3.1.40 \(\int \frac {\arccos (a x)^4}{x^3} \, dx\) [40]

3.1.40.1 Optimal result

Integrand size = 10, antiderivative size = 121 \[ \int \frac {\arccos (a x)^4}{x^3} \, dx=-2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-6 i a^2 \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \]

output

-2*I*a^2*arccos(a*x)^3-1/2*arccos(a*x)^4/x^2+6*a^2*arccos(a*x)^2*ln(1+(a*x 
+I*(-a^2*x^2+1)^(1/2))^2)-6*I*a^2*arccos(a*x)*polylog(2,-(a*x+I*(-a^2*x^2+ 
1)^(1/2))^2)+3*a^2*polylog(3,-(a*x+I*(-a^2*x^2+1)^(1/2))^2)+2*a*arccos(a*x 
)^3*(-a^2*x^2+1)^(1/2)/x
 

3.1.40.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \frac {\arccos (a x)^4}{x^3} \, dx=-\frac {\arccos (a x)^4}{2 x^2}-a^2 \left (-2 \arccos (a x)^2 \left (-i \arccos (a x)+\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a x}+3 \log \left (1+e^{2 i \arccos (a x)}\right )\right )+6 i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )\right ) \]

input

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

output

-1/2*ArcCos[a*x]^4/x^2 - a^2*(-2*ArcCos[a*x]^2*((-I)*ArcCos[a*x] + (Sqrt[1 
 - a^2*x^2]*ArcCos[a*x])/(a*x) + 3*Log[1 + E^((2*I)*ArcCos[a*x])]) + (6*I) 
*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] - 3*PolyLog[3, -E^((2*I)*A 
rcCos[a*x])])
 

3.1.40.3 Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5139, 5187, 5137, 3042, 4202, 2620, 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.

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 5187

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

\(\Big \downarrow \) 5137

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.1.40.3.1 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 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 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]
 

3.1.40.4 Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\arccos \left (a x \right )^{3} \left (-4 i a^{2} x^{2}-4 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-4 i \arccos \left (a x \right )^{3}+6 \arccos \left (a x \right )^{2} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )\right )\)	\(150\)
default	\(a^{2} \left (-\frac {\arccos \left (a x \right )^{3} \left (-4 i a^{2} x^{2}-4 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-4 i \arccos \left (a x \right )^{3}+6 \arccos \left (a x \right )^{2} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )\right )\)	\(150\)

input

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

output

a^2*(-1/2*arccos(a*x)^3*(-4*I*a^2*x^2-4*a*x*(-a^2*x^2+1)^(1/2)+arccos(a*x) 
)/a^2/x^2-4*I*arccos(a*x)^3+6*arccos(a*x)^2*ln(1+(I*(-a^2*x^2+1)^(1/2)+a*x 
)^2)-6*I*arccos(a*x)*polylog(2,-(I*(-a^2*x^2+1)^(1/2)+a*x)^2)+3*polylog(3, 
-(I*(-a^2*x^2+1)^(1/2)+a*x)^2))
 

3.1.40.5 Fricas [F]

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

input

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

output

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

3.1.40.6 Sympy [F]

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

input

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

output

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

3.1.40.7 Maxima [F]

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

input

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

output

-1/2*(arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^4 - 4*a*x^2*integrate(sqr 
t(a*x + 1)*sqrt(-a*x + 1)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3/(a^ 
2*x^4 - x^2), x))/x^2
 

3.1.40.8 Giac [F]

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

input

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

output

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

3.1.40.9 Mupad [F(-1)]

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

input

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

output

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