3.1.23 \(\int x^5 \cot ^{-1}(a x)^3 \, dx\) [23]

Optimal. Leaf size=194 \[ -\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {19 \text {ArcTan}(a x)}{60 a^6}-\frac {23 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^6}+\frac {23 i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^6} \]

[Out]

-19/60*x/a^5+1/60*x^3/a^3-4/15*x^2*arccot(a*x)/a^4+1/20*x^4*arccot(a*x)/a^2+23/30*I*arccot(a*x)^2/a^6+1/2*x*ar
ccot(a*x)^2/a^5-1/6*x^3*arccot(a*x)^2/a^3+1/10*x^5*arccot(a*x)^2/a+1/6*arccot(a*x)^3/a^6+1/6*x^6*arccot(a*x)^3
+19/60*arctan(a*x)/a^6-23/15*arccot(a*x)*ln(2/(1+I*a*x))/a^6+23/30*I*polylog(2,1-2/(1+I*a*x))/a^6

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Rubi [A]
time = 0.45, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {4947, 5037, 308, 209, 327, 5041, 4965, 2449, 2352, 4931, 5005} \begin {gather*} \frac {19 \text {ArcTan}(a x)}{60 a^6}+\frac {23 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{30 a^6}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac {23 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{15 a^6}-\frac {19 x}{60 a^5}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^3}{60 a^3}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {x^5 \cot ^{-1}(a x)^2}{10 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-19*x)/(60*a^5) + x^3/(60*a^3) - (4*x^2*ArcCot[a*x])/(15*a^4) + (x^4*ArcCot[a*x])/(20*a^2) + (((23*I)/30)*Arc
Cot[a*x]^2)/a^6 + (x*ArcCot[a*x]^2)/(2*a^5) - (x^3*ArcCot[a*x]^2)/(6*a^3) + (x^5*ArcCot[a*x]^2)/(10*a) + ArcCo
t[a*x]^3/(6*a^6) + (x^6*ArcCot[a*x]^3)/6 + (19*ArcTan[a*x])/(60*a^6) - (23*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(15
*a^6) + (((23*I)/30)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^6

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

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 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 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 5037

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 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]

Rubi steps

\begin {align*} \int x^5 \cot ^{-1}(a x)^3 \, dx &=\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {1}{2} a \int \frac {x^6 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {\int x^4 \cot ^{-1}(a x)^2 \, dx}{2 a}-\frac {\int \frac {x^4 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a}\\ &=\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {1}{5} \int \frac {x^5 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {\int x^2 \cot ^{-1}(a x)^2 \, dx}{2 a^3}+\frac {\int \frac {x^2 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^3}\\ &=-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {\int \cot ^{-1}(a x)^2 \, dx}{2 a^5}-\frac {\int \frac {\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^5}+\frac {\int x^3 \cot ^{-1}(a x) \, dx}{5 a^2}-\frac {\int \frac {x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\int \frac {x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^2}\\ &=\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3-\frac {\int x \cot ^{-1}(a x) \, dx}{5 a^4}+\frac {\int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^4}-\frac {\int x \cot ^{-1}(a x) \, dx}{3 a^4}+\frac {\int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^4}+\frac {\int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^4}+\frac {\int \frac {x^4}{1+a^2 x^2} \, dx}{20 a}\\ &=-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{5 a^5}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{3 a^5}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{a^5}-\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{10 a^3}-\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{6 a^3}+\frac {\int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx}{20 a}\\ &=-\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3-\frac {23 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^6}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{20 a^5}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{10 a^5}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{6 a^5}-\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^5}-\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^5}-\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^5}\\ &=-\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {19 \tan ^{-1}(a x)}{60 a^6}-\frac {23 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^6}+\frac {i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^6}+\frac {i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^6}+\frac {i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^6}\\ &=-\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {19 \tan ^{-1}(a x)}{60 a^6}-\frac {23 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^6}+\frac {23 i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{30 a^6}\\ \end {align*}

