3.24 \(\int x^4 \cot ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=205 \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{10 a^5}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{5 a^5}+\frac{x^2}{20 a^3}-\frac{\log \left (a^2 x^2+1\right )}{2 a^5}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a} \]

[Out]

x^2/(20*a^3) - (9*x*ArcCot[a*x])/(10*a^4) + (x^3*ArcCot[a*x])/(10*a^2) - (9*ArcCot[a*x]^2)/(20*a^5) - (3*x^2*A
rcCot[a*x]^2)/(10*a^3) + (3*x^4*ArcCot[a*x]^2)/(20*a) + ((I/5)*ArcCot[a*x]^3)/a^5 + (x^5*ArcCot[a*x]^3)/5 - (3
*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^5) - Log[1 + a^2*x^2]/(2*a^5) + (((3*I)/5)*ArcCot[a*x]*PolyLog[2, 1 -
2/(1 + I*a*x)])/a^5 - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(10*a^5)

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Rubi [A]  time = 0.517435, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {4853, 4917, 266, 43, 4847, 260, 4885, 4921, 4855, 4995, 6610} \[ -\frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{10 a^5}+\frac{3 i \cot ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{5 a^5}+\frac{x^2}{20 a^3}-\frac{\log \left (a^2 x^2+1\right )}{2 a^5}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCot[a*x]^3,x]

[Out]

x^2/(20*a^3) - (9*x*ArcCot[a*x])/(10*a^4) + (x^3*ArcCot[a*x])/(10*a^2) - (9*ArcCot[a*x]^2)/(20*a^5) - (3*x^2*A
rcCot[a*x]^2)/(10*a^3) + (3*x^4*ArcCot[a*x]^2)/(20*a) + ((I/5)*ArcCot[a*x]^3)/a^5 + (x^5*ArcCot[a*x]^3)/5 - (3
*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^5) - Log[1 + a^2*x^2]/(2*a^5) + (((3*I)/5)*ArcCot[a*x]*PolyLog[2, 1 -
2/(1 + I*a*x)])/a^5 - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(10*a^5)

Rule 4853

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

Rule 4917

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 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 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 x^4 \cot ^{-1}(a x)^3 \, dx &=\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{1}{5} (3 a) \int \frac{x^5 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3 \int x^3 \cot ^{-1}(a x)^2 \, dx}{5 a}-\frac{3 \int \frac{x^3 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a}\\ &=\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3+\frac{3}{10} \int \frac{x^4 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{3 \int x \cot ^{-1}(a x)^2 \, dx}{5 a^3}+\frac{3 \int \frac{x \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a^3}\\ &=-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3-\frac{3 \int \frac{\cot ^{-1}(a x)^2}{i-a x} \, dx}{5 a^4}+\frac{3 \int x^2 \cot ^{-1}(a x) \, dx}{10 a^2}-\frac{3 \int \frac{x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^2}-\frac{3 \int \frac{x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}\\ &=\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^5}-\frac{3 \int \cot ^{-1}(a x) \, dx}{10 a^4}+\frac{3 \int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^4}-\frac{3 \int \cot ^{-1}(a x) \, dx}{5 a^4}+\frac{3 \int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^4}-\frac{6 \int \frac{\cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^4}+\frac{\int \frac{x^3}{1+a^2 x^2} \, dx}{10 a}\\ &=-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^5}+\frac{3 i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^5}+\frac{(3 i) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^4}-\frac{3 \int \frac{x}{1+a^2 x^2} \, dx}{10 a^3}-\frac{3 \int \frac{x}{1+a^2 x^2} \, dx}{5 a^3}+\frac{\operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )}{20 a}\\ &=-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^5}-\frac{9 \log \left (1+a^2 x^2\right )}{20 a^5}+\frac{3 i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^5}-\frac{3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{10 a^5}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{20 a}\\ &=\frac{x^2}{20 a^3}-\frac{9 x \cot ^{-1}(a x)}{10 a^4}+\frac{x^3 \cot ^{-1}(a x)}{10 a^2}-\frac{9 \cot ^{-1}(a x)^2}{20 a^5}-\frac{3 x^2 \cot ^{-1}(a x)^2}{10 a^3}+\frac{3 x^4 \cot ^{-1}(a x)^2}{20 a}+\frac{i \cot ^{-1}(a x)^3}{5 a^5}+\frac{1}{5} x^5 \cot ^{-1}(a x)^3-\frac{3 \cot ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^5}-\frac{\log \left (1+a^2 x^2\right )}{2 a^5}+\frac{3 i \cot ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^5}-\frac{3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{10 a^5}\\ \end{align*}

