3.1.27 \(\int x \cot ^{-1}(a x)^3 \, dx\) [27]

Optimal. Leaf size=103 \[ \frac {3 i \cot ^{-1}(a x)^2}{2 a^2}+\frac {3 x \cot ^{-1}(a x)^2}{2 a}+\frac {\cot ^{-1}(a x)^3}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^3-\frac {3 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^2}+\frac {3 i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4947, 5037, 4931, 5041, 4965, 2449, 2352, 5005} \begin {gather*} \frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^2}+\frac {\cot ^{-1}(a x)^3}{2 a^2}+\frac {3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^3+\frac {3 x \cot ^{-1}(a x)^2}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 76, normalized size = 0.74 \begin {gather*} \frac {\cot ^{-1}(a x) \left (3 (i+a x) \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2-6 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+3 i \text {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcCot[a*x]*(3*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 - 6*Log[1 - E^((2*I)*ArcCot[a*x])]) + (3*I
)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(2*a^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 4393, normalized size = 42.65

method	result	size
risch	\(-\frac {3 i \pi \ln \left (-i a x +1\right ) x}{8 a}+\frac {3 i \pi \arctan \left (a x \right )}{16 a^{2}}+\frac {21 \arctan \left (a x \right )}{32 a^{2}}+\frac {3 \pi ^{2} x}{8 a}-\frac {i \left (a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )^{3}}{16 a^{2}}-\frac {3 i \pi ^{2} \ln \left (-i a x +1\right ) x^{2}}{16}+\frac {3 i x^{2} \ln \left (-i a x +1\right )}{32}+\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )}{32 a^{2}}+\frac {3 \pi \ln \left (-i a x +1\right ) \left (-i a x +1\right )^{2}}{16 a^{2}}-\frac {3 i \ln \left (-i a x +1\right ) \left (-i a x +1\right )}{4 a^{2}}-\frac {3 i \ln \left (-i a x +1\right ) \ln \left (\frac {1}{2}+\frac {i a x}{2}\right )}{2 a^{2}}+\frac {3 i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i a x}{2}\right )}{2 a^{2}}+\frac {x^{2} \pi ^{3}}{16}+\frac {\pi ^{3}}{16 a^{2}}+\frac {21 \pi \ln \left (a^{2} x^{2}+1\right )}{32 a^{2}}+\left (-\frac {3 i \left (a^{2} x^{2}+1\right ) \ln \left (-i a x +1\right )^{2}}{16 a^{2}}+\frac {3 x \left (\pi a x +2\right ) \ln \left (-i a x +1\right )}{8 a}-\frac {3 \left (-i \pi ^{2} a^{2} x^{2}-4 i \pi a x -2 \ln \left (-i a x +1\right ) \pi -4 i \ln \left (-i a x +1\right )\right )}{16 a^{2}}\right ) \ln \left (i a x +1\right )-\frac {3 i \left (-i a x +1\right )^{2} \ln \left (-i a x +1\right )^{2}}{32 a^{2}}+\frac {i \ln \left (-i a x +1\right )^{3} x^{2}}{16}+\frac {i \ln \left (-i a x +1\right )^{3}}{16 a^{2}}-\frac {9 i \ln \left (-i a x +1\right )^{2}}{32 a^{2}}-\frac {3 i \ln \left (-i a x +1\right )^{2} x^{2}}{32}-\frac {3 \pi \ln \left (-i a x +1\right )^{2} x^{2}}{16}+\frac {3 \pi \ln \left (-i a x +1\right ) x^{2}}{16}-\frac {3 \pi \ln \left (-i a x +1\right )^{2}}{16 a^{2}}-\frac {3 \left (-i x^{2} \ln \left (-i a x +1\right ) a^{2}+\pi \,a^{2} x^{2}-i \ln \left (-i a x +1\right )+2 a x +\pi -2 i\right ) \ln \left (i a x +1\right )^{2}}{16 a^{2}}+\frac {9 \ln \left (-i a x +1\right ) x}{16 a}-\frac {3 \ln \left (-i a x +1\right )^{2} x}{16 a}+\frac {21 i \ln \left (a^{2} x^{2}+1\right )}{64 a^{2}}+\frac {3 i \pi ^{2}}{8 a^{2}}+\frac {3 i \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{2 a^{2}}-\frac {3 \pi ^{2} \arctan \left (a x \right )}{8 a^{2}}\)	\(622\)
derivativedivides	\(\text {Expression too large to display}\)	\(4393\)
default	\(\text {Expression too large to display}\)	\(4393\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(1/2*a^2*x^2*arccot(a*x)^3+3/2*a*x*arccot(a*x)^2+3/8*I*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)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*I*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*a
rccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+3/4*I*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*arccot(a*x)*ln(1-(I+a*
x)/(a^2*x^2+1)^(1/2))+3/8*I*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*arccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+3/8
*I*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)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+3/
4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^
2+1)-1))^3*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*dilog(1+(I+a*x)/(a
^2*x^2+1)^(1/2))-3/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*Pi*csgn(I
/((I+a*x)^2/(a^2*x^2+1)-1))^2*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2
*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^2*dilog((I+a*x)/(a^2*x^2+1)^(1/2)
)+3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2
+1)/((I+a*x)^2/(a^2*x^2+1)-1))^3*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I
+a*x)^2/(a^2*x^2+1)-1))^3*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/
(a^2*x^2+1)-1))^3*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1
)-1))^3*dilog((I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))^3*dilog((I+a*x)/(a^2*x^2+1)^
(1/2))+3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*Pi*csgn(I*(I+a*x)^2/(a
^2*x^2+1))^3*dilog((I+a*x)/(a^2*x^2+1)^(1/2))-3/8*Pi*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*dilog(1+(I+a*x)/(a^2*x^2+
1)^(1/2))+1/2*arccot(a*x)^3+3/2*I*arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*dilog((I+a*x)/(a^2*x^2+1)
^(1/2))-3/4*Pi*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*polylog(2
,-(I+a*x)/(a^2*x^2+1)^(1/2))+3/2*I*arccot(a*x)^2-3/2*I*dilog((I+a*x)/(a^2*x^2+1)^(1/2))+3/2*I*dilog(1+(I+a*x)/
(a^2*x^2+1)^(1/2))+3/2*I*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+3/2*I*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-3/2*
arccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))-3*arccot(a*x)*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+3/4*Pi*arccot(a*x)^2
+3/8*I*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)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*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)^2-3/8*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+3/8*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)^2-3/8*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+3/4*I*Pi*arccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))-3/8*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*dilog((I+a*x)/(a^2*x^2+1)^(1/2))-3/8*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*polylog(2,-(I+a*x)/(a^2*x^2+1)
^(1/2))-3/8*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*dilog((
I+a*x)/(a^2*x^2+1)^(1/2))-3/4*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*dilog((I+a*
x)/(a^2*x^2+1)^(1/2))+3/4*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*dilog(1+(I+a*x)
/(a^2*x^2+1)^(1/2))+3/8*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*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*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*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-3/8*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))*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*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)
)*dilog((I+a*x)/(a^2*x^2+1)^(1/2))+3/8*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))*po
lylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*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))*po
lylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-3/4*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*p
olylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-3/8*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*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-3/8*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*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))+3/8*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*dilog(1+(I+a*x)/(a^2*x^2+1)^(1/2))-3
/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*polylog(2,(I
+a*x)/(a^2*x^2+1)^(1/2))+3/8*Pi*csgn(I/((I+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*arccot(a*x)^3,x, algorithm="maxima")