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Mathematica [A]
time = 0.50, size = 125, normalized size = 0.64 \begin {gather*} \frac {a x \left (-19+a^2 x^2\right )+2 \left (23 i+15 a x-5 a^3 x^3+3 a^5 x^5\right ) \cot ^{-1}(a x)^2+10 \left (1+a^6 x^6\right ) \cot ^{-1}(a x)^3+\cot ^{-1}(a x) \left (-19-16 a^2 x^2+3 a^4 x^4-92 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+46 i \text {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{60 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x*(-19 + a^2*x^2) + 2*(23*I + 15*a*x - 5*a^3*x^3 + 3*a^5*x^5)*ArcCot[a*x]^2 + 10*(1 + a^6*x^6)*ArcCot[a*x]^
3 + ArcCot[a*x]*(-19 - 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^((2*I)*ArcCot[a*x])]) + (46*I)*PolyLog[2, E^((2*I
)*ArcCot[a*x])])/(60*a^6)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 5.21, size = 2455, normalized size = 12.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(1260\)
derivativedivides	\(\text {Expression too large to display}\)	\(2455\)
default	\(\text {Expression too large to display}\)	\(2455\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^6*(-1/6*a^3*x^3*arccot(a*x)^2+1/10*a^5*x^5*arccot(a*x)^2+1/2*a*x*arccot(a*x)^2+1/6*arccot(a*x)^3-1/80*arcc
ot(a*x)*(-4*I*(a^2*x^2+1)^(1/2)*a^2*x^2-2*(a^2*x^2+1)^(1/2)*a^3*x^3+4*I*a^3*x^3+2*a^4*x^4+2*(a^2*x^2+1)^(1/2)*
a*x+2*I*a*x-a^2*x^2-1)+47/240*(I*(a^2*x^2+1)^(1/2)+(a^2*x^2+1)^(1/2)*a*x+I*a*x+a^2*x^2)*arccot(a*x)-41/240*arc
cot(a*x)*(-I*(a^2*x^2+1)^(1/2)+I*a*x-1)+3/40*(2*I*(a^2*x^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3+I*(a^2*x^2+1)^(1/2)+2*
I*a*x+(a^2*x^2+1)^(1/2)*a*x+a^2*x^2+1)*arccot(a*x)-47/120*(a^2*x^2+1+(a^2*x^2+1)^(1/2)*a*x)*arccot(a*x)-23/15*
arccot(a*x)*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))-41/240*arccot(a*x)*(I*(a^2*x^2+1)^(1/2)+I*a*x-1)+1/40*(-2*I*(a^2*x
^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3+I*(a^2*x^2+1)^(1/2)+3*(a^2*x^2+1)^(1/2)*a*x-3*a^2*x^2-1)*arccot(a*x)+1/20*arcc
ot(a*x)*(-2*I*(a^2*x^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3-2*(a^2*x^2+1)^(1/2)*a^3*x^3+2*a^4*x^4-I*(a^2*x^2+1)^(1/2)+
2*I*a*x-(a^2*x^2+1)^(1/2)*a*x+2*a^2*x^2)+1/4*Pi*arccot(a*x)^2-47/120*(a^2*x^2+1-(a^2*x^2+1)^(1/2)*a*x)*arccot(
a*x)+3/40*(-2*I*(a^2*x^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3-I*(a^2*x^2+1)^(1/2)+2*I*a*x-(a^2*x^2+1)^(1/2)*a*x+a^2*x^
2+1)*arccot(a*x)+1/20*arccot(a*x)*(-2*I*(a^2*x^2+1)^(1/2)*a^2*x^2-2*I*a^3*x^3+2*(a^2*x^2+1)^(1/2)*a^3*x^3+2*a^
4*x^4-I*(a^2*x^2+1)^(1/2)-2*I*a*x+(a^2*x^2+1)^(1/2)*a*x+2*a^2*x^2)+1/20*arccot(a*x)*(2*I*(a^2*x^2+1)^(1/2)*a^2
*x^2+2*I*a^3*x^3+2*(a^2*x^2+1)^(1/2)*a^3*x^3+2*a^4*x^4+I*(a^2*x^2+1)^(1/2)+2*I*a*x+(a^2*x^2+1)^(1/2)*a*x+2*a^2
*x^2)-23/15*I*dilog((I+a*x)/(a^2*x^2+1)^(1/2))+23/15*I*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))+23/30*I*arccot(a*x)^
2-3/40*(-2*I*(a^2*x^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3-I*(a^2*x^2+1)^(1/2)+2*I*a*x+(a^2*x^2+1)^(1/2)*a*x-a^2*x^2-1
)*arccot(a*x)+47/240*(-I*(a^2*x^2+1)^(1/2)+I*a*x-(a^2*x^2+1)^(1/2)*a*x+a^2*x^2)*arccot(a*x)-1/4*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)^2-1/8*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+1/8*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)^2-1/8*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)^2-3/40*(a^2*x^2+1)*arccot(a*x)*(-2*(a^2*x^2+1)^(1/2)*a*x+2*a^2*x
