\(\int \frac {x^3 (1-c^2 x^2)^{3/2}}{(a+b \arccos (c x))^2} \, dx\) [392]

Optimal result

Integrand size = 28, antiderivative size = 278 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{(a+b \arccos (c x))^2} \, dx=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c (a+b \arccos (c x))}+\frac {3 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )}{64 b^2 c^4} \] Output:

-x^3*(-c^2*x^2+1)^2/b/c/(a+b*arccos(c*x))+3/64*cos(a/b)*Ci((a+b*arccos(c*x 
))/b)/b^2/c^4+9/64*cos(3*a/b)*Ci(3*(a+b*arccos(c*x))/b)/b^2/c^4-5/64*cos(5 
*a/b)*Ci(5*(a+b*arccos(c*x))/b)/b^2/c^4-7/64*cos(7*a/b)*Ci(7*(a+b*arccos(c 
*x))/b)/b^2/c^4+3/64*sin(a/b)*Si((a+b*arccos(c*x))/b)/b^2/c^4+9/64*sin(3*a 
/b)*Si(3*(a+b*arccos(c*x))/b)/b^2/c^4-5/64*sin(5*a/b)*Si(5*(a+b*arccos(c*x 
))/b)/b^2/c^4-7/64*sin(7*a/b)*Si(7*(a+b*arccos(c*x))/b)/b^2/c^4
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.44 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{(a+b \arccos (c x))^2} \, dx=\frac {64 b c^3 x^3-128 b c^5 x^5+64 b c^7 x^7-3 (a+b \arccos (c x)) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )+9 (a+b \arccos (c x)) \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )+5 a \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )+5 b \arccos (c x) \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-7 a \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )-7 b \arccos (c x) \operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+3 a \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+3 b \arccos (c x) \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )-9 a \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )-9 b \arccos (c x) \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )-5 a \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right )-5 b \arccos (c x) \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arccos (c x)\right )\right )+7 a \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right )+7 b \arccos (c x) \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{64 b^2 c^4 (a+b \arccos (c x))} \] Input:

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

Output:

(64*b*c^3*x^3 - 128*b*c^5*x^5 + 64*b*c^7*x^7 - 3*(a + b*ArcCos[c*x])*CosIn 
tegral[a/b + ArcCos[c*x]]*Sin[a/b] + 9*(a + b*ArcCos[c*x])*CosIntegral[3*( 
a/b + ArcCos[c*x])]*Sin[(3*a)/b] + 5*a*CosIntegral[5*(a/b + ArcCos[c*x])]* 
Sin[(5*a)/b] + 5*b*ArcCos[c*x]*CosIntegral[5*(a/b + ArcCos[c*x])]*Sin[(5*a 
)/b] - 7*a*CosIntegral[7*(a/b + ArcCos[c*x])]*Sin[(7*a)/b] - 7*b*ArcCos[c* 
x]*CosIntegral[7*(a/b + ArcCos[c*x])]*Sin[(7*a)/b] + 3*a*Cos[a/b]*SinInteg 
ral[a/b + ArcCos[c*x]] + 3*b*ArcCos[c*x]*Cos[a/b]*SinIntegral[a/b + ArcCos 
[c*x]] - 9*a*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])] - 9*b*ArcCos[ 
c*x]*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])] - 5*a*Cos[(5*a)/b]*Si 
nIntegral[5*(a/b + ArcCos[c*x])] - 5*b*ArcCos[c*x]*Cos[(5*a)/b]*SinIntegra 
l[5*(a/b + ArcCos[c*x])] + 7*a*Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcCos[c* 
x])] + 7*b*ArcCos[c*x]*Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcCos[c*x])])/(6 
4*b^2*c^4*(a + b*ArcCos[c*x]))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.41, 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.

