\(\int (\pi -c^2 \pi x^2)^{3/2} (a+b \arccos (c x))^2 \, dx\) [44]

Optimal result

Integrand size = 26, antiderivative size = 215 \[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=-\frac {15}{64} b^2 \pi ^{3/2} x \sqrt {1-c^2 x^2}-\frac {1}{32} b^2 \pi ^{3/2} x \left (1-c^2 x^2\right )^{3/2}+\frac {3}{8} b c \pi ^{3/2} x^2 (a+b \arccos (c x))-\frac {b \pi ^{3/2} \left (1-c^2 x^2\right )^2 (a+b \arccos (c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi -c^2 \pi x^2} (a+b \arccos (c x))^2+\frac {1}{4} x \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \arccos (c x))^2-\frac {\pi ^{3/2} (a+b \arccos (c x))^3}{8 b c}+\frac {9 b^2 \pi ^{3/2} \arcsin (c x)}{64 c} \] Output:

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

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94 \[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\frac {\pi ^{3/2} \left (160 a^2 c x \sqrt {1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-32 b^2 \arccos (c x)^3+96 a^2 \arcsin (c x)+64 a b \cos (2 \arccos (c x))-4 a b \cos (4 \arccos (c x))-32 b^2 \sin (2 \arccos (c x))+b^2 \sin (4 \arccos (c x))-8 b \arccos (c x)^2 (12 a-8 b \sin (2 \arccos (c x))+b \sin (4 \arccos (c x)))-4 b \arccos (c x) (-16 b \cos (2 \arccos (c x))+b \cos (4 \arccos (c x))+4 a (-8 \sin (2 \arccos (c x))+\sin (4 \arccos (c x))))\right )}{256 c} \] Input:

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

Output:

(Pi^(3/2)*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64*a^2*c^3*x^3*Sqrt[1 - c^2*x^2 
] - 32*b^2*ArcCos[c*x]^3 + 96*a^2*ArcSin[c*x] + 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]] - 8*b*ArcCos[c*x]^2*(12*a - 8*b*Sin[2*ArcCos[c*x]] + b*Sin[4*ArcCos[ 
c*x]]) - 4*b*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)
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.20, 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.

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

\(\Big \downarrow \) 5159

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5183

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

Input:

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

Output:

(x*(Pi - c^2*Pi*x^2)^(3/2)*(a + b*ArcCos[c*x])^2)/4 + (3*Pi*((x*Sqrt[Pi - 
c^2*Pi*x^2]*(a + b*ArcCos[c*x])^2)/2 - (Sqrt[Pi]*(a + b*ArcCos[c*x])^3)/(6 
*b*c) + b*c*Sqrt[Pi]*((x^2*(a + b*ArcCos[c*x]))/2 + (b*c*(-1/2*(x*Sqrt[1 - 
 c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)))/4 + (b*c*Pi^(3/2)*(-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*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c)))/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 [A] (verified)

Time = 0.44 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {a^{2} x \left (-\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{8 \sqrt {\pi \,c^{2}}}-\frac {b^{2} \pi ^{\frac {3}{2}} \left (16 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} x^{3} c^{3}+8 c^{4} x^{4} \arccos \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-40 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} x c -40 c^{2} x^{2} \arccos \left (c x \right )+17 c x \sqrt {-c^{2} x^{2}+1}+8 \arccos \left (c x \right )^{3}+17 \arccos \left (c x \right )\right )}{64 c}-\frac {a b \,\pi ^{\frac {3}{2}} \left (16 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+4 c^{4} x^{4}-40 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -20 c^{2} x^{2}+12 \arccos \left (c x \right )^{2}+25\right )}{32 c}\)	\(293\)
parts	\(\frac {a^{2} x \left (-\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {-\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \arctan \left (\frac {\sqrt {\pi \,c^{2}}\, x}{\sqrt {-\pi \,c^{2} x^{2}+\pi }}\right )}{8 \sqrt {\pi \,c^{2}}}-\frac {b^{2} \pi ^{\frac {3}{2}} \left (16 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} x^{3} c^{3}+8 c^{4} x^{4} \arccos \left (c x \right )-2 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-40 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} x c -40 c^{2} x^{2} \arccos \left (c x \right )+17 c x \sqrt {-c^{2} x^{2}+1}+8 \arccos \left (c x \right )^{3}+17 \arccos \left (c x \right )\right )}{64 c}-\frac {a b \,\pi ^{\frac {3}{2}} \left (16 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x^{3} c^{3}+4 c^{4} x^{4}-40 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) x c -20 c^{2} x^{2}+12 \arccos \left (c x \right )^{2}+25\right )}{32 c}\)	\(293\)

Input:

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

Output:

