3.1.25 \(\int x^3 \cot ^{-1}(a x)^3 \, dx\) [25]

Optimal. Leaf size=148 \[ \frac {x}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}-\frac {i \cot ^{-1}(a x)^2}{a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3-\frac {\text {ArcTan}(a x)}{4 a^4}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}-\frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4} \]

[Out]

1/4*x/a^3+1/4*x^2*arccot(a*x)/a^2-I*arccot(a*x)^2/a^4-3/4*x*arccot(a*x)^2/a^3+1/4*x^3*arccot(a*x)^2/a-1/4*arcc
ot(a*x)^3/a^4+1/4*x^4*arccot(a*x)^3-1/4*arctan(a*x)/a^4+2*arccot(a*x)*ln(2/(1+I*a*x))/a^4-I*polylog(2,1-2/(1+I
*a*x))/a^4

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Rubi [A]
time = 0.27, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4947, 5037, 327, 209, 5041, 4965, 2449, 2352, 4931, 5005} \begin {gather*} -\frac {\text {ArcTan}(a x)}{4 a^4}-\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{a^4}-\frac {\cot ^{-1}(a x)^3}{4 a^4}-\frac {i \cot ^{-1}(a x)^2}{a^4}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^4}+\frac {x}{4 a^3}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {x^3 \cot ^{-1}(a x)^2}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(4*a^3) + (x^2*ArcCot[a*x])/(4*a^2) - (I*ArcCot[a*x]^2)/a^4 - (3*x*ArcCot[a*x]^2)/(4*a^3) + (x^3*ArcCot[a*x]
^2)/(4*a) - ArcCot[a*x]^3/(4*a^4) + (x^4*ArcCot[a*x]^3)/4 - ArcTan[a*x]/(4*a^4) + (2*ArcCot[a*x]*Log[2/(1 + I*
a*x)])/a^4 - (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4

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

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Mathematica [A]
time = 0.26, size = 96, normalized size = 0.65 \begin {gather*} \frac {a x+\left (-4 i-3 a x+a^3 x^3\right ) \cot ^{-1}(a x)^2+\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^3+\cot ^{-1}(a x) \left (1+a^2 x^2+8 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )-4 i \text {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{4 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x + (-4*I - 3*a*x + a^3*x^3)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x]^3 + ArcCot[a*x]*(1 + a^2*x^2 + 8*Lo
g[1 - E^((2*I)*ArcCot[a*x])]) - (4*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(4*a^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.13, size = 1017, normalized size = 6.87

