3.1.30 \(\int \frac {\arccos (a x)^3}{x^4} \, dx\) [30]

3.1.30.1 Optimal result

Integrand size = 10, antiderivative size = 192 \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=-\frac {a^2 \arccos (a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x^2}-\frac {\arccos (a x)^3}{3 x^3}-i a^3 \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i a^3 \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-a^3 \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+a^3 \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right ) \]

output

-a^2*arccos(a*x)/x-1/3*arccos(a*x)^3/x^3-I*a^3*arccos(a*x)^2*arctan(a*x+I* 
(-a^2*x^2+1)^(1/2))+a^3*arctanh((-a^2*x^2+1)^(1/2))+I*a^3*arccos(a*x)*poly 
log(2,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))-I*a^3*arccos(a*x)*polylog(2,I*(a*x+I* 
(-a^2*x^2+1)^(1/2)))-a^3*polylog(3,-I*(a*x+I*(-a^2*x^2+1)^(1/2)))+a^3*poly 
log(3,I*(a*x+I*(-a^2*x^2+1)^(1/2)))+1/2*a*arccos(a*x)^2*(-a^2*x^2+1)^(1/2) 
/x^2
 

3.1.30.2 Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.86 \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=a^3 \left (-i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )+\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+\operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )\right )-\frac {\arccos (a x) \left (12 a^2 x^2+4 \arccos (a x)^2-3 \arccos (a x) \sin (2 \arccos (a x))\right )}{12 x^3} \]

input

Integrate[ArcCos[a*x]^3/x^4,x]
 

output

a^3*((-I)*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + ArcTanh[Sqrt[1 - a^2*x 
^2]] + I*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - I*ArcCos[a*x]*Po 
lyLog[2, I*E^(I*ArcCos[a*x])] - PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + PolyL 
og[3, I*E^(I*ArcCos[a*x])]) - (ArcCos[a*x]*(12*a^2*x^2 + 4*ArcCos[a*x]^2 - 
 3*ArcCos[a*x]*Sin[2*ArcCos[a*x]]))/(12*x^3)
 

3.1.30.3 Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {5139, 5205, 5139, 243, 73, 221, 5219, 3042, 4669, 3011, 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.

\(\displaystyle \int \frac {\arccos (a x)^3}{x^4} \, dx\)

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5205

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5219

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

\(\Big \downarrow \) 3042

\(\displaystyle -a \left (-\frac {1}{2} a^2 \int \arccos (a x)^2 \csc \left (\arccos (a x)+\frac {\pi }{2}\right )d\arccos (a x)-a \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arccos (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x^2}\right )-\frac {\arccos (a x)^3}{3 x^3}\)

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\arccos (a x)^3}{3 x^3}-a \left (-\frac {1}{2} a^2 \left (2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )d\arccos (a x)\right )-2 i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )\right )-a \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arccos (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x^2}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {\arccos (a x)^3}{3 x^3}-a \left (-\frac {1}{2} a^2 \left (2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}\right )-2 \left (i \arccos (a x) \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\int e^{-i \arccos (a x)} \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )de^{i \arccos (a x)}\right )-2 i \arccos (a x)^2 \arctan \left (e^{i \arccos (a x)}\right )\right )-a \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arccos (a x)}{x}\right )-\frac {\sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x^2}\right )\)

\(\Big \downarrow \) 7143

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

input

Int[ArcCos[a*x]^3/x^4,x]
 

output

-1/3*ArcCos[a*x]^3/x^3 - a*(-1/2*(Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/x^2 - a 
*(-(ArcCos[a*x]/x) + a*ArcTanh[Sqrt[1 - a^2*x^2]]) - (a^2*((-2*I)*ArcCos[a 
*x]^2*ArcTan[E^(I*ArcCos[a*x])] + 2*(I*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*Ar 
cCos[a*x])] - PolyLog[3, (-I)*E^(I*ArcCos[a*x])]) - 2*(I*ArcCos[a*x]*PolyL 
og[2, I*E^(I*ArcCos[a*x])] - PolyLog[3, I*E^(I*ArcCos[a*x])])))/2)
 

3.1.30.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 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 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]
 

rule 5219

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

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]
 

3.1.30.4 Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.33

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arccos \left (a x \right ) \left (-3 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +2 \arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}-\frac {\arccos \left (a x \right )^{2} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}+i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-\operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {\arccos \left (a x \right )^{2} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \arctan \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\)	\(256\)
default	\(a^{3} \left (-\frac {\arccos \left (a x \right ) \left (-3 \sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +2 \arccos \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{6 a^{3} x^{3}}-\frac {\arccos \left (a x \right )^{2} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}+i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-\operatorname {polylog}\left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\frac {\arccos \left (a x \right )^{2} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+\operatorname {polylog}\left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \arctan \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\)	\(256\)

input

int(arccos(a*x)^3/x^4,x,method=_RETURNVERBOSE)
 

output

a^3*(-1/6/a^3/x^3*arccos(a*x)*(-3*(-a^2*x^2+1)^(1/2)*arccos(a*x)*a*x+2*arc 
cos(a*x)^2+6*a^2*x^2)-1/2*arccos(a*x)^2*ln(1+I*(I*(-a^2*x^2+1)^(1/2)+a*x)) 
+I*arccos(a*x)*polylog(2,-I*(I*(-a^2*x^2+1)^(1/2)+a*x))-polylog(3,-I*(I*(- 
a^2*x^2+1)^(1/2)+a*x))+1/2*arccos(a*x)^2*ln(1-I*(I*(-a^2*x^2+1)^(1/2)+a*x) 
)-I*arccos(a*x)*polylog(2,I*(I*(-a^2*x^2+1)^(1/2)+a*x))+polylog(3,I*(I*(-a 
^2*x^2+1)^(1/2)+a*x))-2*I*arctan(I*(-a^2*x^2+1)^(1/2)+a*x))
 

3.1.30.5 Fricas [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arccos(a*x)^3/x^4,x, algorithm="fricas")
 

output

integral(arccos(a*x)^3/x^4, x)
 

3.1.30.6 Sympy [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{4}}\, dx \]

input

integrate(acos(a*x)**3/x**4,x)
 

output

Integral(acos(a*x)**3/x**4, x)
 

3.1.30.7 Maxima [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arccos(a*x)^3/x^4,x, algorithm="maxima")
 

output

1/3*(3*a*x^3*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)* 
sqrt(-a*x + 1), a*x)^2/(a^2*x^5 - x^3), x) - arctan2(sqrt(a*x + 1)*sqrt(-a 
*x + 1), a*x)^3)/x^3
 

3.1.30.8 Giac [F]

\[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{4}} \,d x } \]

input

integrate(arccos(a*x)^3/x^4,x, algorithm="giac")
 

output

integrate(arccos(a*x)^3/x^4, x)
 

3.1.30.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^4} \,d x \]

input

int(acos(a*x)^3/x^4,x)
 

output

int(acos(a*x)^3/x^4, x)