\(\int \frac {x^2 (a+b \arccos (c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [249]

Optimal result

Integrand size = 29, antiderivative size = 250 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}} \] Output:

x*(a+b*arccos(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)-I*(-c^2*x^2+1)^(1/2)*(a+b 
*arccos(c*x))^2/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/3*(-c^2*x^2+1)^(1/2)*(a+b*arc 
cos(c*x))^3/b/c^3/d/(-c^2*d*x^2+d)^(1/2)+2*b*(-c^2*x^2+1)^(1/2)*(a+b*arcco 
s(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-I*b^ 
2*(-c^2*x^2+1)^(1/2)*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c^3/d/(-c^2* 
d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {a^2 x \sqrt {-d \left (-1+c^2 x^2\right )}}{c^2 d^2 \left (-1+c^2 x^2\right )}+\frac {a^2 \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c^3 d^{3/2}}-\frac {a b \left (-2 c x \arccos (c x)-\sqrt {1-c^2 x^2} \left (\arccos (c x)^2-2 \log \left (\sqrt {1-c^2 x^2}\right )\right )\right )}{c^3 d \sqrt {d \left (1-c^2 x^2\right )}}-\frac {b^2 \left (-\arccos (c x) \left (3 c x \arccos (c x)+\sqrt {1-c^2 x^2} \left (\arccos (c x) (3 i+\arccos (c x))-6 \log \left (1-e^{2 i \arccos (c x)}\right )\right )\right )-3 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )}{3 c^3 d \sqrt {d \left (1-c^2 x^2\right )}} \] Input:

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

Output:

-((a^2*x*Sqrt[-(d*(-1 + c^2*x^2))])/(c^2*d^2*(-1 + c^2*x^2))) + (a^2*ArcTa 
n[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^3*d^(3/2)) 
 - (a*b*(-2*c*x*ArcCos[c*x] - Sqrt[1 - c^2*x^2]*(ArcCos[c*x]^2 - 2*Log[Sqr 
t[1 - c^2*x^2]])))/(c^3*d*Sqrt[d*(1 - c^2*x^2)]) - (b^2*(-(ArcCos[c*x]*(3* 
c*x*ArcCos[c*x] + Sqrt[1 - c^2*x^2]*(ArcCos[c*x]*(3*I + ArcCos[c*x]) - 6*L 
og[1 - E^((2*I)*ArcCos[c*x])]))) - (3*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(( 
2*I)*ArcCos[c*x])]))/(3*c^3*d*Sqrt[d*(1 - c^2*x^2)])
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.78, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5207, 5153, 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.

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(x*(a + b*ArcCos[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2] 
*(a + b*ArcCos[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^ 
2*x^2]*(((-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^3*d*Sqrt[d - c^2*d*x^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 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 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 5207

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (250 ) = 500\).

Time = 0.58 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.20

method	result	size
default	\(\frac {a^{2} x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{3 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \arccos \left (c x \right )^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\)	\(550\)
parts	\(\frac {a^{2} x}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{c^{2} d \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{3}}{3 d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \arccos \left (c x \right )^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right )^{2}+\operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2}}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arccos \left (c x \right ) x}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right )}{d^{2} c^{3} \left (c^{2} x^{2}-1\right )}\)	\(550\)

Input:

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

Output:

a^2*x/c^2/d/(-c^2*d*x^2+d)^(1/2)-a^2/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1 
/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(-1/3*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^ 
(1/2)/d^2/c^3/(c^2*x^2-1)*arccos(c*x)^3-(-d*(c^2*x^2-1))^(1/2)*(-I*(-c^2*x 
^2+1)^(1/2)+c*x)*arccos(c*x)^2/d^2/c^3/(c^2*x^2-1)-2*I*(-c^2*x^2+1)^(1/2)* 
(-d*(c^2*x^2-1))^(1/2)*(I*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+I*arc 
cos(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)))/d^2/c^3/(c^2*x^2-1)) 
-a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*arccos( 
c*x)^2-2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^3/(c^2*x^2- 
1)*arccos(c*x)-2*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^2/(c^2*x^2-1)*arccos(c*x 
)*x+2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*ln 
((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)
 

Fricas [F]

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

integrate(x^2*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

integral((b^2*x^2*arccos(c*x)^2 + 2*a*b*x^2*arccos(c*x) + a^2*x^2)*sqrt(-c 
^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

a^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + sqrt(d) 
*integrate((b^2*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*x 
^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1 
)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt( - c**2*x**2 + 1)*acos(c*x)**3*b**2 + 3*sqrt( - c**2*x**2 + 1)*acos( 
c*x)**2*a*b - 3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a**2 - 6*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*c - 3*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*c + 3*a**2*c*x)/(3*s 
qrt(d)*sqrt( - c**2*x**2 + 1)*c**3*d)