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

\(\Big \downarrow \) 5215

\(\displaystyle -\frac {3 \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \arccos (c x)}dx}{b c}+\frac {7 c \int \frac {x^4 \left (1-c^2 x^2\right )}{a+b \arccos (c x)}dx}{b}+\frac {x^3 \left (1-c^2 x^2\right )^2}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 5225

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4906

\(\displaystyle \frac {7 \int \left (-\frac {\sin \left (\frac {7 a}{b}-\frac {7 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}-\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {3 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{64 (a+b \arccos (c x))}+\frac {3 \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{64 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b^2 c^4}-\frac {3 \int \left (-\frac {\sin \left (\frac {5 a}{b}-\frac {5 (a+b \arccos (c x))}{b}\right )}{16 (a+b \arccos (c x))}+\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{16 (a+b \arccos (c x))}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{8 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b^2 c^4}+\frac {x^3 \left (1-c^2 x^2\right )^2}{b c (a+b \arccos (c x))}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (-\frac {1}{8} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{16} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )+\frac {1}{8} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {1}{16} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )-\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )\right )}{b^2 c^4}-\frac {7 \left (-\frac {3}{64} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {3}{64} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {1}{64} \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arccos (c x))}{b}\right )+\frac {1}{64} \sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arccos (c x))}{b}\right )+\frac {3}{64} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {3}{64} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arccos (c x))}{b}\right )-\frac {1}{64} \cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arccos (c x))}{b}\right )\right )}{b^2 c^4}+\frac {x^3 \left (1-c^2 x^2\right )^2}{b c (a+b \arccos (c x))}\)

Input:

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

Output:

(x^3*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcCos[c*x])) + (3*(-1/8*(CosIntegral[(a 
 + b*ArcCos[c*x])/b]*Sin[a/b]) - (CosIntegral[(3*(a + b*ArcCos[c*x]))/b]*S 
in[(3*a)/b])/16 + (CosIntegral[(5*(a + b*ArcCos[c*x]))/b]*Sin[(5*a)/b])/16 
 + (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/8 + (Cos[(3*a)/b]*SinInte 
gral[(3*(a + b*ArcCos[c*x]))/b])/16 - (Cos[(5*a)/b]*SinIntegral[(5*(a + b* 
ArcCos[c*x]))/b])/16))/(b^2*c^4) - (7*((-3*CosIntegral[(a + b*ArcCos[c*x]) 
/b]*Sin[a/b])/64 - (3*CosIntegral[(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 + (CosInteg 
ral[(7*(a + b*ArcCos[c*x]))/b]*Sin[(7*a)/b])/64 + (3*Cos[a/b]*SinIntegral[ 
(a + b*ArcCos[c*x])/b])/64 + (3*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcCos[ 
c*x]))/b])/64 - (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)
 

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 = 454, normalized size of antiderivative = 1.63

method	result	size
default	\(\frac {3 \arccos \left (c x \right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -3 \arccos \left (c x \right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -5 \arccos \left (c x \right ) \operatorname {Si}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +5 \arccos \left (c x \right ) \operatorname {Ci}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b -9 \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 +9 \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 +7 \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 -7 \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 +3 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -3 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -5 \,\operatorname {Si}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a +5 \,\operatorname {Ci}\left (5 \arccos \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a -9 \,\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +9 \,\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +7 \,\operatorname {Si}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -7 \,\operatorname {Ci}\left (7 \arccos \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +3 c x b -\cos \left (5 \arccos \left (c x \right )\right ) b -3 \cos \left (3 \arccos \left (c x \right )\right ) b +\cos \left (7 \arccos \left (c x \right )\right ) b}{64 c^{4} \left (a +b \arccos \left (c x \right )\right ) b^{2}}\)	\(454\)

Input:

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

Output:

1/64/c^4*(3*arccos(c*x)*Si(arccos(c*x)+a/b)*cos(a/b)*b-3*arccos(c*x)*Ci(ar 
ccos(c*x)+a/b)*sin(a/b)*b-5*arccos(c*x)*Si(5*arccos(c*x)+5*a/b)*cos(5*a/b) 
*b+5*arccos(c*x)*Ci(5*arccos(c*x)+5*a/b)*sin(5*a/b)*b-9*arccos(c*x)*Si(3*a 
rccos(c*x)+3*a/b)*cos(3*a/b)*b+9*arccos(c*x)*Ci(3*arccos(c*x)+3*a/b)*sin(3 
*a/b)*b+7*arccos(c*x)*Si(7*arccos(c*x)+7*a/b)*cos(7*a/b)*b-7*arccos(c*x)*C 
i(7*arccos(c*x)+7*a/b)*sin(7*a/b)*b+3*Si(arccos(c*x)+a/b)*cos(a/b)*a-3*Ci( 
arccos(c*x)+a/b)*sin(a/b)*a-5*Si(5*arccos(c*x)+5*a/b)*cos(5*a/b)*a+5*Ci(5* 
arccos(c*x)+5*a/b)*sin(5*a/b)*a-9*Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)*a+9*C 
i(3*arccos(c*x)+3*a/b)*sin(3*a/b)*a+7*Si(7*arccos(c*x)+7*a/b)*cos(7*a/b)*a 
-7*Ci(7*arccos(c*x)+7*a/b)*sin(7*a/b)*a+3*c*x*b-cos(5*arccos(c*x))*b-3*cos 
(3*arccos(c*x))*b+cos(7*arccos(c*x))*b)/(a+b*arccos(c*x))/b^2
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

(c^4*x^7 - 2*c^2*x^5 + x^3 - (b^2*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), 
c*x) + a*b*c)*integrate((7*c^4*x^6 - 10*c^2*x^4 + 3*x^2)/(b^2*c*arctan2(sq 
rt(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. 2080 vs. \(2 (261) = 522\).

Time = 0.26 (sec) , antiderivative size = 2080, normalized size of antiderivative = 7.48 \[ \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{(a+b \arccos (c x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

b*c^7*x^7/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 2*b*c^5*x^5/(b^3*c^4*arccos( 
c*x) + a*b^2*c^4) - 7*b*arccos(c*x)*cos(a/b)^6*cos_integral(7*a/b + 7*arcc 
os(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 7*b*arccos(c*x)*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) - 7*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) + 7*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) + 35/4*b*arccos(c*x)*cos(a/b)^4*cos_ 
integral(7*a/b + 7*arccos(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) 
 + 5/4*b*arccos(c*x)*cos(a/b)^4*cos_integral(5*a/b + 5*arccos(c*x))*sin(a/ 
b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 49/4*b*arccos(c*x)*cos(a/b)^5*sin_i 
ntegral(7*a/b + 7*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 5/4*b*a 
rccos(c*x)*cos(a/b)^5*sin_integral(5*a/b + 5*arccos(c*x))/(b^3*c^4*arccos( 
c*x) + a*b^2*c^4) + b*c^3*x^3/(b^3*c^4*arccos(c*x) + a*b^2*c^4) + 35/4*a*c 
os(a/b)^4*cos_integral(7*a/b + 7*arccos(c*x))*sin(a/b)/(b^3*c^4*arccos(c*x 
) + a*b^2*c^4) + 5/4*a*cos(a/b)^4*cos_integral(5*a/b + 5*arccos(c*x))*sin( 
a/b)/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 49/4*a*cos(a/b)^5*sin_integral(7* 
a/b + 7*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 5/4*a*cos(a/b)^5* 
sin_integral(5*a/b + 5*arccos(c*x))/(b^3*c^4*arccos(c*x) + a*b^2*c^4) - 21 
/8*b*arccos(c*x)*cos(a/b)^2*cos_integral(7*a/b + 7*arccos(c*x))*sin(a/b)/( 
b^3*c^4*arccos(c*x) + a*b^2*c^4) - 15/16*b*arccos(c*x)*cos(a/b)^2*cos_i...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

 - 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)