^2+1)-3/40*(a^2*x^2+1)*arccot(a*x)*(2*(a^2*x^2+1)^(1/2)*a*x+2*a^2*x^2+1)+1/8*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+1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*arccot(a*x)^2-1/80*arccot
(a*x)*(4*I*(a^2*x^2+1)^(1/2)*a^2*x^2+4*I*a^3*x^3+2*(a^2*x^2+1)^(1/2)*a^3*x^3+2*a^4*x^4+2*I*a*x-2*(a^2*x^2+1)^(
1/2)*a*x-a^2*x^2-1)+1/40*(2*I*(a^2*x^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3-I*(a^2*x^2+1)^(1/2)-3*(a^2*x^2+1)^(1/2)*a*
x-3*a^2*x^2-1)*arccot(a*x)-1/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*arccot(a*x)^2+1/8*Pi*csgn(I*(I+a*x)^2/(a
^2*x^2+1))^3*arccot(a*x)^2+1/15*I*(a^2*x^2+1)^(3/2)/(24*I*(a^2*x^2+1)^(1/2)*a*x+24*I*a^2*x^2+16*(a^2*x^2+1)^(1
/2)*a^2*x^2+16*a^3*x^3+8*I-8*(a^2*x^2+1)^(1/2))-1/5*I*(a^2*x^2+1)/(-16*I*(a^2*x^2+1)^(1/2)+16*I*a*x-16*(a^2*x^
2+1)^(1/2)*a*x+16*a^2*x^2)-41/15*I*(a^2*x^2+1)^(1/2)/(8*a*x+8*I-8*(a^2*x^2+1)^(1/2))-1/15*I*(a^2*x^2+1)^(3/2)/
(-24*I*(a^2*x^2+1)^(1/2)*a*x+24*I*a^2*x^2-16*(a^2*x^2+1)^(1/2)*a^2*x^2+16*a^3*x^3+8*I+8*(a^2*x^2+1)^(1/2))+41/
15*I*(a^2*x^2+1)^(1/2)/(8*(a^2*x^2+1)^(1/2)+8*a*x+8*I)+1/2*I*arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))+1/6*a
^6*x^6*arccot(a*x)^3-1/5*I*(a^2*x^2+1)/(16*I*(a^2*x^2+1)^(1/2)+16*I*a*x+16*(a^2*x^2+1)^(1/2)*a*x+16*a^2*x^2)+1
/8*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+1/20*arccot(a*x)*(2*I*(a^2*x^2+1)^(1/2)*a^2*x^2-2*I*a^3*x^3-2*(a^2*x^2+1)^(1/2)*
a^3*x^3+2*a^4*x^4+I*(a^2*x^2+1)^(1/2)-2*I*a*x-(a^2*x^2+1)^(1/2)*a*x+2*a^2*x^2)-1/2*arccot(a*x)^2*arctan(a*x)-3
/40*(2*I*(a^2*x^2+1)^(1/2)*a^2*x^2+2*I*a^3*x^3+I*(a^2*x^2+1)^(1/2)+2*I*a*x-(a^2*x^2+1)^(1/2)*a*x-a^2*x^2-1)*ar
ccot(a*x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(40*a^6*x^6*arctan2(1, a*x)^3 + 12*a^5*x^5*arctan2(1, a*x)^2 - 20*a^3*x^3*arctan2(1, a*x)^2 + 20*(5760*a
^7*integrate(1/480*x^7*arctan(1/(a*x))^3/(a^7*x^2 + a^5), x) + 1440*a^6*integrate(1/480*x^6*arctan(1/(a*x))^2/
(a^7*x^2 + a^5), x) + 360*a^6*integrate(1/480*x^6*log(a^2*x^2 + 1)^2/(a^7*x^2 + a^5), x) + 288*a^6*integrate(1
/480*x^6*log(a^2*x^2 + 1)/(a^7*x^2 + a^5), x) + 5760*a^5*integrate(1/480*x^5*arctan(1/(a*x))^3/(a^7*x^2 + a^5)
, x) + 576*a^5*integrate(1/480*x^5*arctan(1/(a*x))/(a^7*x^2 + a^5), x) - 480*a^4*integrate(1/480*x^4*log(a^2*x
^2 + 1)/(a^7*x^2 + a^5), x) - 960*a^3*integrate(1/480*x^3*arctan(1/(a*x))/(a^7*x^2 + a^5), x) + 1440*a^2*integ
rate(1/480*x^2*log(a^2*x^2 + 1)/(a^7*x^2 + a^5), x) + 2880*a*integrate(1/480*x*arctan(1/(a*x))/(a^7*x^2 + a^5)
, x) + arctan(a*x)^3/a^6 + 3*arctan(a*x)^2*arctan(1/(a*x))/a^6 + 3*arctan(a*x)*arctan(1/(a*x))^2/a^6 + 360*int
egrate(1/480*log(a^2*x^2 + 1)^2/(a^7*x^2 + a^5), x))*a^6 + 60*a*x*arctan2(1, a*x)^2 + 40*arctan2(1, a*x)^3 - (
3*a^5*x^5 - 5*a^3*x^3 + 15*a*x)*log(a^2*x^2 + 1)^2)/a^6

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

[Out]

integral(x^5*arccot(a*x)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \operatorname {acot}^{3}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*acot(a*x)**3, 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(x^5*arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^5*arccot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^5*acot(a*x)^3, x)

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