3.1.33 \(\int \frac {\cot ^{-1}(a x)^3}{x^5} \, dx\) [33]

Optimal. Leaf size=152 \[ \frac {a^3}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac {a \cot ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \text {ArcTan}(a x)-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a^4 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4947, 5039, 331, 209, 5045, 4989, 2497, 5005} \begin {gather*} \frac {1}{4} a^4 \text {ArcTan}(a x)-i a^4 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )+\frac {1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-2 a^4 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac {a^3}{4 x}-\frac {3 a^3 \cot ^{-1}(a x)^2}{4 x}-\frac {a^2 \cot ^{-1}(a x)}{4 x^2}-\frac {\cot ^{-1}(a x)^3}{4 x^4}+\frac {a \cot ^{-1}(a x)^2}{4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^3/(4*x) - (a^2*ArcCot[a*x])/(4*x^2) - I*a^4*ArcCot[a*x]^2 + (a*ArcCot[a*x]^2)/(4*x^3) - (3*a^3*ArcCot[a*x]^2
)/(4*x) + (a^4*ArcCot[a*x]^3)/4 - ArcCot[a*x]^3/(4*x^4) + (a^4*ArcTan[a*x])/4 - 2*a^4*ArcCot[a*x]*Log[2 - 2/(1
- I*a*x)] - I*a^4*PolyLog[2, -1 + 2/(1 - I*a*x)]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4989

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcCot[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] + Dist[b*c*(p/d), Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))
]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5005

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(a + b*ArcCot[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5039

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcCot[c*x])^p/(d + e*x
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 5045

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[I*((a + b*ArcCot[
c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcCot[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c
, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.20, size = 126, normalized size = 0.83 \begin {gather*} \frac {a^3 x^3+\left (a x-3 a^3 x^3+4 i a^4 x^4\right ) \cot ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^3-a^2 x^2 \cot ^{-1}(a x) \left (1+a^2 x^2+8 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )+4 i a^4 x^4 \text {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^3 + (a*x - 3*a^3*x^3 + (4*I)*a^4*x^4)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x]^3 - a^2*x^2*ArcCot[a*x
]*(1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]) + (4*I)*a^4*x^4*PolyLog[2, -E^((2*I)*ArcCot[a*x])])
/(4*x^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.89, size = 855, normalized size = 5.62 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*(-1/4*arccot(a*x)^3/a^4/x^4-3/4*arccot(a*x)^2/a/x+3/4*I*arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))+1/4*ar
ccot(a*x)^2/a^3/x^3-2*arccot(a*x)*ln(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))-2*arccot(a*x)*ln(1+I*(I+a*x)/(a^2*x^2+1)^(
1/2))+1/4*arccot(a*x)^3-1/4*arccot(a*x)*(I+a*x)^2/a^2/x^2+1/4*I/a/x*(a*x-I)+3/8*Pi*arccot(a*x)^2-3/8*Pi*csgn(I
*(I+a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(I+a*x)^2/(a^2*x^2+1))^2*arccot(a*x)^2-3/16*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)
)*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^2*arccot(a*x)^2+3/16*Pi*csgn(I*(I+a*x)/(a^2*x^2+1)^(
1/2))^2*csgn(I*(I+a*x)^2/(a^2*x^2+1))*arccot(a*x)^2-3/16*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*(I+a*x)^2
/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))^2*arccot(a*x)^2+3/16*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x
^2+1)-1))^3*arccot(a*x)^2+3/8*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*arccot(a*x)^2-3/8*Pi*csgn(I/((I+a*x)^2/(a
^2*x^2+1)-1))^2*arccot(a*x)^2+3/16*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*arccot(a*x)^2+3/16*Pi*csgn(I/((I+a*x)^2/
(a^2*x^2+1)-1))*csgn(I*(I+a*x)^2/(a^2*x^2+1))*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1))*arccot(a
*x)^2-3/4*arccot(a*x)^2*arctan(a*x)+1/2*arccot(a*x)*(I+a*x)/a/x+2*I*dilog(1-I*(I+a*x)/(a^2*x^2+1)^(1/2))+2*I*d
ilog(1+I*(I+a*x)/(a^2*x^2+1)^(1/2))+I*arccot(a*x)^2-1/2*arccot(a*x)*(a*x-I)*(I+a*x)/a^2/x^2)

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

integral(arccot(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 {acot}^{3}{\left (a x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(a*x)**3/x**5, x)

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Giac [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(arccot(a*x)^3/x^5,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^3/x^5, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acot(a*x)^3/x^5, x)

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