\(\int \frac {x^3 (1-c^2 x^2)^{5/2}}{(a+b \arccos (c x))^2} \, dx\) [401]

Optimal result

Integrand size = 28, antiderivative size = 278 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arccos (c x))^2} \, dx=-\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arccos (c x))}+\frac {3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \cos \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {9 (a+b \arccos (c x))}{b}\right )}{256 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{32 b^2 c^4}-\frac {21 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )}{256 b^2 c^4}-\frac {9 \sin \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arccos (c x))}{b}\right )}{256 b^2 c^4} \] Output:

-x^3*(-c^2*x^2+1)^3/b/c/(a+b*arccos(c*x))+3/128*cos(a/b)*Ci((a+b*arccos(c* 
x))/b)/b^2/c^4+3/32*cos(3*a/b)*Ci(3*(a+b*arccos(c*x))/b)/b^2/c^4-21/256*co 
s(7*a/b)*Ci(7*(a+b*arccos(c*x))/b)/b^2/c^4-9/256*cos(9*a/b)*Ci(9*(a+b*arcc 
os(c*x))/b)/b^2/c^4+3/128*sin(a/b)*Si((a+b*arccos(c*x))/b)/b^2/c^4+3/32*si 
n(3*a/b)*Si(3*(a+b*arccos(c*x))/b)/b^2/c^4-21/256*sin(7*a/b)*Si(7*(a+b*arc 
cos(c*x))/b)/b^2/c^4-9/256*sin(9*a/b)*Si(9*(a+b*arccos(c*x))/b)/b^2/c^4
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arccos (c x))^2} \, dx=\frac {256 b c^3 x^3-768 b c^5 x^5+768 b c^7 x^7-256 b c^9 x^9-6 (a+b \arccos (c x)) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )+24 (a+b \arccos (c x)) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-21 a \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )-21 b \arccos (c x) \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+9 a \operatorname {CosIntegral}\left (9 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {9 a}{b}\right )+9 b \arccos (c x) \operatorname {CosIntegral}\left (9 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {9 a}{b}\right )+6 a \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+6 b \arccos (c x) \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )-24 a \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )-24 b \arccos (c x) \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )+21 a \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right )+21 b \arccos (c x) \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right )-9 a \cos \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\arccos (c x)\right )\right )-9 b \arccos (c x) \cos \left (\frac {9 a}{b}\right ) \text {Si}\left (9 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{256 b^2 c^4 (a+b \arccos (c x))} \] Input:

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

Output:

(256*b*c^3*x^3 - 768*b*c^5*x^5 + 768*b*c^7*x^7 - 256*b*c^9*x^9 - 6*(a + b* 
ArcCos[c*x])*CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b] + 24*(a + b*ArcCos[c* 
x])*CosIntegral[3*(a/b + ArcCos[c*x])]*Sin[(3*a)/b] - 21*a*CosIntegral[7*( 
a/b + ArcCos[c*x])]*Sin[(7*a)/b] - 21*b*ArcCos[c*x]*CosIntegral[7*(a/b + A 
rcCos[c*x])]*Sin[(7*a)/b] + 9*a*CosIntegral[9*(a/b + ArcCos[c*x])]*Sin[(9* 
a)/b] + 9*b*ArcCos[c*x]*CosIntegral[9*(a/b + ArcCos[c*x])]*Sin[(9*a)/b] + 
6*a*Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]] + 6*b*ArcCos[c*x]*Cos[a/b]*Sin 
Integral[a/b + ArcCos[c*x]] - 24*a*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCo 
s[c*x])] - 24*b*ArcCos[c*x]*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x]) 
] + 21*a*Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcCos[c*x])] + 21*b*ArcCos[c*x 
]*Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcCos[c*x])] - 9*a*Cos[(9*a)/b]*SinIn 
tegral[9*(a/b + ArcCos[c*x])] - 9*b*ArcCos[c*x]*Cos[(9*a)/b]*SinIntegral[9 
*(a/b + ArcCos[c*x])])/(256*b^2*c^4*(a + b*ArcCos[c*x]))
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5215, 5225, 25, 4906, 2009}

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.

