\(\int (d-c^2 d x^2)^{3/2} (a+b \arccos (c x))^2 \, dx\) [58]

Optimal result

Integrand size = 26, antiderivative size = 296 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=-\frac {15}{64} b^2 d x \sqrt {d-c^2 d x^2}-\frac {1}{32} b^2 x \left (d-c^2 d x^2\right )^{3/2}+\frac {3 b c d x^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 \sqrt {1-c^2 x^2}}-\frac {b d \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{8 c}+\frac {3}{8} d x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2-\frac {d \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{8 b c \sqrt {1-c^2 x^2}}+\frac {9 b^2 d \sqrt {d-c^2 d x^2} \arcsin (c x)}{64 c \sqrt {1-c^2 x^2}} \] Output:

-15/64*b^2*d*x*(-c^2*d*x^2+d)^(1/2)-1/32*b^2*x*(-c^2*d*x^2+d)^(3/2)+3/8*b* 
c*d*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/(-c^2*x^2+1)^(1/2)-1/8*b*d* 
(-c^2*x^2+1)^(3/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))/c+3/8*d*x*(-c^2* 
d*x^2+d)^(1/2)*(a+b*arccos(c*x))^2+1/4*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos( 
c*x))^2-1/8*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arccos(c*x))^3/b/c/(-c^2*x^2+1)^(1 
/2)+9/64*b^2*d*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.92 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.11 \[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {-32 b^2 d \sqrt {d-c^2 d x^2} \arccos (c x)^3-96 a^2 d^{3/2} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-8 b d \sqrt {d-c^2 d x^2} \arccos (c x)^2 (12 a-8 b \sin (2 \arccos (c x))+b \sin (4 \arccos (c x)))+d \sqrt {d-c^2 d x^2} \left (160 a^2 c x \sqrt {1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+64 a b \cos (2 \arccos (c x))-4 a b \cos (4 \arccos (c x))-32 b^2 \sin (2 \arccos (c x))+b^2 \sin (4 \arccos (c x))\right )-4 b d \sqrt {d-c^2 d x^2} \arccos (c x) (-16 b \cos (2 \arccos (c x))+b \cos (4 \arccos (c x))+4 a (-8 \sin (2 \arccos (c x))+\sin (4 \arccos (c x))))}{256 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-32*b^2*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^3 - 96*a^2*d^(3/2)*Sqrt[1 - c^2 
*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 8*b*d*S 
qrt[d - c^2*d*x^2]*ArcCos[c*x]^2*(12*a - 8*b*Sin[2*ArcCos[c*x]] + b*Sin[4* 
ArcCos[c*x]]) + d*Sqrt[d - c^2*d*x^2]*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64* 
a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 64*a*b*Cos[2*ArcCos[c*x]] - 4*a*b*Cos[4*Ar 
cCos[c*x]] - 32*b^2*Sin[2*ArcCos[c*x]] + b^2*Sin[4*ArcCos[c*x]]) - 4*b*d*S 
qrt[d - c^2*d*x^2]*ArcCos[c*x]*(-16*b*Cos[2*ArcCos[c*x]] + b*Cos[4*ArcCos[ 
c*x]] + 4*a*(-8*Sin[2*ArcCos[c*x]] + Sin[4*ArcCos[c*x]])))/(256*c*Sqrt[1 - 
 c^2*x^2])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5159, 5157, 5139, 262, 223, 5153, 5183, 211, 211, 223}

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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

\(\displaystyle \frac {3}{4} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {(a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5153

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \int x \left (1-c^2 x^2\right ) (a+b \arccos (c x))dx}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 5183

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {b c d \sqrt {d-c^2 d x^2} \left (-\frac {\left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {3}{4} d \left (\frac {b c \sqrt {d-c^2 d x^2} \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2\right )+\frac {1}{4} x \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2\)

Input:

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

Output:

