\(\int \cot ^{-1}(a x)^3 \, dx\) [28]

Optimal result

Integrand size = 6, antiderivative size = 96 \[ \int \cot ^{-1}(a x)^3 \, dx=\frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4931, 5041, 4965, 5005, 5115, 6745} \[ \int \cot ^{-1}(a x)^3 \, dx=-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}+\frac {3 i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right ) \cot ^{-1}(a x)}{a}+x \cot ^{-1}(a x)^3+\frac {i \cot ^{-1}(a x)^3}{a}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a} \]

[In]

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

[Out]

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

Rule 4931

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

Rule 4965

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] - Dist[b*c*(p/e), Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[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 5041

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

Rule 5115

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

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = x \cot ^{-1}(a x)^3+(3 a) \int \frac {x \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-3 \int \frac {\cot ^{-1}(a x)^2}{i-a x} \, dx \\ & = \frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-6 \int \frac {\cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = \frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a}+3 i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = \frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a}-\frac {3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \cot ^{-1}(a x)^3 \, dx=-\frac {i \cot ^{-1}(a x)^3}{a}+x \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac {3 i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac {3 \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{2 a} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (89 ) = 178\).

Time = 1.15 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.95

method	result	size
derivativedivides	\(\frac {\operatorname {arccot}\left (a x \right )^{3} \left (a x -i\right )+2 i \operatorname {arccot}\left (a x \right )^{3}-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}\)	\(187\)
default	\(\frac {\operatorname {arccot}\left (a x \right )^{3} \left (a x -i\right )+2 i \operatorname {arccot}\left (a x \right )^{3}-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-3 \operatorname {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a}\)	\(187\)

[In]

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

[Out]

1/a*(arccot(a*x)^3*(a*x-I)+2*I*arccot(a*x)^3-3*arccot(a*x)^2*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+6*I*arccot(a*x)*p
olylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,(I+a*x)/(a^2*x^2+1)^(1/2))-3*arccot(a*x)^2*ln(1+(I+a*x)/(a^2*x
^2+1)^(1/2))+6*I*arccot(a*x)*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,-(I+a*x)/(a^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int \cot ^{-1}(a x)^3 \, dx=\int { \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral(arccot(a*x)^3, x)

Sympy [F]

\[ \int \cot ^{-1}(a x)^3 \, dx=\int \operatorname {acot}^{3}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(acot(a*x)**3, x)

Maxima [F]

\[ \int \cot ^{-1}(a x)^3 \, dx=\int { \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

1/8*x*arctan2(1, a*x)^3 - 3/32*x*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + 21/16*arctan(a*x)^2*arctan(1/(a*x))^2/a
+ 7/8*arctan(a*x)*arctan(1/(a*x))^3/a + 28*a^2*integrate(1/32*x^2*arctan(1/(a*x))^3/(a^2*x^2 + 1), x) + 3*a^2*
integrate(1/32*x^2*arctan(1/(a*x))*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12*a^2*integrate(1/32*x^2*arctan(1/(
a*x))*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) + 12*a*integrate(1/32*x*arctan(1/(a*x))^2/(a^2*x^2 + 1), x) - 3*a*int
egrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 7/32*(a*arctan(a*x)^4 + 4*a*arctan(a*x)^3*arctan(1/(a*x))
)/a^2 + 3*integrate(1/32*arctan(1/(a*x))*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x)

Giac [F]

\[ \int \cot ^{-1}(a x)^3 \, dx=\int { \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

integrate(arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \cot ^{-1}(a x)^3 \, dx=\int {\mathrm {acot}\left (a\,x\right )}^3 \,d x \]

[In]

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

[Out]

int(acot(a*x)^3, x)