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

\(\Big \downarrow \) 5215

\(\displaystyle -\frac {3 \int \frac {x^2 \left (1-c^2 x^2\right )^2}{a+b \arccos (c x)}dx}{b c}+\frac {9 c \int \frac {x^4 \left (1-c^2 x^2\right )^2}{a+b \arccos (c x)}dx}{b}+\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 5225

\(\displaystyle -\frac {9 \int -\frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^5\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^4}+\frac {3 \int -\frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^5\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^4}+\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {9 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^5\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^4}-\frac {3 \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right ) \sin ^5\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^4}+\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {9 \int \left (\frac {\sin \left (\frac {9 a}{b}-\frac {9 (a+b \arccos (c x))}{b}\right )}{256 (a+b \arccos (c x))}-\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arccos (c x))}{b}\right )}{256 (a+b \arccos (c x))}-\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {3 \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{128 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b^2 c^4}-\frac {3 \int \left (\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}-\frac {3 \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {5 \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{64 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b^2 c^4}+\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (-\frac {5}{64} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{64} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {3}{64} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )-\frac {1}{64} \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right )+\frac {5}{64} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {1}{64} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )-\frac {3}{64} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )+\frac {1}{64} \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )\right )}{b^2 c^4}-\frac {9 \left (-\frac {3}{128} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{64} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {1}{64} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )+\frac {1}{256} \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right )-\frac {1}{256} \sin \left (\frac {9 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {9 (a+b \arccos (c x))}{b}\right )+\frac {3}{128} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {1}{64} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )-\frac {1}{256} \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )+\frac {1}{256} \cos \left (\frac {9 a}{b}\right ) \text {Si}\left (\frac {9 (a+b \arccos (c x))}{b}\right )\right )}{b^2 c^4}+\frac {x^3 \left (1-c^2 x^2\right )^3}{b c (a+b \arccos (c x))}\)

Input:

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

Output:

(x^3*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcCos[c*x])) + (3*((-5*CosIntegral[(a + 
 b*ArcCos[c*x])/b]*Sin[a/b])/64 - (CosIntegral[(3*(a + b*ArcCos[c*x]))/b]* 
Sin[(3*a)/b])/64 + (3*CosIntegral[(5*(a + b*ArcCos[c*x]))/b]*Sin[(5*a)/b]) 
/64 - (CosIntegral[(7*(a + b*ArcCos[c*x]))/b]*Sin[(7*a)/b])/64 + (5*Cos[a/ 
b]*SinIntegral[(a + b*ArcCos[c*x])/b])/64 + (Cos[(3*a)/b]*SinIntegral[(3*( 
a + b*ArcCos[c*x]))/b])/64 - (3*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcCos[ 
c*x]))/b])/64 + (Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcCos[c*x]))/b])/64)) 
/(b^2*c^4) - (9*((-3*CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/128 - (C 
osIntegral[(3*(a + b*ArcCos[c*x]))/b]*Sin[(3*a)/b])/64 + (CosIntegral[(5*( 
a + b*ArcCos[c*x]))/b]*Sin[(5*a)/b])/64 + (CosIntegral[(7*(a + b*ArcCos[c* 
x]))/b]*Sin[(7*a)/b])/256 - (CosIntegral[(9*(a + b*ArcCos[c*x]))/b]*Sin[(9 
*a)/b])/256 + (3*Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/128 + (Cos[( 
3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/64 - (Cos[(5*a)/b]*SinInte 
gral[(5*(a + b*ArcCos[c*x]))/b])/64 - (Cos[(7*a)/b]*SinIntegral[(7*(a + b* 
ArcCos[c*x]))/b])/256 + (Cos[(9*a)/b]*SinIntegral[(9*(a + b*ArcCos[c*x]))/ 
b])/256))/(b^2*c^4)
 

Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5215

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(f*x)^m)*Sqrt[1 - c^2*x^2]*(d + e*x^2) 
^p*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f*( 
n + 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] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 
 1, 0] && IGtQ[m, -3]
 

rule 5225

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.64

method	result	size
default	\(\frac {6 \arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -6 \arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -24 \arccos \left (c x \right ) \operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +24 \arccos \left (c x \right ) \operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +21 \arccos \left (c x \right ) \operatorname {Si}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b -21 \arccos \left (c x \right ) \operatorname {Ci}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b -9 \arccos \left (c x \right ) \operatorname {Si}\left (9 \arccos \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right ) b +9 \arccos \left (c x \right ) \operatorname {Ci}\left (9 \arccos \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right ) b +6 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -6 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -24 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +24 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +21 \,\operatorname {Si}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -21 \,\operatorname {Ci}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a -9 \,\operatorname {Si}\left (9 \arccos \left (c x \right )+\frac {9 a}{b}\right ) \cos \left (\frac {9 a}{b}\right ) a +9 \,\operatorname {Ci}\left (9 \arccos \left (c x \right )+\frac {9 a}{b}\right ) \sin \left (\frac {9 a}{b}\right ) a +6 c x b -8 \cos \left (3 \arccos \left (c x \right )\right ) b +3 \cos \left (7 \arccos \left (c x \right )\right ) b -\cos \left (9 \arccos \left (c x \right )\right ) b}{256 c^{4} \left (a +b \arccos \left (c x \right )\right ) b^{2}}\)	\(455\)

Input:

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)
 

Output:

1/256/c^4*(6*arccos(c*x)*Si(arccos(c*x)+a/b)*cos(a/b)*b-6*arccos(c*x)*Ci(a 
rccos(c*x)+a/b)*sin(a/b)*b-24*arccos(c*x)*Si(3*arccos(c*x)+3*a/b)*cos(3*a/ 
b)*b+24*arccos(c*x)*Ci(3*arccos(c*x)+3*a/b)*sin(3*a/b)*b+21*arccos(c*x)*Si 
(7*arccos(c*x)+7*a/b)*cos(7*a/b)*b-21*arccos(c*x)*Ci(7*arccos(c*x)+7*a/b)* 
sin(7*a/b)*b-9*arccos(c*x)*Si(9*arccos(c*x)+9*a/b)*cos(9*a/b)*b+9*arccos(c 
*x)*Ci(9*arccos(c*x)+9*a/b)*sin(9*a/b)*b+6*Si(arccos(c*x)+a/b)*cos(a/b)*a- 
6*Ci(arccos(c*x)+a/b)*sin(a/b)*a-24*Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)*a+2 
4*Ci(3*arccos(c*x)+3*a/b)*sin(3*a/b)*a+21*Si(7*arccos(c*x)+7*a/b)*cos(7*a/ 
b)*a-21*Ci(7*arccos(c*x)+7*a/b)*sin(7*a/b)*a-9*Si(9*arccos(c*x)+9*a/b)*cos 
(9*a/b)*a+9*Ci(9*arccos(c*x)+9*a/b)*sin(9*a/b)*a+6*c*x*b-8*cos(3*arccos(c* 
x))*b+3*cos(7*arccos(c*x))*b-cos(9*arccos(c*x))*b)/(a+b*arccos(c*x))/b^2
 

Fricas [F]

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

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arccos(c*x))^2,x, algorithm="fricas" 
)
 

Output:

integral((c^4*x^7 - 2*c^2*x^5 + x^3)*sqrt(-c^2*x^2 + 1)/(b^2*arccos(c*x)^2 
 + 2*a*b*arccos(c*x) + a^2), x)
 

Sympy [F]

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

integrate(x**3*(-c**2*x**2+1)**(5/2)/(a+b*acos(c*x))**2,x)
 

Output:

Integral(x**3*(-(c*x - 1)*(c*x + 1))**(5/2)/(a + b*acos(c*x))**2, x)
 

Maxima [F]

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

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arccos(c*x))^2,x, algorithm="maxima" 
)
 

Output:

-(c^6*x^9 - 3*c^4*x^7 + 3*c^2*x^5 - x^3 - (b^2*c*arctan2(sqrt(c*x + 1)*sqr 
t(-c*x + 1), c*x) + a*b*c)*integrate(3*(3*c^6*x^8 - 7*c^4*x^6 + 5*c^2*x^4 
- x^2)/(b^2*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c), x))/(b^ 
2*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2528 vs. \(2 (260) = 520\).