[Out]

1/32*(8*a^2*x^2*arctan2(1, a*x)^3 + 12*a*x*arctan2(1, a*x)^2 - 3*a*x*log(a^2*x^2 + 1)^2 + 4*(128*a^3*integrate
(1/32*x^3*arctan(1/(a*x))^3/(a^3*x^2 + a), x) + 96*a^2*integrate(1/32*x^2*arctan(1/(a*x))^2/(a^3*x^2 + a), x)
+ 24*a^2*integrate(1/32*x^2*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) + 96*a^2*integrate(1/32*x^2*log(a^2*x^2 + 1)/
(a^3*x^2 + a), x) + 128*a*integrate(1/32*x*arctan(1/(a*x))^3/(a^3*x^2 + a), x) + 192*a*integrate(1/32*x*arctan
(1/(a*x))/(a^3*x^2 + a), x) + arctan(a*x)^3/a^2 + 3*arctan(a*x)^2*arctan(1/(a*x))/a^2 + 3*arctan(a*x)*arctan(1
/(a*x))^2/a^2 + 24*integrate(1/32*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x))*a^2 + 8*arctan2(1, a*x)^3)/a^2

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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*arccot(a*x)^3,x, algorithm="fricas")

[Out]

integral(x*arccot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x*arccot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*acot(a*x)^3, x)

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