Mathematica [A]  time = 0.585929, size = 184, normalized size = 0.9 \[ \frac{-24 i \cot ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-12 \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )+2 a^2 x^2+40 \log \left (\frac{1}{a x \sqrt{\frac{1}{a^2 x^2}+1}}\right )+8 a^5 x^5 \cot ^{-1}(a x)^3+6 a^4 x^4 \cot ^{-1}(a x)^2+4 a^3 x^3 \cot ^{-1}(a x)-12 a^2 x^2 \cot ^{-1}(a x)^2-36 a x \cot ^{-1}(a x)-8 i \cot ^{-1}(a x)^3-18 \cot ^{-1}(a x)^2-24 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )+i \pi ^3+2}{40 a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*ArcCot[a*x]^3,x]

[Out]

(2 + I*Pi^3 + 2*a^2*x^2 - 36*a*x*ArcCot[a*x] + 4*a^3*x^3*ArcCot[a*x] - 18*ArcCot[a*x]^2 - 12*a^2*x^2*ArcCot[a*
x]^2 + 6*a^4*x^4*ArcCot[a*x]^2 - (8*I)*ArcCot[a*x]^3 + 8*a^5*x^5*ArcCot[a*x]^3 - 24*ArcCot[a*x]^2*Log[1 - E^((
-2*I)*ArcCot[a*x])] + 40*Log[1/(a*Sqrt[1 + 1/(a^2*x^2)]*x)] - (24*I)*ArcCot[a*x]*PolyLog[2, E^((-2*I)*ArcCot[a
*x])] - 12*PolyLog[3, E^((-2*I)*ArcCot[a*x])])/(40*a^5)

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Maple [C]  time = 3.753, size = 2731, normalized size = 13.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccot(a*x)^3,x)

[Out]

