3.28 \(\int \cot ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=96 \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.149912, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4847, 4921, 4855, 4885, 4995, 6610} \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a}+x \cot ^{-1}(a x)^3+\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a} \]

Antiderivative was successfully verified.

[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 4847

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

Rule 4921

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 4855

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])^p*Lo
g[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 4885

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 4995

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Cot[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 6610

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*} \int \cot ^{-1}(a x)^3 \, dx &=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) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}+3 i \int \frac{\text{Li}_2\left (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) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a}-\frac{3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.104136, size = 90, normalized size = 0.94 \[ -\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )}{a}-\frac{3 \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{2 a}+x \cot ^{-1}(a x)^3-\frac{i \cot ^{-1}(a x)^3}{a}-\frac{3 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )}{a} \]

Warning: Unable to verify antiderivative.

[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]  time = 0.163, size = 199, normalized size = 2.1 \begin{align*} x \left ({\rm arccot} \left (ax\right ) \right ) ^{3}+{\frac{i \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{a}}-3\,{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{a}\ln \left ( 1-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-3\,{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{a}\ln \left ( 1+{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{6\,i{\rm arccot} \left (ax\right )}{a}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{6\,i{\rm arccot} \left (ax\right )}{a}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }-6\,{\frac{1}{a}{\it polylog} \left ( 3,-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-6\,{\frac{1}{a}{\it polylog} \left ( 3,{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \, x \arctan \left (1, a x\right )^{3} - \frac{3}{32} \, x \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \frac{21 \, \arctan \left (a x\right )^{2} \arctan \left (\frac{1}{a x}\right )^{2}}{16 \, a} + \frac{7 \, \arctan \left (a x\right ) \arctan \left (\frac{1}{a x}\right )^{3}}{8 \, a} + 28 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right )^{3}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{2} \int \frac{x^{2} \arctan \left (\frac{1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a \int \frac{x \arctan \left (\frac{1}{a x}\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} - 3 \, a \int \frac{x \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac{7 \,{\left (a \arctan \left (a x\right )^{4} + 4 \, a \arctan \left (a x\right )^{3} \arctan \left (\frac{1}{a x}\right )\right )}}{32 \, a^{2}} + 3 \, \int \frac{\arctan \left (\frac{1}{a x}\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{arccot}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(a*x)^3, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acot}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(a*x)**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arccot}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x)^3, x)