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

Optimal. Leaf size=169 \[ \frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \text {ArcCos}(a x)}{4 x^2}-\frac {1}{2} i a^4 \text {ArcCos}(a x)^2+\frac {a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{2 x}-\frac {\text {ArcCos}(a x)^3}{4 x^4}+a^4 \text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{2} i a^4 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right ) \]

[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

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Rubi [A]
time = 0.19, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {4724, 4790, 4772, 4722, 3800, 2221, 2317, 2438, 270} \begin {gather*} -\frac {1}{2} i a^4 \text {Li}_2\left (-e^{2 i \text {ArcCos}(a x)}\right )-\frac {1}{2} i a^4 \text {ArcCos}(a x)^2+a^4 \text {ArcCos}(a x) \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )-\frac {a^2 \text {ArcCos}(a x)}{4 x^2}+\frac {a \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{2 x}+\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {\text {ArcCos}(a x)^3}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\cos ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\cos ^{-1}(a x)^3}{4 x^4}-\frac {1}{4} (3 a) \int \frac {\cos ^{-1}(a x)^2}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}-\frac {\cos ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\cos ^{-1}(a x)}{x^3} \, dx-\frac {1}{2} a^3 \int \frac {\cos ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \cos ^{-1}(a x)}{4 x^2}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(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 {\cos ^{-1}(a x)}{x} \, dx\\ &=\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \cos ^{-1}(a x)}{4 x^2}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{4 x^4}-a^4 \text {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \cos ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \cos ^{-1}(a x)^2+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(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,\cos ^{-1}(a x)\right )\\ &=\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \cos ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \cos ^{-1}(a x)^2+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{4 x^4}+a^4 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-a^4 \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \cos ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \cos ^{-1}(a x)^2+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{4 x^4}+a^4 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(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 \cos ^{-1}(a x)}\right )\\ &=\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \cos ^{-1}(a x)}{4 x^2}-\frac {1}{2} i a^4 \cos ^{-1}(a x)^2+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{4 x^4}+a^4 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {1}{2} i a^4 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 149, normalized size = 0.88 \begin {gather*} -\frac {\text {ArcCos}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \sqrt {1-a^2 x^2} \left (\frac {1+2 \text {ArcCos}(a x)^2+\frac {\text {ArcCos}(a x)^2}{a^2 x^2}}{a x}+\frac {\text {ArcCos}(a x) \left (-\frac {1}{a^2 x^2}-2 i \text {ArcCos}(a x)+4 \log \left (1+e^{2 i \text {ArcCos}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}-\frac {2 i \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(a x)}\right )}{\sqrt {1-a^2 x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.45, size = 190, normalized size = 1.12

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccos(a*x)^3/x^5, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acos(a*x)**3/x**5, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acos(a*x)^3/x^5, x)

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