\(\int \frac {(a+b \arccos (c x))^2}{(\pi -c^2 \pi x^2)^{5/2}} \, dx\) [48]

Optimal result

Integrand size = 26, antiderivative size = 226 \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b^2 x}{3 \pi ^{5/2} \sqrt {1-c^2 x^2}}+\frac {b (a+b \arccos (c x))}{3 c \pi ^{5/2} \left (1-c^2 x^2\right )}+\frac {2 i (a+b \arccos (c x))^2}{3 c \pi ^{5/2}}+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -c^2 \pi x^2\right )^{3/2}}+\frac {2 x (a+b \arccos (c x))^2}{3 \pi ^2 \sqrt {\pi -c^2 \pi x^2}}-\frac {2 b (2 a+b \pi -b (\pi -2 \arccos (c x))) \log \left (1-e^{2 i \arccos (c x)}\right )}{3 c \pi ^{5/2}}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{3 c \pi ^{5/2}} \] Output:

1/3*b^2*x/Pi^(5/2)/(-c^2*x^2+1)^(1/2)+1/3*b*(a+b*arccos(c*x))/c/Pi^(5/2)/( 
-c^2*x^2+1)+2/3*I*(a+b*arccos(c*x))^2/c/Pi^(5/2)+1/3*x*(a+b*arccos(c*x))^2 
/Pi/(-Pi*c^2*x^2+Pi)^(3/2)+2/3*x*(a+b*arccos(c*x))^2/Pi^2/(-Pi*c^2*x^2+Pi) 
^(1/2)-2/3*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^(5/2)+2/3*I*b^2*polylog(2,(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c/Pi^(5 
/2)
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {-3 a^2 c x-b^2 c x+2 a^2 c^3 x^3+b^2 c^3 x^3-a b \sqrt {1-c^2 x^2}+b^2 \left (-3 c x+2 c^3 x^3-2 i \sqrt {1-c^2 x^2}+2 i c^2 x^2 \sqrt {1-c^2 x^2}\right ) \arccos (c x)^2+b \arccos (c x) \left (-6 a c x+4 a c^3 x^3-b \sqrt {1-c^2 x^2}+4 b \left (1-c^2 x^2\right )^{3/2} \log \left (1-e^{2 i \arccos (c x)}\right )\right )+2 a b \sqrt {1-c^2 x^2} \log \left (-1+c^2 x^2\right )-2 a b c^2 x^2 \sqrt {1-c^2 x^2} \log \left (-1+c^2 x^2\right )-2 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{3 c \pi ^{5/2} \left (1-c^2 x^2\right )^{3/2}} \] Input:

Integrate[(a + b*ArcCos[c*x])^2/(Pi - c^2*Pi*x^2)^(5/2),x]
 

Output:

-1/3*(-3*a^2*c*x - b^2*c*x + 2*a^2*c^3*x^3 + b^2*c^3*x^3 - a*b*Sqrt[1 - c^ 
2*x^2] + b^2*(-3*c*x + 2*c^3*x^3 - (2*I)*Sqrt[1 - c^2*x^2] + (2*I)*c^2*x^2 
*Sqrt[1 - c^2*x^2])*ArcCos[c*x]^2 + b*ArcCos[c*x]*(-6*a*c*x + 4*a*c^3*x^3 
- b*Sqrt[1 - c^2*x^2] + 4*b*(1 - c^2*x^2)^(3/2)*Log[1 - E^((2*I)*ArcCos[c* 
x])]) + 2*a*b*Sqrt[1 - c^2*x^2]*Log[-1 + c^2*x^2] - 2*a*b*c^2*x^2*Sqrt[1 - 
 c^2*x^2]*Log[-1 + c^2*x^2] - (2*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, E^( 
(2*I)*ArcCos[c*x])])/(c*Pi^(5/2)*(1 - c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5163, 5161, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838, 5183, 208}

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.

\(\displaystyle \int \frac {(a+b \arccos (c x))^2}{\left (\pi -\pi c^2 x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5163

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{3/2}}dx}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5161

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {2 b c \int \frac {x (a+b \arccos (c x))}{1-c^2 x^2}dx}{\pi ^{3/2}}+\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5181

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \int \frac {c x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}d\arccos (c x)}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \int -\left ((a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )\right )d\arccos (c x)}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {2 b \int (a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{\pi ^{3/2} c}+\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 4200

\(\displaystyle \frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (2 i \int -\frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1-e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 b c \int \frac {x (a+b \arccos (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {2 b c \left (\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 \pi ^{5/2}}+\frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {2 \left (\frac {x (a+b \arccos (c x))^2}{\pi \sqrt {\pi -\pi c^2 x^2}}-\frac {2 b \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{\pi ^{3/2} c}\right )}{3 \pi }+\frac {x (a+b \arccos (c x))^2}{3 \pi \left (\pi -\pi c^2 x^2\right )^{3/2}}+\frac {2 b c \left (\frac {a+b \arccos (c x)}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b x}{2 c \sqrt {1-c^2 x^2}}\right )}{3 \pi ^{5/2}}\)

Input:

Int[(a + b*ArcCos[c*x])^2/(Pi - c^2*Pi*x^2)^(5/2),x]
 

Output:

(x*(a + b*ArcCos[c*x])^2)/(3*Pi*(Pi - c^2*Pi*x^2)^(3/2)) + (2*b*c*((b*x)/( 
2*c*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*c^2*(1 - c^2*x^2))))/(3*Pi 
^(5/2)) + (2*((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))))/(3*Pi)
 

Defintions of rubi rules used

rule 25

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

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, 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 5163

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(2*d*(p + 1 
))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b*Ar 
cCos[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 
*x^2)^p]   Int[x*(1 - c^2*x^2)^(p + 1/2)*(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] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

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]
 

rule 5183

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1686 vs. \(2 (216 ) = 432\).