-9/10*x*arccot(a*x)/a^4+1/10*x^3*arccot(a*x)/a^2-3/10*x^2*arccot(a*x)^2/a^3+3/20*x^4*arccot(a*x)^2/a+1/20/a^5-
6/5/a^5*polylog(3,(a*x+I)/(a^2*x^2+1)^(1/2))-6/5/a^5*polylog(3,-(a*x+I)/(a^2*x^2+1)^(1/2))+1/a^5*ln(1+(a*x+I)/
(a^2*x^2+1)^(1/2))+1/a^5*ln((a*x+I)/(a^2*x^2+1)^(1/2)-1)-3/10*I/a^5*arccot(a*x)^2*Pi+6/5*I/a^5*arccot(a*x)*pol
ylog(2,(a*x+I)/(a^2*x^2+1)^(1/2))+6/5*I/a^5*arccot(a*x)*polylog(2,-(a*x+I)/(a^2*x^2+1)^(1/2))+1/5*I*arccot(a*x
)^3/a^5+1/20*x^2/a^3-9/20*arccot(a*x)^2/a^5+1/5*x^5*arccot(a*x)^3+9/160/a^4*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4
/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)^3*x+9/160/a^4*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)
^3*x+21/160*I/a^5*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)^3-3/160*I/a^5*
arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^3-3/20*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1
)/((a*x+I)^2/(a^2*x^2+1)-1)^2)^3+9/160/a^4*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2
*x^2+1)-I)*csgn(I*(a*x+I)^2/(a^2*x^2+1)-I)^2*x-9/80/a^4*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^2
*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1))*x+9/160/a^4*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*((a
*x+I)^2/(a^2*x^2+1)-1))^2*x+3/80/a^2*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*((a*x+I)^2/
(a^2*x^2+1)-1))*x^3-3/160/a^2*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*((a*x+I)^2/(a^2*x^2+
1)-1))^2*x^3-3/80/a^2*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)^2*csgn(I*(
a*x+I)^2/(a^2*x^2+1)-I)*x^3-3/160/a^2*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+
1)-I)*csgn(I*(a*x+I)^2/(a^2*x^2+1)-I)^2*x^3+9/160*I/a^3*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(
a*x+I)^2/(a^2*x^2+1)-I)^3*x^2+9/160*I/a^3*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^3*x^2+21/80*I/a
^5*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)^2*csgn(I*(a*x+I)^2/(a^2*x^2+1
)-I)+21/160*I/a^5*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)*csgn(I*(a*x+I)
^2/(a^2*x^2+1)-I)^2+3/80*I/a^5*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*((a*x+I)^2/(a^2*x
^2+1)-1))-3/160*I/a^5*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1))^2
-3/20*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*(a*x+I)^2/(a^2
*x^2+1))-3/20*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2*x^2+1)-1)^2)^2*csgn(I/((a*x+
I)^2/(a^2*x^2+1)-1)^2)-3/10*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1))^2*csgn(I*(a*x+I)/(a^2*x^2+1)^
(1/2))+3/20*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1))*csgn(I*(a*x+I)/(a^2*x^2+1)^(1/2))^2+9/80/a^4*
arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)^2*csgn(I*(a*x+I)^2/(a^2*x^2+1)-I
)*x+3/20*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1))^3-3/160/a^2*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(
a^2*x^2+1)-1)^2)^3*x^3-9/80*I/a^3*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*((a*x+I)^2/(a^
2*x^2+1)-1))*x^2+9/160*I/a^3*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*((a*x+I)^2/(a^2*x^2+1
)-1))^2*x^2+3/20*I/a^5*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*(a*x+
I)^2/(a^2*x^2+1))*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)^2)+9/80*I/a^3*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1
)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)^2*csgn(I*(a*x+I)^2/(a^2*x^2+1)-I)*x^2+9/160*I/a^3*arccot(a*x)^2*Pi*csgn(-I*(a
*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^2+1)-I)*csgn(I*(a*x+I)^2/(a^2*x^2+1)-I)^2*x^2+3/10*I/a^5*arccot(a*x
)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2*x^2+1)-1)^2)^2+3/5/a^5*arccot(a*x)^2*ln((a*x+I)^2/(a^2*x^2
+1)-1)+3/10/a^5*arccot(a*x)^2*ln(a^2*x^2+1)-3/5/a^5*arccot(a*x)^2*ln(1-(a*x+I)/(a^2*x^2+1)^(1/2))-3/5/a^5*arcc
ot(a*x)^2*ln(1+(a*x+I)/(a^2*x^2+1)^(1/2))-3/5/a^5*arccot(a*x)^2*ln((a*x+I)/(a^2*x^2+1)^(1/2))-3/5/a^5*arccot(a
*x)^2*ln(2)-I/a^5*arccot(a*x)-3/160/a^2*arccot(a*x)^2*Pi*csgn(-I*(a*x+I)^4/(a^2*x^2+1)^2+2*I*(a*x+I)^2/(a^2*x^
2+1)-I)^3*x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{40} \, x^{5} \arctan \left (1, a x\right )^{3} - \frac{3}{160} \, x^{5} \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{140 \, a^{2} x^{6} \arctan \left (1, a x\right )^{3} + 12 \, a^{2} x^{6} \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 12 \, a x^{5} \arctan \left (1, a x\right )^{2} + 140 \, x^{4} \arctan \left (1, a x\right )^{3} + 3 \,{\left (5 \, a^{2} x^{6} \arctan \left (1, a x\right ) - a x^{5} + 5 \, x^{4} \arctan \left (1, a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x)^3,x, algorithm="maxima")

[Out]

1/40*x^5*arctan2(1, a*x)^3 - 3/160*x^5*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + integrate(1/160*(140*a^2*x^6*arcta
n2(1, a*x)^3 + 12*a^2*x^6*arctan2(1, a*x)*log(a^2*x^2 + 1) + 12*a*x^5*arctan2(1, a*x)^2 + 140*x^4*arctan2(1, a
*x)^3 + 3*(5*a^2*x^6*arctan2(1, a*x) - a*x^5 + 5*x^4*arctan2(1, a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{arccot}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^4*arccot(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{acot}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acot(a*x)**3,x)

[Out]

Integral(x**4*acot(a*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arccot}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^4*arccot(a*x)^3, x)