\(\int \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx\) [59]

Optimal result

Integrand size = 26, antiderivative size = 192 \[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx=-\frac {1}{4} b^2 x \sqrt {d-c^2 d x^2}+\frac {b c x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {b^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{4 c \sqrt {1-c^2 x^2}} \] Output:

-1/4*b^2*x*(-c^2*d*x^2+d)^(1/2)+1/2*b*c*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
os(c*x))/(-c^2*x^2+1)^(1/2)+1/2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2 
-1/6*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^3/b/c/(-c^2*x^2+1)^(1/2)+1/4*b 
^2*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.14 \[ \int \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2 \, dx=\frac {1}{2} a^2 x \sqrt {d-c^2 d x^2}-\frac {a^2 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{2 c}-\frac {b^2 \sqrt {d-c^2 d x^2} \left (4 \arccos (c x)^3-6 \arccos (c x) \cos (2 \arccos (c x))+\left (3-6 \arccos (c x)^2\right ) \sin (2 \arccos (c x))\right )}{24 c \sqrt {1-c^2 x^2}}+\frac {a b \sqrt {d-c^2 d x^2} (\cos (2 \arccos (c x))+2 \arccos (c x) (-\arccos (c x)+\sin (2 \arccos (c x))))}{4 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(a^2*x*Sqrt[d - c^2*d*x^2])/2 - (a^2*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^ 
2])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*c) - (b^2*Sqrt[d - c^2*d*x^2]*(4*ArcCos[ 
c*x]^3 - 6*ArcCos[c*x]*Cos[2*ArcCos[c*x]] + (3 - 6*ArcCos[c*x]^2)*Sin[2*Ar 
cCos[c*x]]))/(24*c*Sqrt[1 - c^2*x^2]) + (a*b*Sqrt[d - c^2*d*x^2]*(Cos[2*Ar 
cCos[c*x]] + 2*ArcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])))/(4*c*Sqrt 
[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.88, 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.

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

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

Input:

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

Output:

(x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/2 - (Sqrt[d - c^2*d*x^2]*(a 
+ b*ArcCos[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*( 
(x^2*(a + b*ArcCos[c*x]))/2 + (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSi 
n[c*x]/(2*c^3)))/2))/Sqrt[1 - c^2*x^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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.77

method	result	size
default	\(\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{6 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}\right )\)	\(531\)
parts	\(\frac {a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a^{2} d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{6 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (2 \arccos \left (c x \right )^{2}-1+2 i \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (2 \arccos \left (c x \right )^{2}-1-2 i \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{4 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arccos \left (c x \right )\right )}{16 \left (c^{2} x^{2}-1\right ) c}\right )\)	\(531\)

Input:

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

Output:

1/2*a^2*x*(-c^2*d*x^2+d)^(1/2)+1/2*a^2*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2 
)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(1/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/ 
2)/(c^2*x^2-1)/c*arccos(c*x)^3+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c* 
x+2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1)^(1/2))*(2*arccos(c*x)^2-1+ 
2*I*arccos(c*x))/(c^2*x^2-1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2 
+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*(2*arccos(c*x)^2-1 
-2*I*arccos(c*x))/(c^2*x^2-1)/c)+2*a*b*(1/4*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x 
^2+1)^(1/2)/(c^2*x^2-1)/c*arccos(c*x)^2+1/16*(-d*(c^2*x^2-1))^(1/2)*(2*c^3 
*x^3-2*c*x+2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-I*(-c^2*x^2+1)^(1/2))*(I+2*arcco 
s(c*x))/(c^2*x^2-1)/c+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2) 
*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*(-I+2*arccos(c*x))/(c^2*x^2 
-1)/c)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(sqrt(-c^2*d*x^2 + d)*x + sqrt(d)*arcsin(c*x)/c)*a^2 + sqrt(d)*integra 
te((b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*arctan2(sqrt( 
c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(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)