\(\int \frac {\arccos (a x)^3}{x^5} \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 169 \[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arccos (a x)}{4 x^2}-\frac {1}{2} i a^4 \arccos (a x)^2+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{4 x^4}+a^4 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {1}{2} i a^4 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \] Output:

1/4*a^3*(-a^2*x^2+1)^(1/2)/x-1/4*a^2*arccos(a*x)/x^2-1/2*I*a^4*arccos(a*x) 
^2+1/4*a*(-a^2*x^2+1)^(1/2)*arccos(a*x)^2/x^3+1/2*a^3*(-a^2*x^2+1)^(1/2)*a 
rccos(a*x)^2/x-1/4*arccos(a*x)^3/x^4+a^4*arccos(a*x)*ln(1+(a*x+I*(-a^2*x^2 
+1)^(1/2))^2)-1/2*I*a^4*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\frac {a^3 x^3 \sqrt {1-a^2 x^2}+a x \left (-2 i a^3 x^3+\sqrt {1-a^2 x^2}+2 a^2 x^2 \sqrt {1-a^2 x^2}\right ) \arccos (a x)^2-\arccos (a x)^3+a^2 x^2 \arccos (a x) \left (-1+4 a^2 x^2 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-2 i a^4 x^4 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )}{4 x^4} \] Input:

Integrate[ArcCos[a*x]^3/x^5,x]
 

Output:

(a^3*x^3*Sqrt[1 - a^2*x^2] + a*x*((-2*I)*a^3*x^3 + Sqrt[1 - a^2*x^2] + 2*a 
^2*x^2*Sqrt[1 - a^2*x^2])*ArcCos[a*x]^2 - ArcCos[a*x]^3 + a^2*x^2*ArcCos[a 
*x]*(-1 + 4*a^2*x^2*Log[1 + E^((2*I)*ArcCos[a*x])]) - (2*I)*a^4*x^4*PolyLo 
g[2, -E^((2*I)*ArcCos[a*x])])/(4*x^4)
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5139, 5205, 5139, 242, 5187, 5137, 3042, 4202, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {\arccos (a x)^3}{x^5} \, dx\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5187

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\arccos (a x)^3}{4 x^4}-\frac {3}{4} a \left (\frac {2}{3} a^2 \left (-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{x}+2 a \left (\frac {1}{2} i \arccos (a x)^2-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )\right )\right )\right )-\frac {2}{3} a \left (\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arccos (a x)}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{3 x^3}\right )\)

Input:

Int[ArcCos[a*x]^3/x^5,x]
 

Output:

-1/4*ArcCos[a*x]^3/x^4 - (3*a*(-1/3*(Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/x^3 
- (2*a*((a*Sqrt[1 - a^2*x^2])/(2*x) - ArcCos[a*x]/(2*x^2)))/3 + (2*a^2*(-( 
(Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/x) + 2*a*((I/2)*ArcCos[a*x]^2 - (2*I)*(( 
-1/2*I)*ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - PolyLog[2, -E^((2*I)* 
ArcCos[a*x])]/4))))/3))/4
 

Defintions of rubi rules used

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

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

rule 5139

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

rule 5187

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x 
^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*A 
rcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 
2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 5205

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b 
*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) 
)   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Simp[b* 
c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)*( 
1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, 
 c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(a^{4} \left (-\frac {-2 i \arccos \left (a x \right )^{2} a^{4} x^{4}-2 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-i a^{4} x^{4}-\arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x -a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )^{3}+a^{2} x^{2} \arccos \left (a x \right )}{4 a^{4} x^{4}}-i \arccos \left (a x \right )^{2}+\arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(190\)
default	\(a^{4} \left (-\frac {-2 i \arccos \left (a x \right )^{2} a^{4} x^{4}-2 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-i a^{4} x^{4}-\arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x -a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )^{3}+a^{2} x^{2} \arccos \left (a x \right )}{4 a^{4} x^{4}}-i \arccos \left (a x \right )^{2}+\arccos \left (a x \right ) \ln \left (1+\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (a x +i \sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(190\)

Input:

int(arccos(a*x)^3/x^5,x,method=_RETURNVERBOSE)
 

Output:

a^4*(-1/4*(-2*I*arccos(a*x)^2*a^4*x^4-2*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a 
^3*x^3-I*a^4*x^4-arccos(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x-a^3*x^3*(-a^2*x^2+1) 
^(1/2)+arccos(a*x)^3+a^2*x^2*arccos(a*x))/a^4/x^4-I*arccos(a*x)^2+arccos(a 
*x)*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-1/2*I*polylog(2,-(a*x+I*(-a^2*x^2+1 
)^(1/2))^2))
 

Fricas [F]

\[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{5}} \,d x } \] Input:

integrate(arccos(a*x)^3/x^5,x, algorithm="fricas")
 

Output:

integral(arccos(a*x)^3/x^5, x)
 

Sympy [F]

\[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{5}}\, dx \] Input:

integrate(acos(a*x)**3/x**5,x)
 

Output:

Integral(acos(a*x)**3/x**5, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{5}} \,d x } \] Input:

integrate(arccos(a*x)^3/x^5,x, algorithm="maxima")
 

Output:

1/4*(12*a*x^4*integrate(1/4*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x 
+ 1)*sqrt(-a*x + 1), a*x)^2/(a^2*x^6 - x^4), x) - arctan2(sqrt(a*x + 1)*sq 
rt(-a*x + 1), a*x)^3)/x^4
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(arccos(a*x)^3/x^5,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^5} \,d x \] Input:

int(acos(a*x)^3/x^5,x)
 

Output:

int(acos(a*x)^3/x^5, x)
 

Reduce [F]

\[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\int \frac {\mathit {acos} \left (a x \right )^{3}}{x^{5}}d x \] Input:

int(acos(a*x)^3/x^5,x)
 

Output:

int(acos(a*x)**3/x**5,x)