\(\int \frac {(a+b \arccos (c x))^2}{(\pi -c^2 \pi x^2)^{3/2}} \, dx\) [47]

Optimal result

Integrand size = 26, antiderivative size = 124 \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\frac {i (a+b \arccos (c x))^2}{c \pi ^{3/2}}+\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -c^2 \pi x^2}}-\frac {b (2 a+b \pi -b (\pi -2 \arccos (c x))) \log \left (1-e^{2 i \arccos (c x)}\right )}{c \pi ^{3/2}}+\frac {i b^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{c \pi ^{3/2}} \] Output:

I*(a+b*arccos(c*x))^2/c/Pi^(3/2)+x*(a+b*arccos(c*x))^2/Pi/(-Pi*c^2*x^2+Pi) 
^(1/2)-b*(2*a+b*Pi-b*(Pi-2*arccos(c*x)))*ln(1-(c*x+I*(-c^2*x^2+1)^(1/2))^2 
)/c/Pi^(3/2)+I*b^2*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c/Pi^(3/2)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\frac {b^2 \left (c x+i \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+2 b \arccos (c x) \left (a c x-b \sqrt {1-c^2 x^2} \log \left (1-e^{2 i \arccos (c x)}\right )\right )+a \left (a c x-b \sqrt {1-c^2 x^2} \log \left (-1+c^2 x^2\right )\right )+i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{c \pi ^{3/2} \sqrt {1-c^2 x^2}} \] Input:

Integrate[(a + b*ArcCos[c*x])^2/(Pi - c^2*Pi*x^2)^(3/2),x]
 

Output:

(b^2*(c*x + I*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + 2*b*ArcCos[c*x]*(a*c*x - 
b*Sqrt[1 - c^2*x^2]*Log[1 - E^((2*I)*ArcCos[c*x])]) + a*(a*c*x - b*Sqrt[1 
- c^2*x^2]*Log[-1 + c^2*x^2]) + I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I 
)*ArcCos[c*x])])/(c*Pi^(3/2)*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5161, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5161

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

\(\Big \downarrow \) 5181

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \int \frac {c x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{\pi ^{3/2} c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \int -\left ((a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )\right )d\arccos (c x)}{\pi ^{3/2} c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 b \int (a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{\pi ^{3/2} c}+\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}\)

\(\Big \downarrow \) 4200

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (2 i \int -\frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\)

Input:

Int[(a + b*ArcCos[c*x])^2/(Pi - c^2*Pi*x^2)^(3/2),x]
 

Output:

(x*(a + b*ArcCos[c*x])^2)/(Pi*Sqrt[Pi - c^2*Pi*x^2]) - (2*b*(((-1/2*I)*(a 
+ b*ArcCos[c*x])^2)/b - (2*I)*((I/2)*(a + b*ArcCos[c*x])*Log[1 - E^((2*I)* 
ArcCos[c*x])] + (b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/4)))/(c*Pi^(3/2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5161

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[x*((a + b*ArcCos[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp[b 
*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcCos[c*x 
])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d 
 + e, 0] && GtQ[n, 0]
 

rule 5181

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.17

method	result	size
default	\(\frac {a^{2} x}{\pi \sqrt {-\pi \,c^{2} x^{2}+\pi }}+b^{2} \left (-\frac {\left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \arccos \left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 i \left (i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{c \,\pi ^{\frac {3}{2}}}\right )-\frac {a b \left (\ln \left (-c^{2} x^{2}+1\right ) x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -\ln \left (-c^{2} x^{2}+1\right )\right )}{c \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}-1\right )}\)	\(269\)
parts	\(\frac {a^{2} x}{\pi \sqrt {-\pi \,c^{2} x^{2}+\pi }}+b^{2} \left (-\frac {\left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \arccos \left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}-1\right )}+\frac {2 i \left (i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{c \,\pi ^{\frac {3}{2}}}\right )-\frac {a b \left (\ln \left (-c^{2} x^{2}+1\right ) x^{2} c^{2}+2 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -\ln \left (-c^{2} x^{2}+1\right )\right )}{c \,\pi ^{\frac {3}{2}} \left (c^{2} x^{2}-1\right )}\)	\(269\)

Input:

int((a+b*arccos(c*x))^2/(-Pi*c^2*x^2+Pi)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a^2/Pi*x/(-Pi*c^2*x^2+Pi)^(1/2)+b^2*(-1/Pi^(3/2)*(I*c^2*x^2+c*x*(-c^2*x^2+ 
1)^(1/2)-I)*arccos(c*x)^2/c/(c^2*x^2-1)+2*I*(I*arccos(c*x)*ln(1+c*x+I*(-c^ 
2*x^2+1)^(1/2))+I*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))+arccos(c*x)^2 
+polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))) 
/c/Pi^(3/2))-a*b/c/Pi^(3/2)*(ln(-c^2*x^2+1)*x^2*c^2+2*(-c^2*x^2+1)^(1/2)*a 
rccos(c*x)*x*c-ln(-c^2*x^2+1))/(c^2*x^2-1)
 

Fricas [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*arccos(c*x))^2/(-pi*c^2*x^2+pi)^(3/2),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)/(pi^2*c^4*x^4 - 2*pi^2*c^2*x^2 + pi^2), x)
 

Sympy [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a^{2}}{- c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{- c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {acos}{\left (c x \right )}}{- c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \] Input:

integrate((a+b*acos(c*x))**2/(-pi*c**2*x**2+pi)**(3/2),x)
 

Output:

(Integral(a**2/(-c**2*x**2*sqrt(-c**2*x**2 + 1) + sqrt(-c**2*x**2 + 1)), x 
) + Integral(b**2*acos(c*x)**2/(-c**2*x**2*sqrt(-c**2*x**2 + 1) + sqrt(-c* 
*2*x**2 + 1)), x) + Integral(2*a*b*acos(c*x)/(-c**2*x**2*sqrt(-c**2*x**2 + 
 1) + sqrt(-c**2*x**2 + 1)), x))/pi**(3/2)
 

Maxima [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*arccos(c*x))^2/(-pi*c^2*x^2+pi)^(3/2),x, algorithm="maxima" 
)
 

Output:

2*a*b*x*arccos(c*x)/(pi*sqrt(pi - pi*c^2*x^2)) - b^2*integrate(-arctan2(sq 
rt(c*x + 1)*sqrt(-c*x + 1), c*x)^2/((pi - pi*c^2*x^2)*sqrt(c*x + 1)*sqrt(- 
c*x + 1)), x)/sqrt(pi) + a^2*x/(pi*sqrt(pi - pi*c^2*x^2)) + a*b*log(x^2 - 
1/c^2)/(pi^(3/2)*c)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (\Pi -\Pi \,c^2\,x^2\right )}^{3/2}} \,d x \] Input:

int((a + b*acos(c*x))^2/(Pi - Pi*c^2*x^2)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*acos(c*x))^2/(Pi - Pi*c^2*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) a b -\sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2}+a^{2} x}{\sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \pi } \] Input:

int((a+b*acos(c*x))^2/(-Pi*c^2*x^2+Pi)^(3/2),x)
 

Output:

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