\(\int \frac {x^3}{\arccos (a x)^3} \, dx\) [61]

Optimal result

Integrand size = 10, antiderivative size = 83 \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}-\frac {3 x^2}{2 a^2 \arccos (a x)}+\frac {2 x^4}{\arccos (a x)}+\frac {\text {Si}(2 \arccos (a x))}{2 a^4}+\frac {\text {Si}(4 \arccos (a x))}{a^4} \] Output:

1/2*x^3*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^2-3/2*x^2/a^2/arccos(a*x)+2*x^4/a 
rccos(a*x)+1/2*Si(2*arccos(a*x))/a^4+Si(4*arccos(a*x))/a^4
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {\frac {a^2 x^2 \left (a x \sqrt {1-a^2 x^2}+\left (-3+4 a^2 x^2\right ) \arccos (a x)\right )}{\arccos (a x)^2}+\text {Si}(2 \arccos (a x))+2 \text {Si}(4 \arccos (a x))}{2 a^4} \] Input:

Integrate[x^3/ArcCos[a*x]^3,x]
 

Output:

((a^2*x^2*(a*x*Sqrt[1 - a^2*x^2] + (-3 + 4*a^2*x^2)*ArcCos[a*x]))/ArcCos[a 
*x]^2 + SinIntegral[2*ArcCos[a*x]] + 2*SinIntegral[4*ArcCos[a*x]])/(2*a^4)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5145, 5223, 5147, 4906, 27, 2009, 3042, 3780}

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.

\(\displaystyle \int \frac {x^3}{\arccos (a x)^3} \, dx\)

\(\Big \downarrow \) 5145

\(\displaystyle -\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx}{2 a}+2 a \int \frac {x^4}{\sqrt {1-a^2 x^2} \arccos (a x)^2}dx+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\)

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5147

\(\displaystyle -\frac {3 \left (\frac {2 \int \frac {a x \sqrt {1-a^2 x^2}}{\arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )}{2 a}+2 a \left (\frac {4 \int \frac {a^3 x^3 \sqrt {1-a^2 x^2}}{\arccos (a x)}d\arccos (a x)}{a^5}+\frac {x^4}{a \arccos (a x)}\right )+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\)

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \left (\frac {\int \frac {\sin (2 \arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )}{2 a}+2 a \left (\frac {4 \left (\frac {1}{4} \text {Si}(2 \arccos (a x))+\frac {1}{8} \text {Si}(4 \arccos (a x))\right )}{a^5}+\frac {x^4}{a \arccos (a x)}\right )+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (\frac {\int \frac {\sin (2 \arccos (a x))}{\arccos (a x)}d\arccos (a x)}{a^3}+\frac {x^2}{a \arccos (a x)}\right )}{2 a}+2 a \left (\frac {4 \left (\frac {1}{4} \text {Si}(2 \arccos (a x))+\frac {1}{8} \text {Si}(4 \arccos (a x))\right )}{a^5}+\frac {x^4}{a \arccos (a x)}\right )+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\)

\(\Big \downarrow \) 3780

\(\displaystyle 2 a \left (\frac {4 \left (\frac {1}{4} \text {Si}(2 \arccos (a x))+\frac {1}{8} \text {Si}(4 \arccos (a x))\right )}{a^5}+\frac {x^4}{a \arccos (a x)}\right )-\frac {3 \left (\frac {\text {Si}(2 \arccos (a x))}{a^3}+\frac {x^2}{a \arccos (a x)}\right )}{2 a}+\frac {x^3 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)^2}\)

Input:

Int[x^3/ArcCos[a*x]^3,x]
 

Output:

(x^3*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - (3*(x^2/(a*ArcCos[a*x]) + Si 
nIntegral[2*ArcCos[a*x]]/a^3))/(2*a) + 2*a*(x^4/(a*ArcCos[a*x]) + (4*(SinI 
ntegral[2*ArcCos[a*x]]/4 + SinIntegral[4*ArcCos[a*x]]/8))/a^5)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

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 5145

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[( 
-x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] + ( 
-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n + 1)/ 
Sqrt[1 - c^2*x^2]), x], x] + Simp[m/(b*c*(n + 1))   Int[x^(m - 1)*((a + b*A 
rcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && I 
GtQ[m, 0] && LtQ[n, -2]
 

rule 5147

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

rule 5223

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\operatorname {Si}\left (4 \arccos \left (a x \right )\right )}{a^{4}}\)	\(82\)
default	\(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}+\frac {\cos \left (2 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\frac {\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{2}}+\frac {\cos \left (4 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}+\operatorname {Si}\left (4 \arccos \left (a x \right )\right )}{a^{4}}\)	\(82\)

Input:

int(x^3/arccos(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

1/a^4*(1/8*sin(2*arccos(a*x))/arccos(a*x)^2+1/4*cos(2*arccos(a*x))/arccos( 
a*x)+1/2*Si(2*arccos(a*x))+1/16/arccos(a*x)^2*sin(4*arccos(a*x))+1/4*cos(4 
*arccos(a*x))/arccos(a*x)+Si(4*arccos(a*x)))
 

Fricas [F]

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

integrate(x^3/arccos(a*x)^3,x, algorithm="fricas")
 

Output:

integral(x^3/arccos(a*x)^3, x)
 

Sympy [F]

\[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\int \frac {x^{3}}{\operatorname {acos}^{3}{\left (a x \right )}}\, dx \] Input:

integrate(x**3/acos(a*x)**3,x)
 

Output:

Integral(x**3/acos(a*x)**3, x)
 

Maxima [F]

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

integrate(x^3/arccos(a*x)^3,x, algorithm="maxima")
 

Output:

1/2*(sqrt(a*x + 1)*sqrt(-a*x + 1)*a*x^3 - 2*arctan2(sqrt(a*x + 1)*sqrt(-a* 
x + 1), a*x)^2*integrate((8*a^2*x^3 - 3*x)/arctan2(sqrt(a*x + 1)*sqrt(-a*x 
 + 1), a*x), x) + (4*a^2*x^4 - 3*x^2)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1) 
, a*x))/(a^2*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\arccos (a x)^3} \, dx=\frac {2 \, x^{4}}{\arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{2 \, a \arccos \left (a x\right )^{2}} - \frac {3 \, x^{2}}{2 \, a^{2} \arccos \left (a x\right )} + \frac {\operatorname {Si}\left (4 \, \arccos \left (a x\right )\right )}{a^{4}} + \frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} \] Input:

integrate(x^3/arccos(a*x)^3,x, algorithm="giac")
 

Output:

2*x^4/arccos(a*x) + 1/2*sqrt(-a^2*x^2 + 1)*x^3/(a*arccos(a*x)^2) - 3/2*x^2 
/(a^2*arccos(a*x)) + sin_integral(4*arccos(a*x))/a^4 + 1/2*sin_integral(2* 
arccos(a*x))/a^4
 

Mupad [F(-1)]

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

int(x^3/acos(a*x)^3,x)
 

Output:

int(x^3/acos(a*x)^3, x)
 

Reduce [F]

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

int(x^3/acos(a*x)^3,x)
 

Output:

int(x**3/acos(a*x)**3,x)