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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {4724, 4790, 4772, 4722, 3800, 2221, 2317, 2438, 270} \[ \int \frac {\arccos (a x)^3}{x^5} \, dx=-\frac {1}{2} i a^4 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {1}{2} i a^4 \arccos (a x)^2+a^4 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {a^2 \arccos (a x)}{4 x^2}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}+\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {\arccos (a x)^3}{4 x^4} \]

[In]

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

[Out]

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

Rule 270

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

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[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] &&
IGtQ[m, 0]

Rule 4722

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 4724

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCo
s[c*x])^n/(d*(m + 1))), x] + Dist[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 4772

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] + Dist[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, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rule 4790

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] + (Dist[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] + Dist[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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^3}{4 x^4}-\frac {1}{4} (3 a) \int \frac {\arccos (a x)^2}{x^4 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {a \sqrt {1-a^2 x^2} \arccos (a x)^2}{4 x^3}-\frac {\arccos (a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\arccos (a x)}{x^3} \, dx-\frac {1}{2} a^3 \int \frac {\arccos (a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a^2 \arccos (a x)}{4 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}-\frac {1}{4} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+a^4 \int \frac {\arccos (a x)}{x} \, dx \\ & = \frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arccos (a x)}{4 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 \text {Subst}(\int x \tan (x) \, dx,x,\arccos (a x)) \\ & = \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}+\left (2 i a^4\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\arccos (a x)\right ) \\ & = \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 )-a^4 \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = \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} \left (i a^4\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arccos (a x)}\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int \frac {\arccos (a x)^3}{x^5} \, dx=\frac {a^3 x^3 \sqrt {1-a^2 x^2}+a x \left (-2 i a^3 x^3+\sqrt {1-a^2 x^2}+2 a^2 x^2 \sqrt {1-a^2 x^2}\right ) \arccos (a x)^2-\arccos (a x)^3+a^2 x^2 \arccos (a x) \left (-1+4 a^2 x^2 \log \left (1+e^{2 i \arccos (a x)}\right )\right )-2 i a^4 x^4 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )}{4 x^4} \]

[In]

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

[Out]

(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*Poly
Log[2, -E^((2*I)*ArcCos[a*x])])/(4*x^4)

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(arccos(a*x)^3/x^5, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acos(a*x)**3/x**5, x)

Maxima [F]

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

[In]

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

[Out]

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)*sqrt(-a*x + 1), a*x)^3)/x^4

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(acos(a*x)^3/x^5, x)