\(\int \frac {x^4}{\arccos (a x)^4} \, dx\) [67]

Optimal result

Integrand size = 10, antiderivative size = 158 \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}-\frac {2 x^3}{3 a^2 \arccos (a x)^2}+\frac {5 x^5}{6 \arccos (a x)^2}+\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arccos (a x)}-\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arccos (a x)}+\frac {\operatorname {CosIntegral}(\arccos (a x))}{48 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{32 a^5}+\frac {125 \operatorname {CosIntegral}(5 \arccos (a x))}{96 a^5} \] Output:

1/3*x^4*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^3-2/3*x^3/a^2/arccos(a*x)^2+5/6*x 
^5/arccos(a*x)^2+2*x^2*(-a^2*x^2+1)^(1/2)/a^3/arccos(a*x)-25/6*x^4*(-a^2*x 
^2+1)^(1/2)/a/arccos(a*x)+1/48*Ci(arccos(a*x))/a^5+27/32*Ci(3*arccos(a*x)) 
/a^5+125/96*Ci(5*arccos(a*x))/a^5
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {32 a^4 x^4 \sqrt {1-a^2 x^2}-64 a^3 x^3 \arccos (a x)+80 a^5 x^5 \arccos (a x)+192 a^2 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^2-400 a^4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^2+2 \arccos (a x)^3 \operatorname {CosIntegral}(\arccos (a x))+81 \arccos (a x)^3 \operatorname {CosIntegral}(3 \arccos (a x))+125 \arccos (a x)^3 \operatorname {CosIntegral}(5 \arccos (a x))}{96 a^5 \arccos (a x)^3} \] Input:

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

Output:

(32*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcCos[a*x] + 80*a^5*x^5*ArcCos 
[a*x] + 192*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2 - 400*a^4*x^4*Sqrt[1 - 
 a^2*x^2]*ArcCos[a*x]^2 + 2*ArcCos[a*x]^3*CosIntegral[ArcCos[a*x]] + 81*Ar 
cCos[a*x]^3*CosIntegral[3*ArcCos[a*x]] + 125*ArcCos[a*x]^3*CosIntegral[5*A 
rcCos[a*x]])/(96*a^5*ArcCos[a*x]^3)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5145, 5223, 5143, 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.

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

\(\Big \downarrow \) 5145

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

\(\Big \downarrow \) 5223

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

\(\Big \downarrow \) 5143

\(\displaystyle \frac {5}{3} a \left (\frac {x^5}{2 a \arccos (a x)^2}-\frac {5 \left (\frac {\int \left (-\frac {a x}{8 \arccos (a x)}-\frac {9 \cos (3 \arccos (a x))}{16 \arccos (a x)}-\frac {5 \cos (5 \arccos (a x))}{16 \arccos (a x)}\right )d\arccos (a x)}{a^5}+\frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)}\right )}{2 a}\right )-\frac {4 \left (\frac {x^3}{2 a \arccos (a x)^2}-\frac {3 \left (\frac {\int \left (-\frac {a x}{4 \arccos (a x)}-\frac {3 \cos (3 \arccos (a x))}{4 \arccos (a x)}\right )d\arccos (a x)}{a^3}+\frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)}\right )}{2 a}\right )}{3 a}+\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arccos (a x)^3}+\frac {5}{3} a \left (\frac {x^5}{2 a \arccos (a x)^2}-\frac {5 \left (\frac {-\frac {1}{8} \operatorname {CosIntegral}(\arccos (a x))-\frac {9}{16} \operatorname {CosIntegral}(3 \arccos (a x))-\frac {5}{16} \operatorname {CosIntegral}(5 \arccos (a x))}{a^5}+\frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)}\right )}{2 a}\right )-\frac {4 \left (\frac {x^3}{2 a \arccos (a x)^2}-\frac {3 \left (\frac {-\frac {1}{4} \operatorname {CosIntegral}(\arccos (a x))-\frac {3}{4} \operatorname {CosIntegral}(3 \arccos (a x))}{a^3}+\frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)}\right )}{2 a}\right )}{3 a}\)

