\(\int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx\) [21]

Optimal result

Integrand size = 10, antiderivative size = 113 \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \arctan (a x)+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 5039, 331, 209, 5045, 4989, 2497} \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=-\frac {1}{3} a^3 \arctan (a x)+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2+\frac {2}{3} a^3 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {a^2}{3 x}-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {a \cot ^{-1}(a x)}{3 x^2} \]

[In]

Int[ArcCot[a*x]^2/x^4,x]

[Out]

-1/3*a^2/x + (a*ArcCot[a*x])/(3*x^2) + (I/3)*a^3*ArcCot[a*x]^2 - ArcCot[a*x]^2/(3*x^3) - (a^3*ArcTan[a*x])/3 +
(2*a^3*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)])/3 + (I/3)*a^3*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 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*} \text {integral}& = -\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cot ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\cot ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx \\ & = \frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 i a^3\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx \\ & = -\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{3} a^4 \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \arctan (a x)+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\frac {-a^2 x^2+\left (-1-i a^3 x^3\right ) \cot ^{-1}(a x)^2+a x \cot ^{-1}(a x) \left (1+a^2 x^2+2 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )-i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )}{3 x^3} \]

[In]

Integrate[ArcCot[a*x]^2/x^4,x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (95 ) = 190\).

Time = 0.72 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.22

method	result	size
parts	\(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{3 x^{3}}-\frac {2 a^{3} \left (\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\operatorname {arccot}\left (a x \right )}{2 a^{2} x^{2}}-\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4}+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}+\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}-\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\right )}{3}\)	\(251\)
derivativedivides	\(a^{3} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\operatorname {arccot}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \,\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}-\frac {\arctan \left (a x \right )}{3}-\frac {1}{3 a x}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{3}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{3}\right )\)	\(252\)
default	\(a^{3} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\operatorname {arccot}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \,\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}-\frac {\arctan \left (a x \right )}{3}-\frac {1}{3 a x}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{3}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{3}\right )\)	\(252\)

[In]

int(arccot(a*x)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}} \,d x } \]

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^2/x^4, x)

Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

integrate(acot(a*x)**2/x**4,x)

[Out]

Integral(acot(a*x)**2/x**4, x)

Maxima [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}} \,d x } \]

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="maxima")

[Out]

1/48*(48*x^3*integrate(1/48*(36*a^2*x^2*arctan2(1, a*x)^2 - 4*a^2*x^2*log(a^2*x^2 + 1) - 8*a*x*arctan2(1, a*x)
+ 3*(a^2*x^2 + 1)*log(a^2*x^2 + 1)^2 + 36*arctan2(1, a*x)^2)/(a^2*x^6 + x^4), x) - 4*arctan2(1, a*x)^2 + log(
a^2*x^2 + 1)^2)/x^3

Giac [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}} \,d x } \]

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^2/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^4} \,d x \]

[In]

int(acot(a*x)^2/x^4,x)

[Out]

int(acot(a*x)^2/x^4, x)