\(\int \frac {(c-a^2 c x^2)^3}{\arccos (a x)} \, dx\) [313]

Optimal result

Integrand size = 20, antiderivative size = 67 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\frac {35 c^3 \operatorname {CosIntegral}(\arccos (a x))}{64 a}+\frac {21 c^3 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a}+\frac {7 c^3 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a}+\frac {c^3 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a} \] Output:

35/64*c^3*Ci(arccos(a*x))/a+21/64*c^3*Ci(3*arccos(a*x))/a+7/64*c^3*Ci(5*ar 
ccos(a*x))/a+1/64*c^3*Ci(7*arccos(a*x))/a
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\frac {c^3 (-35 \text {Si}(\arccos (a x))+21 \text {Si}(3 \arccos (a x))-7 \text {Si}(5 \arccos (a x))+\text {Si}(7 \arccos (a x)))}{64 a} \] Input:

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

Output:

(c^3*(-35*SinIntegral[ArcCos[a*x]] + 21*SinIntegral[3*ArcCos[a*x]] - 7*Sin 
Integral[5*ArcCos[a*x]] + SinIntegral[7*ArcCos[a*x]]))/(64*a)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5169, 3042, 3793, 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.

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

\(\Big \downarrow \) 5169

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c^3 \int \frac {\sin (\arccos (a x))^7}{\arccos (a x)}d\arccos (a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {c^3 \int \left (-\frac {21 \sin (3 \arccos (a x))}{64 \arccos (a x)}+\frac {7 \sin (5 \arccos (a x))}{64 \arccos (a x)}-\frac {\sin (7 \arccos (a x))}{64 \arccos (a x)}+\frac {35 \sqrt {1-a^2 x^2}}{64 \arccos (a x)}\right )d\arccos (a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c^3 \left (\frac {35}{64} \text {Si}(\arccos (a x))-\frac {21}{64} \text {Si}(3 \arccos (a x))+\frac {7}{64} \text {Si}(5 \arccos (a x))-\frac {1}{64} \text {Si}(7 \arccos (a x))\right )}{a}\)

Input:

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

Output:

-((c^3*((35*SinIntegral[ArcCos[a*x]])/64 - (21*SinIntegral[3*ArcCos[a*x]]) 
/64 + (7*SinIntegral[5*ArcCos[a*x]])/64 - SinIntegral[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5169

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[(-(b*c)^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Subst[ 
Int[x^n*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, 0]
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {c^{3} \left (21 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )-7 \,\operatorname {Si}\left (5 \arccos \left (a x \right )\right )+\operatorname {Si}\left (7 \arccos \left (a x \right )\right )-35 \,\operatorname {Si}\left (\arccos \left (a x \right )\right )\right )}{64 a}\)	\(42\)
default	\(\frac {c^{3} \left (21 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )-7 \,\operatorname {Si}\left (5 \arccos \left (a x \right )\right )+\operatorname {Si}\left (7 \arccos \left (a x \right )\right )-35 \,\operatorname {Si}\left (\arccos \left (a x \right )\right )\right )}{64 a}\)	\(42\)

Input:

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

Output:

1/64/a*c^3*(21*Si(3*arccos(a*x))-7*Si(5*arccos(a*x))+Si(7*arccos(a*x))-35* 
Si(arccos(a*x)))
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-integrate((a^2*c*x^2 - c)^3/arccos(a*x), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\frac {c^{3} \operatorname {Si}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a} - \frac {7 \, c^{3} \operatorname {Si}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a} + \frac {21 \, c^{3} \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (\arccos \left (a x\right )\right )}{64 \, a} \] Input:

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

Output:

1/64*c^3*sin_integral(7*arccos(a*x))/a - 7/64*c^3*sin_integral(5*arccos(a* 
x))/a + 21/64*c^3*sin_integral(3*arccos(a*x))/a - 35/64*c^3*sin_integral(a 
rccos(a*x))/a
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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