Input:

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

Output:

(x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - (4*(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)))/(3*a) + (5*a*(x^5/( 
2*a*ArcCos[a*x]^2) - (5*((x^4*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) + (-1/8*C 
osIntegral[ArcCos[a*x]] - (9*CosIntegral[3*ArcCos[a*x]])/16 - (5*CosIntegr 
al[5*ArcCos[a*x]])/16)/a^5))/(2*a)))/3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )^{3}}+\frac {a x}{48 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{48}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{3}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{32}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{48 \arccos \left (a x \right )^{3}}+\frac {5 \cos \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )^{2}}-\frac {25 \sin \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )}+\frac {125 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{96}}{a^{5}}\)	\(171\)
default	\(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arccos \left (a x \right )^{3}}+\frac {a x}{48 \arccos \left (a x \right )^{2}}-\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arccos \left (a x \right )}+\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{48}+\frac {\sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )^{3}}+\frac {3 \cos \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{32}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{48 \arccos \left (a x \right )^{3}}+\frac {5 \cos \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )^{2}}-\frac {25 \sin \left (5 \arccos \left (a x \right )\right )}{96 \arccos \left (a x \right )}+\frac {125 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{96}}{a^{5}}\)	\(171\)

Input:

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

Output:

1/a^5*(1/24/arccos(a*x)^3*(-a^2*x^2+1)^(1/2)+1/48*a*x/arccos(a*x)^2-1/48/a 
rccos(a*x)*(-a^2*x^2+1)^(1/2)+1/48*Ci(arccos(a*x))+1/16/arccos(a*x)^3*sin( 
3*arccos(a*x))+3/32*cos(3*arccos(a*x))/arccos(a*x)^2-9/32/arccos(a*x)*sin( 
3*arccos(a*x))+27/32*Ci(3*arccos(a*x))+1/48/arccos(a*x)^3*sin(5*arccos(a*x 
))+5/96*cos(5*arccos(a*x))/arccos(a*x)^2-25/96/arccos(a*x)*sin(5*arccos(a* 
x))+125/96*Ci(5*arccos(a*x)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^4/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*(125 
*a^4*x^5 - 136*a^2*x^3 + 24*x)*sqrt(a*x + 1)*sqrt(-a*x + 1)/((a^5*x^2 - a^ 
3)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)), x) + (2*a^2*x^4 - (25*a^2* 
x^4 - 12*x^2)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2)*sqrt(a*x + 1)* 
sqrt(-a*x + 1) + (5*a^3*x^5 - 4*a*x^3)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1 
), a*x))/(a^3*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.87 \[ \int \frac {x^4}{\arccos (a x)^4} \, dx=\frac {5 \, x^{5}}{6 \, \arccos \left (a x\right )^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{6 \, a \arccos \left (a x\right )} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{3 \, a \arccos \left (a x\right )^{3}} - \frac {2 \, x^{3}}{3 \, a^{2} \arccos \left (a x\right )^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{3} \arccos \left (a x\right )} + \frac {125 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{96 \, a^{5}} + \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{32 \, a^{5}} + \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{48 \, a^{5}} \] Input:

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

Output:

5/6*x^5/arccos(a*x)^2 - 25/6*sqrt(-a^2*x^2 + 1)*x^4/(a*arccos(a*x)) + 1/3* 
sqrt(-a^2*x^2 + 1)*x^4/(a*arccos(a*x)^3) - 2/3*x^3/(a^2*arccos(a*x)^2) + 2 
*sqrt(-a^2*x^2 + 1)*x^2/(a^3*arccos(a*x)) + 125/96*cos_integral(5*arccos(a 
*x))/a^5 + 27/32*cos_integral(3*arccos(a*x))/a^5 + 1/48*cos_integral(arcco 
s(a*x))/a^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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