method	result	size
risch	\(\frac {x}{4 a^{3}}-\frac {511 \arctan \left (a x \right )}{768 a^{4}}+\frac {3 i x^{4} \ln \left (-i a x +1\right )}{256}+\left (-\frac {3 i \left (a^{4} x^{4}-1\right ) \ln \left (-i a x +1\right )^{2}}{32 a^{4}}+\frac {x \left (3 \pi \,a^{3} x^{3}+2 a^{2} x^{2}-6\right ) \ln \left (-i a x +1\right )}{16 a^{3}}-\frac {-3 i \pi ^{2} a^{4} x^{4}-4 i \pi \,a^{3} x^{3}-4 i a^{2} x^{2}+12 i \pi a x +6 \ln \left (-i a x +1\right ) \pi +16 i \ln \left (-i a x +1\right )}{32 a^{4}}\right ) \ln \left (i a x +1\right )+\frac {i}{4 a^{4}}-\frac {7 i \pi \arctan \left (a x \right )}{64 a^{4}}-\frac {i \left (a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{3}}{32 a^{4}}+\frac {3 i \ln \left (-i a x +1\right )^{2} x^{2}}{64 a^{2}}-\frac {13 i \ln \left (-i a x +1\right ) x^{2}}{128 a^{2}}-\frac {3 i \pi ^{2} \ln \left (-i a x +1\right ) x^{4}}{32}+\frac {i \ln \left (-i a x +1\right ) \left (-i a x +1\right )}{2 a^{4}}-\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i a x}{2}\right )}{a^{4}}-\frac {5 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )}{32 a^{4}}+\frac {i \ln \left (-i a x +1\right ) \left (-i a x +1\right )^{3}}{12 a^{4}}-\frac {3 \pi \ln \left (-i a x +1\right ) \left (-i a x +1\right )^{2}}{16 a^{4}}-\frac {3 \pi \ln \left (-i a x +1\right ) \left (-i a x +1\right )^{4}}{64 a^{4}}+\frac {\pi \ln \left (-i a x +1\right ) \left (-i a x +1\right )^{3}}{8 a^{4}}-\frac {3 \pi \ln \left (-i a x +1\right ) x^{2}}{32 a^{2}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{a^{4}}+\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )^{2}}{32 a^{4}}-\frac {i \left (-i a x +1\right )^{3} \ln \left (-i a x +1\right )^{2}}{16 a^{4}}-\frac {3 i \left (-i a x +1\right )^{4} \ln \left (-i a x +1\right )}{256 a^{4}}+\frac {3 i \left (-i a x +1\right )^{4} \ln \left (-i a x +1\right )^{2}}{128 a^{4}}-\frac {i \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{a^{4}}-\frac {319 i \ln \left (a^{2} x^{2}+1\right )}{1536 a^{4}}-\frac {\left (-3 i x^{4} \ln \left (-i a x +1\right ) a^{4}+3 \pi \,a^{4} x^{4}+2 a^{3} x^{3}+3 i \ln \left (-i a x +1\right )-6 a x -3 \pi +8 i\right ) \ln \left (i a x +1\right )^{2}}{32 a^{4}}+\frac {3 \pi \ln \left (-i a x +1\right )^{2}}{32 a^{4}}-\frac {3 \pi \ln \left (-i a x +1\right )^{2} x^{4}}{32}+\frac {3 \pi \ln \left (-i a x +1\right ) x^{4}}{64}-\frac {\ln \left (-i a x +1\right )^{2} x^{3}}{32 a}+\frac {7 \ln \left (-i a x +1\right ) x^{3}}{192 a}+\frac {3 \ln \left (-i a x +1\right )^{2} x}{32 a^{3}}-\frac {25 \ln \left (-i a x +1\right ) x}{64 a^{3}}-\frac {i \pi ^{2}}{4 a^{4}}-\frac {i \ln \left (-i a x +1\right )^{3}}{32 a^{4}}+\frac {25 i \ln \left (-i a x +1\right )^{2}}{128 a^{4}}-\frac {3 i \ln \left (-i a x +1\right )^{2} x^{4}}{128}+\frac {i \ln \left (-i a x +1\right )^{3} x^{4}}{32}-\frac {57 \pi \ln \left (a^{2} x^{2}+1\right )}{128 a^{4}}+\frac {3 \pi ^{2} \arctan \left (a x \right )}{16 a^{4}}+\frac {3 i \pi \ln \left (-i a x +1\right ) x}{16 a^{3}}-\frac {i \pi \ln \left (-i a x +1\right ) x^{3}}{16 a}+\frac {\pi }{8 a^{4}}-\frac {\pi ^{3}}{32 a^{4}}+\frac {x^{4} \pi ^{3}}{32}-\frac {3 \pi ^{2} x}{16 a^{3}}+\frac {\pi ^{2} x^{3}}{16 a}+\frac {\pi \,x^{2}}{8 a^{2}}\)	\(950\)
derivativedivides	\(\text {Expression too large to display}\)	\(1017\)
default	\(\text {Expression too large to display}\)	\(1017\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(1/4*a^4*x^4*arccot(a*x)^3+1/4*a^3*x^3*arccot(a*x)^2-3/4*a*x*arccot(a*x)^2-1/2*I*(a^2*x^2+1)^(1/2)/(2*(a
^2*x^2+1)^(1/2)+2*a*x+2*I)-3/4*I*arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))-1/4*arccot(a*x)^3-1/8*(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)+1/8*arccot(a*x)*(-I*(a^2*x^2+1)^(1/2)+I*a*x-1)+1/4*
(a^2*x^2+1+(a^2*x^2+1)^(1/2)*a*x)*arccot(a*x)+2*arccot(a*x)*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+1/8*arccot(a*x)*(I
*(a^2*x^2+1)^(1/2)+I*a*x-1)-3/8*Pi*arccot(a*x)^2+1/4*(a^2*x^2+1-(a^2*x^2+1)^(1/2)*a*x)*arccot(a*x)-1/8*(-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)+3/8*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+3/16*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)^2-3/16*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+3/16*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+1/2*I*(a^2*x^2+1)^(1/2)/(2*a*x+2*I-2*(a^2*x^2+1)^(1/2))+2*I*dilog((I+a*x)/(a^2*x^2+1)^(
1/2))-I*arccot(a*x)^2-2*I*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))-3/16*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)^2-3/8*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*arccot(a*x)^2+3/8*Pi*csgn(I/((I+a*x)
^2/(a^2*x^2+1)-1))^2*arccot(a*x)^2-3/16*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*arccot(a*x)^2-3/16*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))*arc
cot(a*x)^2+3/4*arccot(a*x)^2*arctan(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^3*arccot(a*x)^3,x, algorithm="maxima")

[Out]

1/64*(8*a^4*x^4*arctan2(1, a*x)^3 + 4*a^3*x^3*arctan2(1, a*x)^2 + 4*(512*a^5*integrate(1/64*x^5*arctan(1/(a*x)
)^3/(a^5*x^2 + a^3), x) + 192*a^4*integrate(1/64*x^4*arctan(1/(a*x))^2/(a^5*x^2 + a^3), x) + 48*a^4*integrate(
1/64*x^4*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x) + 64*a^4*integrate(1/64*x^4*log(a^2*x^2 + 1)/(a^5*x^2 + a^3),
x) + 512*a^3*integrate(1/64*x^3*arctan(1/(a*x))^3/(a^5*x^2 + a^3), x) + 128*a^3*integrate(1/64*x^3*arctan(1/(a
*x))/(a^5*x^2 + a^3), x) - 192*a^2*integrate(1/64*x^2*log(a^2*x^2 + 1)/(a^5*x^2 + a^3), x) - 384*a*integrate(1
/64*x*arctan(1/(a*x))/(a^5*x^2 + a^3), x) - arctan(a*x)^3/a^4 - 3*arctan(a*x)^2*arctan(1/(a*x))/a^4 - 3*arctan
(a*x)*arctan(1/(a*x))^2/a^4 - 48*integrate(1/64*log(a^2*x^2 + 1)^2/(a^5*x^2 + a^3), x))*a^4 - 12*a*x*arctan2(1
, a*x)^2 - 8*arctan2(1, a*x)^3 - (a^3*x^3 - 3*a*x)*log(a^2*x^2 + 1)^2)/a^4

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

[Out]

integral(x^3*arccot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x^3*arccot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*acot(a*x)^3, x)

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