Time = 0.50 (sec) , antiderivative size = 1687, normalized size of antiderivative = 7.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1687\)
parts	\(\text {Expression too large to display}\)	\(1687\)

Input:

int((a+b*arccos(c*x))^2/(-Pi*c^2*x^2+Pi)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-4*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*arc 
cos(c*x)^2*x-b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^4*(-c^2*x^ 
2+1)^(1/2)*x^5+7/3*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^2*(- 
c^2*x^2+1)^(1/2)*x^3+16/3*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1) 
*c*arccos(c*x)*x^2-4/3*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^ 
7*arccos(c*x)*x^8+16/3*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^ 
5*arccos(c*x)*x^6-8*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^3*a 
rccos(c*x)*x^4+2/3*I*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^7* 
x^8-10/3*I*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^5*x^6+6*I*b^ 
2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c^3*x^4-14/3*I*b^2/Pi^(5/2) 
/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*c*x^2+8/3*I*b^2/Pi^(5/2)/(3*c^4*x^4-7 
*c^2*x^2+4)/(c^2*x^2-1)/c*arccos(c*x)^2-4/3*b^2/c/Pi^(5/2)*arccos(c*x)*ln( 
1+c*x+I*(-c^2*x^2+1)^(1/2))-4/3*b^2/c/Pi^(5/2)*arccos(c*x)*ln(1-c*x-I*(-c^ 
2*x^2+1)^(1/2))+4/3*I*b^2/c/Pi^(5/2)*arccos(c*x)^2+4/3*I*b^2/c/Pi^(5/2)*po 
lylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))+4/3*I*b^2/c/Pi^(5/2)*polylog(2,c*x+I*(- 
c^2*x^2+1)^(1/2))+a^2*(1/3/Pi*x/(-Pi*c^2*x^2+Pi)^(3/2)+2/3/Pi^2*x/(-Pi*c^2 
*x^2+Pi)^(1/2))-4/3*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)*(-c^2 
*x^2+1)^(1/2)*x-4/3*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)/c*arc 
cos(c*x)+4/3*I*b^2/Pi^(5/2)/(3*c^4*x^4-7*c^2*x^2+4)/(c^2*x^2-1)/c-1/3*a*b/ 
c/Pi^(5/2)*(2*ln(-c^2*x^2+1)*x^4*c^4+4*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x...
 

Fricas [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccos(c*x))^2/(-pi*c^2*x^2+pi)^(5/2),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^3*c^6*x^6 - 3*pi^3*c^4*x^4 + 3*pi^3*c^2*x^2 - pi^3), x)
 

Sympy [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{4} \sqrt {- c^{2} x^{2} + 1} - 2 c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {acos}^{2}{\left (c x \right )}}{c^{4} x^{4} \sqrt {- c^{2} x^{2} + 1} - 2 c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} \sqrt {- c^{2} x^{2} + 1} - 2 c^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} + \sqrt {- c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \] Input:

integrate((a+b*acos(c*x))**2/(-pi*c**2*x**2+pi)**(5/2),x)
 

Output:

(Integral(a**2/(c**4*x**4*sqrt(-c**2*x**2 + 1) - 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**4*x** 
4*sqrt(-c**2*x**2 + 1) - 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**4*x**4*sqrt(-c**2*x**2 + 1) - 2 
*c**2*x**2*sqrt(-c**2*x**2 + 1) + sqrt(-c**2*x**2 + 1)), x))/pi**(5/2)
 

Maxima [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{{\left (\pi - \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*arccos(c*x))^2/(-pi*c^2*x^2+pi)^(5/2),x, algorithm="maxima" 
)
 

Output:

-1/3*a*b*c*(1/(pi^(5/2)*c^4*x^2 - pi^(5/2)*c^2) + 2*log(c*x + 1)/(pi^(5/2) 
*c^2) + 2*log(c*x - 1)/(pi^(5/2)*c^2)) + 2/3*a*b*(x/(pi*(pi - pi*c^2*x^2)^ 
(3/2)) + 2*x/(pi^2*sqrt(pi - pi*c^2*x^2)))*arccos(c*x) + 1/3*a^2*(x/(pi*(p 
i - pi*c^2*x^2)^(3/2)) + 2*x/(pi^2*sqrt(pi - pi*c^2*x^2))) + b^2*integrate 
(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2/((pi^2*c^4*x^4 - 2*pi^2*c^2* 
x^2 + pi^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(pi)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arccos(c*x))^2/(-pi*c^2*x^2+pi)^(5/2),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. \[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (\Pi -\Pi \,c^2\,x^2\right )}^{5/2}} \,d x \] Input:

int((a + b*acos(c*x))^2/(Pi - Pi*c^2*x^2)^(5/2),x)
 

Output:

int((a + b*acos(c*x))^2/(Pi - Pi*c^2*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \arccos (c x))^2}{\left (\pi -c^2 \pi x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2}+2 a^{2} c^{2} x^{3}-3 a^{2} x}{3 \sqrt {\pi }\, \sqrt {-c^{2} x^{2}+1}\, \pi ^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((a+b*acos(c*x))^2/(-Pi*c^2*x^2+Pi)^(5/2),x)
 

Output:

(6*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 
- 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*a*b*c**2 
*x**2 - 6*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c** 
4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*a 
*b + 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)**2/(sqrt( - c**2*x**2 + 1)*c** 
4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b 
**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)**2/(sqrt( - c**2*x* 
*2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 
 + 1)),x)*b**2 + 2*a**2*c**2*x**3 - 3*a**2*x)/(3*sqrt(pi)*sqrt( - c**2*x** 
2 + 1)*pi**2*(c**2*x**2 - 1))