3.1.38 \(\int \frac {\arccos (a x)^4}{x} \, dx\) [38]

3.1.38.1 Optimal result

Integrand size = 10, antiderivative size = 119 \[ \int \frac {\arccos (a x)^4}{x} \, dx=-\frac {1}{5} i \arccos (a x)^5+\arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )-2 i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 \arccos (a x)^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+3 i \arccos (a x) \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (5,-e^{2 i \arccos (a x)}\right ) \]

output

-1/5*I*arccos(a*x)^5+arccos(a*x)^4*ln(1+(a*x+I*(-a^2*x^2+1)^(1/2))^2)-2*I* 
arccos(a*x)^3*polylog(2,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+3*arccos(a*x)^2*pol 
ylog(3,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+3*I*arccos(a*x)*polylog(4,-(a*x+I*(- 
a^2*x^2+1)^(1/2))^2)-3/2*polylog(5,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)
 

3.1.38.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)^4}{x} \, dx=-\frac {1}{5} i \arccos (a x)^5+\arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )-2 i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 \arccos (a x)^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+3 i \arccos (a x) \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (5,-e^{2 i \arccos (a x)}\right ) \]

input

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

output

(-1/5*I)*ArcCos[a*x]^5 + ArcCos[a*x]^4*Log[1 + E^((2*I)*ArcCos[a*x])] - (2 
*I)*ArcCos[a*x]^3*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + 3*ArcCos[a*x]^2*Pol 
yLog[3, -E^((2*I)*ArcCos[a*x])] + (3*I)*ArcCos[a*x]*PolyLog[4, -E^((2*I)*A 
rcCos[a*x])] - (3*PolyLog[5, -E^((2*I)*ArcCos[a*x])])/2
 

3.1.38.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5137, 3042, 4202, 2620, 3011, 7163, 7163, 2720, 7143}

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.

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

\(\displaystyle -\int \arccos (a x)^4 \tan (\arccos (a x))d\arccos (a x)\)

\(\Big \downarrow \) 4202

\(\displaystyle 2 i \int \frac {e^{2 i \arccos (a x)} \arccos (a x)^4}{1+e^{2 i \arccos (a x)}}d\arccos (a x)-\frac {1}{5} i \arccos (a x)^5\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \int \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )d\arccos (a x)\right )-\frac {1}{2} i \arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{5} i \arccos (a x)^5\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \left (i \int \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )d\arccos (a x)-\frac {1}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )\right )\right )-\frac {1}{2} i \arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{5} i \arccos (a x)^5\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \left (i \left (\frac {1}{2} i \int \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )d\arccos (a x)-\frac {1}{2} i \arccos (a x) \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )\right )\right )-\frac {1}{2} i \arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{5} i \arccos (a x)^5\)

\(\Big \downarrow \) 2720

\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \left (i \left (\frac {1}{4} \int e^{-2 i \arccos (a x)} \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )de^{2 i \arccos (a x)}-\frac {1}{2} i \arccos (a x) \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )\right )\right )-\frac {1}{2} i \arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{5} i \arccos (a x)^5\)

\(\Big \downarrow \) 7143

\(\displaystyle 2 i \left (2 i \left (\frac {1}{2} i \arccos (a x)^3 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \left (i \left (\frac {1}{4} \operatorname {PolyLog}\left (5,-e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \arccos (a x) \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )\right )-\frac {1}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )\right )\right )-\frac {1}{2} i \arccos (a x)^4 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-\frac {1}{5} i \arccos (a x)^5\)

input

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

output

(-1/5*I)*ArcCos[a*x]^5 + (2*I)*((-1/2*I)*ArcCos[a*x]^4*Log[1 + E^((2*I)*Ar 
cCos[a*x])] + (2*I)*((I/2)*ArcCos[a*x]^3*PolyLog[2, -E^((2*I)*ArcCos[a*x]) 
] - ((3*I)/2)*((-1/2*I)*ArcCos[a*x]^2*PolyLog[3, -E^((2*I)*ArcCos[a*x])] + 
 I*((-1/2*I)*ArcCos[a*x]*PolyLog[4, -E^((2*I)*ArcCos[a*x])] + PolyLog[5, - 
E^((2*I)*ArcCos[a*x])]/4))))
 

3.1.38.3.1 Defintions of rubi rules used

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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

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

3.1.38.4 Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(-\frac {i \arccos \left (a x \right )^{5}}{5}+\arccos \left (a x \right )^{4} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-2 i \arccos \left (a x \right )^{3} \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 \arccos \left (a x \right )^{2} \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 i \arccos \left (a x \right ) \operatorname {polylog}\left (4, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 \operatorname {polylog}\left (5, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\)	\(168\)
default	\(-\frac {i \arccos \left (a x \right )^{5}}{5}+\arccos \left (a x \right )^{4} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-2 i \arccos \left (a x \right )^{3} \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 \arccos \left (a x \right )^{2} \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 i \arccos \left (a x \right ) \operatorname {polylog}\left (4, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 \operatorname {polylog}\left (5, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\)	\(168\)

input

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

output

-1/5*I*arccos(a*x)^5+arccos(a*x)^4*ln(1+(I*(-a^2*x^2+1)^(1/2)+a*x)^2)-2*I* 
arccos(a*x)^3*polylog(2,-(I*(-a^2*x^2+1)^(1/2)+a*x)^2)+3*arccos(a*x)^2*pol 
ylog(3,-(I*(-a^2*x^2+1)^(1/2)+a*x)^2)+3*I*arccos(a*x)*polylog(4,-(I*(-a^2* 
x^2+1)^(1/2)+a*x)^2)-3/2*polylog(5,-(I*(-a^2*x^2+1)^(1/2)+a*x)^2)
 

3.1.38.5 Fricas [F]

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

input

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

output

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

3.1.38.6 Sympy [F]

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

input

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

output

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

3.1.38.7 Maxima [F]

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

input

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

output

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

3.1.38.8 Giac [F]

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

input

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

output

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

3.1.38.9 Mupad [F(-1)]

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

input

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

output

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