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

Optimal result

Integrand size = 10, antiderivative size = 194 \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=-\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \cot ^{-1}(a x)^2}{10 a}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {19 \arctan (a x)}{60 a^6}-\frac {23 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^6}+\frac {23 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^6} \]

[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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {4947, 5037, 308, 209, 327, 5041, 4965, 2449, 2352, 4931, 5005} \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\frac {19 \arctan (a x)}{60 a^6}+\frac {23 i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{30 a^6}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac {23 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{15 a^6}-\frac {19 x}{60 a^5}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^3}{60 a^3}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {x^5 \cot ^{-1}(a x)^2}{10 a} \]

[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*} \text {integral}& = \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 \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 {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 \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 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{30 a^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.64 \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\frac {a x \left (-19+a^2 x^2\right )+2 \left (23 i+15 a x-5 a^3 x^3+3 a^5 x^5\right ) \cot ^{-1}(a x)^2+10 \left (1+a^6 x^6\right ) \cot ^{-1}(a x)^3+\cot ^{-1}(a x) \left (-19-16 a^2 x^2+3 a^4 x^4-92 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+46 i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{60 a^6} \]

[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)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (164 ) = 328\).

Time = 29.34 (sec) , antiderivative size = 1104, normalized size of antiderivative = 5.69

method	result	size
risch	\(\text {Expression too large to display}\)	\(1104\)
parts	\(\text {Expression too large to display}\)	\(2453\)
derivativedivides	\(\text {Expression too large to display}\)	\(2455\)
default	\(\text {Expression too large to display}\)	\(2455\)

[In]

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

[Out]

23/120*I/a^6*Pi^2+8929/57600*I/a^6*ln(a^2*x^2+1)-1/4*I/a^5*Pi*ln(1-I*a*x)*x-61/1920*I/a^2*ln(1-I*a*x)*x^4+151/
960*I/a^4*ln(1-I*a*x)*x^2+37/480*I/a^6*Pi*arctan(a*x)+1/40/a^2*Pi*x^4+1/40/a*Pi^2*x^5+1/12*I/a^3*Pi*ln(1-I*a*x
)*x^3+23/30*I/a^6*dilog(1/2-1/2*I*a*x)-1291/3600*I/a^6*ln(1-I*a*x)-1/8/a^6*Pi^2*arctan(a*x)+331/960/a^6*Pi*ln(
a^2*x^2+1)-1/240*(-15*I*x^6*ln(1-I*a*x)*a^6+15*Pi*a^6*x^6+6*a^5*x^5-10*a^3*x^3-15*I*ln(1-I*a*x)+30*a*x+15*Pi-4
6*I)/a^6*ln(1+I*a*x)^2+1/48/a^3*ln(1-I*a*x)^2*x^3+7/1440/a^3*ln(1-I*a*x)*x^3-1/16/a^5*ln(1-I*a*x)^2*x-1/1200/a
*ln(1-I*a*x)*x^5-37/480/a^5*ln(1-I*a*x)*x-1/80/a*ln(1-I*a*x)^2*x^5-1/3*I/a^6+23/30*I/a^6*ln(1/2-1/2*I*a*x)*ln(
1/2+1/2*I*a*x)-23/30*I/a^6*ln(1-I*a*x)*ln(1/2+1/2*I*a*x)+1/48*x^6*Pi^3+1/48/a^6*Pi^3-19/120/a^6*Pi+(-1/16*I*(a
^6*x^6+1)/a^6*ln(1-I*a*x)^2+1/120*x*(15*Pi*a^5*x^5+6*a^4*x^4-10*a^2*x^2+30)/a^5*ln(1-I*a*x)-1/240*(-15*I*Pi^2*
a^6*x^6-12*I*Pi*a^5*x^5-6*I*a^4*x^4+20*I*Pi*a^3*x^3+32*I*a^2*x^2-60*I*Pi*a*x-30*ln(1-I*a*x)*Pi-92*I*ln(1-I*a*x
))/a^6)*ln(1+I*a*x)-1/16*I*Pi^2*ln(1-I*a*x)*x^6-1/20*I/a*Pi*ln(1-I*a*x)*x^5-1/32*I/a^4*ln(1-I*a*x)^2*x^2+1/64*
I/a^2*ln(1-I*a*x)^2*x^4-1/48*I*(a^6*x^6+1)/a^6*ln(1+I*a*x)^3-1/24/a^3*Pi^2*x^3+1/8/a^5*Pi^2*x-1/16/a^6*Pi*ln(1
-I*a*x)^2+37/480/a^6*Pi*ln(1-I*a*x)-1/16*Pi*ln(1-I*a*x)^2*x^6-2/15/a^4*Pi*x^2+1/48*I/a^6*ln(1-I*a*x)^3-49/320*
I/a^6*ln(1-I*a*x)^2+1/48*I*ln(1-I*a*x)^3*x^6-1/96*I*ln(1-I*a*x)^2*x^6+1/288*I*ln(1-I*a*x)*x^6-1/50*I/a^6*(1-I*
a*x)^5*ln(1-I*a*x)+1/8*I/a^6*(1-I*a*x)^3*ln(1-I*a*x)^2-3/32*I/a^6*(1-I*a*x)^2*ln(1-I*a*x)^2+3/32*I/a^6*(1-I*a*
x)^2*ln(1-I*a*x)+7/128*I/a^6*(1-I*a*x)^4*ln(1-I*a*x)-1/96*I/a^6*(1-I*a*x)^6*ln(1-I*a*x)^2-1/12*I/a^6*(1-I*a*x)
^3*ln(1-I*a*x)+1/20*I/a^6*(1-I*a*x)^5*ln(1-I*a*x)^2+1/288*I/a^6*(1-I*a*x)^6*ln(1-I*a*x)-7/64*I/a^6*(1-I*a*x)^4
*ln(1-I*a*x)^2-19/60*x/a^5+1/60*x^3/a^3+18049/28800*arctan(a*x)/a^6

Fricas [F]

\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integral(x^5*arccot(a*x)^3, x)

Sympy [F]

\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int x^{5} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral(x**5*acot(a*x)**3, x)

Maxima [F]

\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[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

Giac [F]

\[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{3} \,d x } \]

[In]

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

[Out]

integrate(x^5*arccot(a*x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^5 \cot ^{-1}(a x)^3 \, dx=\int x^5\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]

[In]

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

[Out]

int(x^5*acot(a*x)^3, x)