(x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/4 + (3*d*((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*ArcC 
os[c*x]))/2 + (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3))) 
/2))/Sqrt[1 - c^2*x^2]))/4 + (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/4*((1 - c^2*x^ 
2)^2*(a + b*ArcCos[c*x]))/c^2 - (b*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqr 
t[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 - c^2*x^2])
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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]
 

rule 5159

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCos[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos[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] && GtQ[p, 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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 987, normalized size of antiderivative = 3.33

method	result	size
default	\(\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \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} d}{8 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right ) d}{512 \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 ) d}{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 ) d}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right ) d}{512 \left (c^{2} x^{2}-1\right ) c}\right )+2 a b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} d}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+4 \arccos \left (c x \right )\right ) d}{256 \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 ) d}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (5 i+12 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d}{256 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (17 i+28 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d}{256 \left (c^{2} x^{2}-1\right ) c}\right )\)	\(987\)
parts	\(\frac {a^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \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} d}{8 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right ) d}{512 \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 ) d}{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 ) d}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-12 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}+4 c x \right ) \left (-4 i \arccos \left (c x \right )+8 \arccos \left (c x \right )^{2}-1\right ) d}{512 \left (c^{2} x^{2}-1\right ) c}\right )+2 a b \left (\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} d}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}+4 c x -8 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+i \sqrt {-c^{2} x^{2}+1}\right ) \left (i+4 \arccos \left (c x \right )\right ) d}{256 \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 ) d}{16 \left (c^{2} x^{2}-1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (5 i+12 \arccos \left (c x \right )\right ) \cos \left (3 \arccos \left (c x \right )\right ) d}{256 \left (c^{2} x^{2}-1\right ) c}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (17 i+28 \arccos \left (c x \right )\right ) \sin \left (3 \arccos \left (c x \right )\right ) d}{256 \left (c^{2} x^{2}-1\right ) c}\right )\)	\(987\)

Input:

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

Output:

1/4*a^2*x*(-c^2*d*x^2+d)^(3/2)+3/8*a^2*d*x*(-c^2*d*x^2+d)^(1/2)+3/8*a^2*d^ 
2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b^2*(1/8*(-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*d-1/512* 
(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^4*c^ 
4+4*c*x-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I*(-c^2*x^2+1)^(1/2))*(4*I*arccos(c 
*x)+8*arccos(c*x)^2-1)*d/(c^2*x^2-1)/c+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))*d/(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*arcco 
s(c*x)^2-1-2*I*arccos(c*x))*d/(c^2*x^2-1)/c-1/512*(-d*(c^2*x^2-1))^(1/2)*( 
-8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-1 
2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*(-4*I*arccos(c*x)+8*arccos(c*x)^2-1) 
*d/(c^2*x^2-1)/c)+2*a*b*(3/16*(-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*d-1/256*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3 
*x^3+8*I*(-c^2*x^2+1)^(1/2)*x^4*c^4+4*c*x-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+I 
*(-c^2*x^2+1)^(1/2))*(I+4*arccos(c*x))*d/(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))*d/(c^2*x^2-1)/c-3/256*(-d*(c^2*x^2-1))^(1/2)*(-I* 
(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(5*I+12*arccos(c*x))*cos(3*arccos(c*x))* 
d/(c^2*x^2-1)/c-1/256*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2+c*x*(-c^2*x^2+1...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/8*(2*(-c^2*d*x^2 + d)^(3/2)*x + 3*sqrt(-c^2*d*x^2 + d)*d*x + 3*d^(3/2)*a 
rcsin(c*x)/c)*a^2 + sqrt(d)*integrate(-((b^2*c^2*d*x^2 - b^2*d)*arctan2(sq 
rt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arctan2(sqr 
t(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {\sqrt {d}\, d \left (3 \mathit {asin} \left (c x \right ) a^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+5 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x -16 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+16 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )d x \right ) a b c -8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{8 c} \] Input:

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

Output:

(sqrt(d)*d*(3*asin(c*x)*a**2 - 2*sqrt( - c**2*x**2 + 1)*a**2*c**3*x**3 + 5 
*sqrt( - c**2*x**2 + 1)*a**2*c*x - 16*int(sqrt( - c**2*x**2 + 1)*acos(c*x) 
*x**2,x)*a*b*c**3 + 16*int(sqrt( - c**2*x**2 + 1)*acos(c*x),x)*a*b*c - 8*i 
nt(sqrt( - c**2*x**2 + 1)*acos(c*x)**2*x**2,x)*b**2*c**3 + 8*int(sqrt( - c 
**2*x**2 + 1)*acos(c*x)**2,x)*b**2*c))/(8*c)