Time = 0.26 (sec) , antiderivative size = 2528, normalized size of antiderivative = 9.09 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arccos (c x))^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(-c^2*x^2+1)^(5/2)/(a+b*arccos(c*x))^2,x, algorithm="giac")
 

Output:

-b*c^9*x^9/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 3*b*c^7*x^7/(b^3*c^4*arccos 
(c*x) + a*b^2*c^4) + 9*b*arccos(c*x)*cos(a/b)^8*cos_integral(9*a/b + 9*arc 
cos(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 9*b*arccos(c*x)*cos 
(a/b)^9*sin_integral(9*a/b + 9*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c 
^4) - 3*b*c^5*x^5/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 9*a*cos(a/b)^8*cos_i 
ntegral(9*a/b + 9*arccos(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) 
- 9*a*cos(a/b)^9*sin_integral(9*a/b + 9*arccos(c*x))/(b^3*c^4*arccos(c*x) 
+ a*b^2*c^4) - 63/4*b*arccos(c*x)*cos(a/b)^6*cos_integral(9*a/b + 9*arccos 
(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 21/4*b*arccos(c*x)*cos 
(a/b)^6*cos_integral(7*a/b + 7*arccos(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x) 
+ a*b^2*c^4) + 81/4*b*arccos(c*x)*cos(a/b)^7*sin_integral(9*a/b + 9*arccos 
(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 21/4*b*arccos(c*x)*cos(a/b)^7*s 
in_integral(7*a/b + 7*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 63/ 
4*a*cos(a/b)^6*cos_integral(9*a/b + 9*arccos(c*x))*sin(a/b)/(b^3*c^4*arcco 
s(c*x) + a*b^2*c^4) - 21/4*a*cos(a/b)^6*cos_integral(7*a/b + 7*arccos(c*x) 
)*sin(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 81/4*a*cos(a/b)^7*sin_integ 
ral(9*a/b + 9*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 21/4*a*cos( 
a/b)^7*sin_integral(7*a/b + 7*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^ 
4) + 135/16*b*arccos(c*x)*cos(a/b)^4*cos_integral(9*a/b + 9*arccos(c*x))*s 
in(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 105/16*b*arccos(c*x)*cos(a/...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arccos (c x))^2} \, dx=\int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \] Input:

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*acos(c*x))^2,x)
 

Output:

int((x^3*(1 - c^2*x^2)^(5/2))/(a + b*acos(c*x))^2, x)
 

Reduce [F]

\[ \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{(a+b \arccos (c x))^2} \, dx=\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{7}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{5}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {-c^{2} x^{2}+1}\, x^{3}}{\mathit {acos} \left (c x \right )^{2} b^{2}+2 \mathit {acos} \left (c x \right ) a b +a^{2}}d x \] Input:

int(x^3*(-c^2*x^2+1)^(5/2)/(a+b*acos(c*x))^2,x)
 

Output:

int((sqrt( - c**2*x**2 + 1)*x**7)/(acos(c*x)**2*b**2 + 2*acos(c*x)*a*b + a 
**2),x)*c**4 - 2*int((sqrt( - c**2*x**2 + 1)*x**5)/(acos(c*x)**2*b**2 + 2* 
acos(c*x)*a*b + a**2),x)*c**2 + int((sqrt( - c**2*x**2 + 1)*x**3)/(acos(c* 
x)**2*b**2 + 2*acos(c*x)*a*b + a**2),x)