1/4*a^2*x*(-Pi*c^2*x^2+Pi)^(3/2)+3/8*a^2*Pi*x*(-Pi*c^2*x^2+Pi)^(1/2)+3/8*a 
^2*Pi^2/(Pi*c^2)^(1/2)*arctan((Pi*c^2)^(1/2)*x/(-Pi*c^2*x^2+Pi)^(1/2))-1/6 
4*b^2*Pi^(3/2)*(16*(-c^2*x^2+1)^(1/2)*arccos(c*x)^2*x^3*c^3+8*c^4*x^4*arcc 
os(c*x)-2*c^3*x^3*(-c^2*x^2+1)^(1/2)-40*(-c^2*x^2+1)^(1/2)*arccos(c*x)^2*x 
*c-40*c^2*x^2*arccos(c*x)+17*c*x*(-c^2*x^2+1)^(1/2)+8*arccos(c*x)^3+17*arc 
cos(c*x))/c-1/32*a*b*Pi^(3/2)*(16*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x^3*c^3+4 
*c^4*x^4-40*(-c^2*x^2+1)^(1/2)*arccos(c*x)*x*c-20*c^2*x^2+12*arccos(c*x)^2 
+25)/c
 

Fricas [F]

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

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (197) = 394\).

Time = 2.68 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.91 \[ \int \left (\pi -c^2 \pi x^2\right )^{3/2} (a+b \arccos (c x))^2 \, dx=\begin {cases} - \frac {\pi ^{\frac {3}{2}} a^{2} c^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a^{2} x \sqrt {- c^{2} x^{2} + 1}}{8} - \frac {3 \pi ^{\frac {3}{2}} a^{2} \operatorname {acos}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} a b c^{3} x^{4}}{8} - \frac {\pi ^{\frac {3}{2}} a b c^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{2} + \frac {5 \pi ^{\frac {3}{2}} a b c x^{2}}{8} + \frac {5 \pi ^{\frac {3}{2}} a b x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}{\left (c x \right )}}{4} - \frac {3 \pi ^{\frac {3}{2}} a b \operatorname {acos}^{2}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b^{2} c^{3} x^{4} \operatorname {acos}{\left (c x \right )}}{8} - \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{4} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{32} + \frac {5 \pi ^{\frac {3}{2}} b^{2} c x^{2} \operatorname {acos}{\left (c x \right )}}{8} + \frac {5 \pi ^{\frac {3}{2}} b^{2} x \sqrt {- c^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (c x \right )}}{8} - \frac {17 \pi ^{\frac {3}{2}} b^{2} x \sqrt {- c^{2} x^{2} + 1}}{64} - \frac {\pi ^{\frac {3}{2}} b^{2} \operatorname {acos}^{3}{\left (c x \right )}}{8 c} - \frac {17 \pi ^{\frac {3}{2}} b^{2} \operatorname {acos}{\left (c x \right )}}{64 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} x \left (a + \frac {\pi b}{2}\right )^{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-pi**(3/2)*a**2*c**2*x**3*sqrt(-c**2*x**2 + 1)/4 + 5*pi**(3/2)* 
a**2*x*sqrt(-c**2*x**2 + 1)/8 - 3*pi**(3/2)*a**2*acos(c*x)/(8*c) - pi**(3/ 
2)*a*b*c**3*x**4/8 - pi**(3/2)*a*b*c**2*x**3*sqrt(-c**2*x**2 + 1)*acos(c*x 
)/2 + 5*pi**(3/2)*a*b*c*x**2/8 + 5*pi**(3/2)*a*b*x*sqrt(-c**2*x**2 + 1)*ac 
os(c*x)/4 - 3*pi**(3/2)*a*b*acos(c*x)**2/(8*c) - pi**(3/2)*b**2*c**3*x**4* 
acos(c*x)/8 - pi**(3/2)*b**2*c**2*x**3*sqrt(-c**2*x**2 + 1)*acos(c*x)**2/4 
 + pi**(3/2)*b**2*c**2*x**3*sqrt(-c**2*x**2 + 1)/32 + 5*pi**(3/2)*b**2*c*x 
**2*acos(c*x)/8 + 5*pi**(3/2)*b**2*x*sqrt(-c**2*x**2 + 1)*acos(c*x)**2/8 - 
 17*pi**(3/2)*b**2*x*sqrt(-c**2*x**2 + 1)/64 - pi**(3/2)*b**2*acos(c*x)**3 
/(8*c) - 17*pi**(3/2)*b**2*acos(c*x)/(64*c), Ne(c, 0)), (pi**(3/2)*x*(a + 
pi*b/2)**2, True))
 

Maxima [F]

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

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

Output:

1/8*(3*pi*sqrt(pi - pi*c^2*x^2)*x + 2*(pi - pi*c^2*x^2)^(3/2)*x + 3*pi^(3/ 
2)*arcsin(c*x)/c)*a^2 + sqrt(pi)*integrate(-((pi*b^2*c^2*x^2 - pi*b^2)*arc 
tan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*(pi*a*b*c^2*x^2 - pi*a*b)*ar 
ctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)
 

Giac [F(-2)]

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

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

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

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

Output:

(sqrt(pi)*pi*(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 
*int(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)