\(\int \frac {x^2}{\arccos (a x)^4} \, dx\) [69]

Optimal result

Integrand size = 10, antiderivative size = 141 \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {x}{3 a^2 \arccos (a x)^2}+\frac {x^3}{2 \arccos (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arccos (a x)}-\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{24 a^3}+\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{8 a^3} \] Output:

1/3*x^2*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^3-1/3*x/a^2/arccos(a*x)^2+1/2*x^3 
/arccos(a*x)^2+1/3*(-a^2*x^2+1)^(1/2)/a^3/arccos(a*x)-3/2*x^2*(-a^2*x^2+1) 
^(1/2)/a/arccos(a*x)+1/24*Ci(arccos(a*x))/a^3+9/8*Ci(3*arccos(a*x))/a^3
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {\frac {8 a^2 x^2 \sqrt {1-a^2 x^2}}{\arccos (a x)^3}+\frac {4 a x \left (-2+3 a^2 x^2\right )}{\arccos (a x)^2}-\frac {4 \sqrt {1-a^2 x^2} \left (-2+9 a^2 x^2\right )}{\arccos (a x)}-80 \operatorname {CosIntegral}(\arccos (a x))+27 (3 \operatorname {CosIntegral}(\arccos (a x))+\operatorname {CosIntegral}(3 \arccos (a x)))}{24 a^3} \] Input:

Integrate[x^2/ArcCos[a*x]^4,x]
 

Output:

((8*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcCos[a*x]^3 + (4*a*x*(-2 + 3*a^2*x^2))/Ar 
cCos[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*(-2 + 9*a^2*x^2))/ArcCos[a*x] - 80*CosI 
ntegral[ArcCos[a*x]] + 27*(3*CosIntegral[ArcCos[a*x]] + CosIntegral[3*ArcC 
os[a*x]]))/(24*a^3)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5145, 5223, 5133, 5143, 2009, 5225, 3042, 3783}

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.

\(\displaystyle \int \frac {x^2}{\arccos (a x)^4} \, dx\)

\(\Big \downarrow \) 5145

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5133

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

\(\Big \downarrow \) 5143

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5225

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

Input:

Int[x^2/ArcCos[a*x]^4,x]
 

Output:

(x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - (2*(x/(2*a*ArcCos[a*x]^2) - 
(Sqrt[1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a)/(2*a)))/( 
3*a) + a*(x^3/(2*a*ArcCos[a*x]^2) - (3*((x^2*Sqrt[1 - a^2*x^2])/(a*ArcCos[ 
a*x]) + (-1/4*CosIntegral[ArcCos[a*x]] - (3*CosIntegral[3*ArcCos[a*x]])/4) 
/a^3))/(2*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 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 5133

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

rule 5143

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] - S 
imp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Cos[- 
a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos 
[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
 

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 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]
 

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 = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arccos \left (a x \right )^{3}}+\frac {a x}{24 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{24}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{3}}+\frac {\cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{8}}{a^{3}}\)	\(117\)
default	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arccos \left (a x \right )^{3}}+\frac {a x}{24 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{24}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{12 \arccos \left (a x \right )^{3}}+\frac {\cos \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )^{2}}-\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}+\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{8}}{a^{3}}\)	\(117\)

Input:

int(x^2/arccos(a*x)^4,x,method=_RETURNVERBOSE)
 

Output:

1/a^3*(1/12/arccos(a*x)^3*(-a^2*x^2+1)^(1/2)+1/24*a*x/arccos(a*x)^2-1/24/a 
rccos(a*x)*(-a^2*x^2+1)^(1/2)+1/24*Ci(arccos(a*x))+1/12/arccos(a*x)^3*sin( 
3*arccos(a*x))+1/8*cos(3*arccos(a*x))/arccos(a*x)^2-3/8/arccos(a*x)*sin(3* 
arccos(a*x))+9/8*Ci(3*arccos(a*x)))
 

Fricas [F]

\[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{4}} \,d x } \] Input:

integrate(x^2/arccos(a*x)^4,x, algorithm="fricas")
 

Output:

integral(x^2/arccos(a*x)^4, x)
 

Sympy [F]

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

integrate(x**2/acos(a*x)**4,x)
 

Output:

Integral(x**2/acos(a*x)**4, x)
 

Maxima [F]

\[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{4}} \,d x } \] Input:

integrate(x^2/arccos(a*x)^4,x, algorithm="maxima")
 

Output:

1/6*(6*a^3*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3*integrate(1/6*(27* 
a^2*x^3 - 20*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^3*x^2 - a)*arctan2(sqrt(a 
*x + 1)*sqrt(-a*x + 1), a*x)), x) + (2*a^2*x^2 - (9*a^2*x^2 - 2)*arctan2(s 
qrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) + (3*a^3 
*x^3 - 2*a*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))/(a^3*arctan2(sqr 
t(a*x + 1)*sqrt(-a*x + 1), a*x)^3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\frac {x^{3}}{2 \, \arccos \left (a x\right )^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{2 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a \arccos \left (a x\right )^{3}} + \frac {9 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{24 \, a^{3}} - \frac {x}{3 \, a^{2} \arccos \left (a x\right )^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3} \arccos \left (a x\right )} \] Input:

integrate(x^2/arccos(a*x)^4,x, algorithm="giac")
 

Output:

1/2*x^3/arccos(a*x)^2 - 3/2*sqrt(-a^2*x^2 + 1)*x^2/(a*arccos(a*x)) + 1/3*s 
qrt(-a^2*x^2 + 1)*x^2/(a*arccos(a*x)^3) + 9/8*cos_integral(3*arccos(a*x))/ 
a^3 + 1/24*cos_integral(arccos(a*x))/a^3 - 1/3*x/(a^2*arccos(a*x)^2) + 1/3 
*sqrt(-a^2*x^2 + 1)/(a^3*arccos(a*x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\arccos (a x)^4} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^4} \,d x \] Input:

int(x^2/acos(a*x)^4,x)
 

Output:

int(x^2/acos(a*x)^4, x)
 

Reduce [F]

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

int(x^2/acos(a*x)^4,x)
 

Output:

int(x**2/acos(a*x)**4,x)