\(\int \frac {(c-a^2 c x^2)^3}{\arccos (a x)^2} \, dx\) [376]

Optimal result

Integrand size = 20, antiderivative size = 95 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)^2} \, dx=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \arccos (a x)}-\frac {35 c^3 \text {Si}(\arccos (a x))}{64 a}-\frac {63 c^3 \text {Si}(3 \arccos (a x))}{64 a}-\frac {35 c^3 \text {Si}(5 \arccos (a x))}{64 a}-\frac {7 c^3 \text {Si}(7 \arccos (a x))}{64 a} \] Output:

-c^3*(-a^2*x^2+1)^(7/2)/a/arccos(a*x)-35/64*c^3*Si(arccos(a*x))/a-63/64*c^ 
3*Si(3*arccos(a*x))/a-35/64*c^3*Si(5*arccos(a*x))/a-7/64*c^3*Si(7*arccos(a 
*x))/a
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)^2} \, dx=\frac {c^3 \left (64 \left (1-a^2 x^2\right )^{7/2}-35 \arccos (a x) \operatorname {CosIntegral}(\arccos (a x))+63 \arccos (a x) \operatorname {CosIntegral}(3 \arccos (a x))-35 \arccos (a x) \operatorname {CosIntegral}(5 \arccos (a x))+7 \arccos (a x) \operatorname {CosIntegral}(7 \arccos (a x))\right )}{64 a \arccos (a x)} \] Input:

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

Output:

(c^3*(64*(1 - a^2*x^2)^(7/2) - 35*ArcCos[a*x]*CosIntegral[ArcCos[a*x]] + 6 
3*ArcCos[a*x]*CosIntegral[3*ArcCos[a*x]] - 35*ArcCos[a*x]*CosIntegral[5*Ar 
cCos[a*x]] + 7*ArcCos[a*x]*CosIntegral[7*ArcCos[a*x]]))/(64*a*ArcCos[a*x])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5167, 5225, 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.

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

\(\Big \downarrow \) 5167

\(\displaystyle 7 a c^3 \int \frac {x \left (1-a^2 x^2\right )^{5/2}}{\arccos (a x)}dx+\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \arccos (a x)}\)

\(\Big \downarrow \) 5225

\(\displaystyle \frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \arccos (a x)}-\frac {7 c^3 \int \frac {a x \left (1-a^2 x^2\right )^3}{\arccos (a x)}d\arccos (a x)}{a}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \arccos (a x)}-\frac {7 c^3 \int \left (\frac {5 a x}{64 \arccos (a x)}-\frac {9 \cos (3 \arccos (a x))}{64 \arccos (a x)}+\frac {5 \cos (5 \arccos (a x))}{64 \arccos (a x)}-\frac {\cos (7 \arccos (a x))}{64 \arccos (a x)}\right )d\arccos (a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \arccos (a x)}-\frac {7 c^3 \left (\frac {5}{64} \operatorname {CosIntegral}(\arccos (a x))-\frac {9}{64} \operatorname {CosIntegral}(3 \arccos (a x))+\frac {5}{64} \operatorname {CosIntegral}(5 \arccos (a x))-\frac {1}{64} \operatorname {CosIntegral}(7 \arccos (a x))\right )}{a}\)

Input:

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

Output:

(c^3*(1 - a^2*x^2)^(7/2))/(a*ArcCos[a*x]) - (7*c^3*((5*CosIntegral[ArcCos[ 
a*x]])/64 - (9*CosIntegral[3*ArcCos[a*x]])/64 + (5*CosIntegral[5*ArcCos[a* 
x]])/64 - CosIntegral[7*ArcCos[a*x]]/64))/a
 

Defintions of rubi rules used

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 5167

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[(-Sqrt[1 - c^2*x^2])*(d + e*x^2)^p*((a + b*ArcCos[c*x])^(n 
+ 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 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, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, - 
1]
 

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 = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {c^{3} \left (63 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-35 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right ) \arccos \left (a x \right )+7 \,\operatorname {Ci}\left (7 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-35 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right ) \arccos \left (a x \right )-21 \sin \left (3 \arccos \left (a x \right )\right )+7 \sin \left (5 \arccos \left (a x \right )\right )-\sin \left (7 \arccos \left (a x \right )\right )+35 \sqrt {-a^{2} x^{2}+1}\right )}{64 a \arccos \left (a x \right )}\)	\(107\)
default	\(\frac {c^{3} \left (63 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-35 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right ) \arccos \left (a x \right )+7 \,\operatorname {Ci}\left (7 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-35 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right ) \arccos \left (a x \right )-21 \sin \left (3 \arccos \left (a x \right )\right )+7 \sin \left (5 \arccos \left (a x \right )\right )-\sin \left (7 \arccos \left (a x \right )\right )+35 \sqrt {-a^{2} x^{2}+1}\right )}{64 a \arccos \left (a x \right )}\)	\(107\)

Input:

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

Output:

1/64/a*c^3*(63*Ci(3*arccos(a*x))*arccos(a*x)-35*Ci(5*arccos(a*x))*arccos(a 
*x)+7*Ci(7*arccos(a*x))*arccos(a*x)-35*Ci(arccos(a*x))*arccos(a*x)-21*sin( 
3*arccos(a*x))+7*sin(5*arccos(a*x))-sin(7*arccos(a*x))+35*(-a^2*x^2+1)^(1/ 
2))/arccos(a*x)
 

Fricas [F]

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

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

Output:

integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arccos(a*x)^ 
2, x)
 

Sympy [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)^2} \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acos}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acos}^{2}{\left (a x \right )}}\right )\, dx\right ) \] Input:

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

Output:

-c**3*(Integral(3*a**2*x**2/acos(a*x)**2, x) + Integral(-3*a**4*x**4/acos( 
a*x)**2, x) + Integral(a**6*x**6/acos(a*x)**2, x) + Integral(-1/acos(a*x)* 
*2, x))
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.78 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)^2} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} a^{5} c^{3} x^{6}}{\arccos \left (a x\right )} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{4}}{\arccos \left (a x\right )} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} a c^{3} x^{2}}{\arccos \left (a x\right )} + \frac {7 \, c^{3} \operatorname {Ci}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a} + \frac {63 \, c^{3} \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Ci}\left (\arccos \left (a x\right )\right )}{64 \, a} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a \arccos \left (a x\right )} \] Input:

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

Output:

-sqrt(-a^2*x^2 + 1)*a^5*c^3*x^6/arccos(a*x) + 3*sqrt(-a^2*x^2 + 1)*a^3*c^3 
*x^4/arccos(a*x) - 3*sqrt(-a^2*x^2 + 1)*a*c^3*x^2/arccos(a*x) + 7/64*c^3*c 
os_integral(7*arccos(a*x))/a - 35/64*c^3*cos_integral(5*arccos(a*x))/a + 6 
3/64*c^3*cos_integral(3*arccos(a*x))/a - 35/64*c^3*cos_integral(arccos(a*x 
))/a + sqrt(-a^2*x^2 + 1)*c^3/(a*arccos(a*x))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)^2} \, dx=c^{3} \left (-\left (\int \frac {x^{6}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{6}+3 \left (\int \frac {x^{4}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{4}-3 \left (\int \frac {x^{2}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{2}+\int \frac {1}{\mathit {acos} \left (a x \right )^{2}}d x \right ) \] Input:

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

Output:

c**3*( - int(x**6/acos(a*x)**2,x)*a**6 + 3*int(x**4/acos(a*x)**2,x)*a**4 - 
 3*int(x**2/acos(a*x)**2,x)*a**2 + int(1/acos(a*x)**2,x))