqrt {\pi } b c \int x (a+b \arccos (c x))dx+\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{2} \sqrt {\pi } \int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx+\sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5153

\(\displaystyle \sqrt {\pi } b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {\pi -\pi c^2 x^2} (a+b \arccos (c x))^2-\frac {\sqrt {\pi } (a+b \arccos (c x))^3}{6 b c}\)

Input:

Int[Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcCos[c*x])^2,x]
 

Output:

(x*Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcCos[c*x])^2)/2 - (Sqrt[Pi]*(a + b*ArcCo 
s[c*x])^3)/(6*b*c) + b*c*Sqrt[Pi]*((x^2*(a + b*ArcCos[c*x]))/2 + (b*c*(-1/ 
2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)
 

Defintions of rubi rules used

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5139

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^n/(d*(m + 1))), x] + Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCos[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

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 [A] (verified)

Time = 0.38 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {a^{2} x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a^{2} \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}-\frac {b^{2} \sqrt {\pi }\, \left (-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} x c -6 c^{2} x^{2} \arccos \left (c x \right )+2 \arccos \left (c x \right )^{3}+3 c x \sqrt {-c^{2} x^{2}+1}+3 \arccos \left (c x \right )\right )}{12 c}-\frac {a b \sqrt {\pi }\, \left (-2 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -c^{2} x^{2}+\arccos \left (c x \right )^{2}+1\right )}{2 c}\)	\(179\)
parts	\(\frac {a^{2} x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{2}+\frac {a^{2} \pi \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi \,c^{2}}}-\frac {b^{2} \sqrt {\pi }\, \left (-6 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} x c -6 c^{2} x^{2} \arccos \left (c x \right )+2 \arccos \left (c x \right )^{3}+3 c x \sqrt {-c^{2} x^{2}+1}+3 \arccos \left (c x \right )\right )}{12 c}-\frac {a b \sqrt {\pi }\, \left (-2 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -c^{2} x^{2}+\arccos \left (c x \right )^{2}+1\right )}{2 c}\)	\(179\)

Input:

int((-Pi*c^2*x^2+Pi)^(1/2)*(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*a^2*x*(-Pi*c^2*x^2+Pi)^(1/2)+1/2*a^2*Pi/(Pi*c^2)^(1/2)*arctan((Pi*c^2) 
^(1/2)*x/(-Pi*c^2*x^2+Pi)^(1/2))-1/12*b^2*Pi^(1/2)*(-6*(-c^2*x^2+1)^(1/2)* 
arccos(c*x)^2*x*c-6*c^2*x^2*arccos(c*x)+2*arccos(c*x)^3+3*c*x*(-c^2*x^2+1) 
^(1/2)+3*arccos(c*x))/c-1/2*a*b*Pi^(1/2)*(-2*(-c^2*x^2+1)^(1/2)*arccos(c*x 
)*x*c-c^2*x^2+arccos(c*x)^2+1)/c
 

Fricas [F]

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

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

Output:

integral(sqrt(pi - pi*c^2*x^2)*(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^ 
2), x)
 

Sympy [F]

\[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2 \, dx=\sqrt {\pi } \left (\int a^{2} \sqrt {- c^{2} x^{2} + 1}\, dx + \int b^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}\, dx + \int 2 a b \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}\, dx\right ) \] Input:

integrate((-pi*c**2*x**2+pi)**(1/2)*(a+b*acos(c*x))**2,x)
 

Output:

sqrt(pi)*(Integral(a**2*sqrt(-c**2*x**2 + 1), x) + Integral(b**2*sqrt(-c** 
2*x**2 + 1)*acos(c*x)**2, x) + Integral(2*a*b*sqrt(-c**2*x**2 + 1)*acos(c* 
x), x))
 

Maxima [F]

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

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

Output:

1/2*(sqrt(pi - pi*c^2*x^2)*x + sqrt(pi)*arcsin(c*x)/c)*a^2 + sqrt(pi)*inte 
grate((b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*arctan2(sq 
rt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-pi*c^2*x^2+pi)^(1/2)*(a+b*arccos(c*x))^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
 

Mupad [F(-1)]

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

int((a + b*acos(c*x))^2*(Pi - Pi*c^2*x^2)^(1/2),x)
 

Output:

int((a + b*acos(c*x))^2*(Pi - Pi*c^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {\pi }\, \left (\mathit {asin} \left (c x \right ) a^{2}+\sqrt {-c^{2} x^{2}+1}\, a^{2} c x +4 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) a b c +2 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{2 c} \] Input:

int((-Pi*c^2*x^2+Pi)^(1/2)*(a+b*acos(c*x))^2,x)
 

Output:

(sqrt(pi)*(asin(c*x)*a**2 + sqrt( - c**2*x**2 + 1)*a**2*c*x + 4*int(sqrt( 
- c**2*x**2 + 1)*acos(c*x),x)*a*b*c + 2*int(sqrt( - c**2*x**2 + 1)*acos(c* 
x)**2,x)*b**2*c))/(2*c)