3.26 \(\int x^2 \cot ^{-1}(a x)^3 \, dx\)
Optimal. Leaf size=157 \[ \frac {\text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{2 a^3}-\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \cot ^{-1}(a x)}{a^3}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a^3}+\frac {x \cot ^{-1}(a x)}{a^2}+\frac {\log \left (a^2 x^2+1\right )}{2 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {x^2 \cot ^{-1}(a x)^2}{2 a} \]
[Out]
x*arccot(a*x)/a^2+1/2*arccot(a*x)^2/a^3+1/2*x^2*arccot(a*x)^2/a-1/3*I*arccot(a*x)^3/a^3+1/3*x^3*arccot(a*x)^3+
arccot(a*x)^2*ln(2/(1+I*a*x))/a^3+1/2*ln(a^2*x^2+1)/a^3-I*arccot(a*x)*polylog(2,1-2/(1+I*a*x))/a^3+1/2*polylog
(3,1-2/(1+I*a*x))/a^3
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Rubi [A] time = 0.30, antiderivative size = 157, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used
= {4853, 4917, 4847, 260, 4885, 4921, 4855, 4995, 6610} \[ \frac {\text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3}-\frac {i \cot ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3}+\frac {\log \left (a^2 x^2+1\right )}{2 a^3}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {x \cot ^{-1}(a x)}{a^2}+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)^2}{a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {x^2 \cot ^{-1}(a x)^2}{2 a} \]
Antiderivative was successfully verified.
[In]
Int[x^2*ArcCot[a*x]^3,x]
[Out]
(x*ArcCot[a*x])/a^2 + ArcCot[a*x]^2/(2*a^3) + (x^2*ArcCot[a*x]^2)/(2*a) - ((I/3)*ArcCot[a*x]^3)/a^3 + (x^3*Arc
Cot[a*x]^3)/3 + (ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/a^3 + Log[1 + a^2*x^2]/(2*a^3) - (I*ArcCot[a*x]*PolyLog[2,
1 - 2/(1 + I*a*x)])/a^3 + PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^3)
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 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 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 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 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 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 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 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^2 \cot ^{-1}(a x)^3 \, dx &=\frac {1}{3} x^3 \cot ^{-1}(a x)^3+a \int \frac {x^3 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\int x \cot ^{-1}(a x)^2 \, dx}{a}-\frac {\int \frac {x \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}\\ &=\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\int \frac {\cot ^{-1}(a x)^2}{i-a x} \, dx}{a^2}+\int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {\int \cot ^{-1}(a x) \, dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {2 \int \frac {\cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}\\ &=\frac {x \cot ^{-1}(a x)}{a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}-\frac {i \cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^3}-\frac {i \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}+\frac {\int \frac {x}{1+a^2 x^2} \, dx}{a}\\ &=\frac {x \cot ^{-1}(a x)}{a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {\log \left (1+a^2 x^2\right )}{2 a^3}-\frac {i \cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^3}+\frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 149, normalized size = 0.95 \[ \frac {8 a^3 x^3 \cot ^{-1}(a x)^3-24 \log \left (\frac {1}{a x \sqrt {\frac {1}{a^2 x^2}+1}}\right )+12 a^2 x^2 \cot ^{-1}(a x)^2+24 i \cot ^{-1}(a x) \text {Li}_2\left (e^{-2 i \cot ^{-1}(a x)}\right )+12 \text {Li}_3\left (e^{-2 i \cot ^{-1}(a x)}\right )+8 i \cot ^{-1}(a x)^3+12 \cot ^{-1}(a x)^2+24 a x \cot ^{-1}(a x)+24 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-i \pi ^3}{24 a^3} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x^2*ArcCot[a*x]^3,x]
[Out]
((-I)*Pi^3 + 24*a*x*ArcCot[a*x] + 12*ArcCot[a*x]^2 + 12*a^2*x^2*ArcCot[a*x]^2 + (8*I)*ArcCot[a*x]^3 + 8*a^3*x^
3*ArcCot[a*x]^3 + 24*ArcCot[a*x]^2*Log[1 - E^((-2*I)*ArcCot[a*x])] - 24*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])])/(24*a^3)
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arccot}\left (a x\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccot(a*x)^3,x, algorithm="fricas")
[Out]
integral(x^2*arccot(a*x)^3, x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arccot}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccot(a*x)^3,x, algorithm="giac")
[Out]
integrate(x^2*arccot(a*x)^3, x)
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maple [C] time = 4.19, size = 1815, normalized size = 11.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*arccot(a*x)^3,x)
[Out]
x*arccot(a*x)/a^2+1/3*x^3*arccot(a*x)^3+1/2*x^2*arccot(a*x)^2/a-1/3*I*arccot(a*x)^3/a^3-1/2/a^3*arccot(a*x)^2*
ln(a^2*x^2+1)+1/a^3*arccot(a*x)^2*ln((I+a*x)/(a^2*x^2+1)^(1/2))-1/a^3*arccot(a*x)^2*ln((I+a*x)^2/(a^2*x^2+1)-1
)+1/a^3*arccot(a*x)^2*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+1/a^3*arccot(a*x)^2*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+I/a^
3*arccot(a*x)+1/a^3*ln(2)*arccot(a*x)^2-1/a^3*ln((I+a*x)/(a^2*x^2+1)^(1/2)-1)-1/a^3*ln(1+(I+a*x)/(a^2*x^2+1)^(
1/2))+2/a^3*polylog(3,-(I+a*x)/(a^2*x^2+1)^(1/2))+2/a^3*polylog(3,(I+a*x)/(a^2*x^2+1)^(1/2))-1/8/a^2*arccot(a*
x)^2*csgn(-I*(I+a*x)^4/(a^2*x^2+1)^2+2*I*(I+a*x)^2/(a^2*x^2+1)-I)^3*Pi*x+1/2*I/a^3*arccot(a*x)^2*Pi-2*I/a^3*ar
ccot(a*x)*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-2*I/a^3*arccot(a*x)*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-1/8/a
^2*arccot(a*x)^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^3*Pi*x-1/8*I/a^3*arccot(a*x)^2*csgn(-I*(I+a*x)^4/(a^2*x^2
+1)^2+2*I*(I+a*x)^2/(a^2*x^2+1)-I)^3*Pi-1/4*I/a^3*arccot(a*x)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*Pi+1/4*I/a^3*a
rccot(a*x)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^3*Pi+1/8*I/a^3*arccot(a*x)^2*csgn(I*((I
+a*x)^2/(a^2*x^2+1)-1)^2)^3*Pi-1/2*I/a^3*arccot(a*x)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^
2)^2*Pi-1/4/a^2*arccot(a*x)^2*csgn(-I*(I+a*x)^4/(a^2*x^2+1)^2+2*I*(I+a*x)^2/(a^2*x^2+1)-I)^2*csgn(I*(I+a*x)^2/
(a^2*x^2+1)-I)*Pi*x-1/8/a^2*arccot(a*x)^2*csgn(-I*(I+a*x)^4/(a^2*x^2+1)^2+2*I*(I+a*x)^2/(a^2*x^2+1)-I)*csgn(I*
(I+a*x)^2/(a^2*x^2+1)-I)^2*Pi*x+1/4/a^2*arccot(a*x)^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*((I+a*x)^2/
(a^2*x^2+1)-1))*Pi*x-1/8/a^2*arccot(a*x)^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1
))^2*Pi*x-1/4*I/a^3*arccot(a*x)^2*csgn(-I*(I+a*x)^4/(a^2*x^2+1)^2+2*I*(I+a*x)^2/(a^2*x^2+1)-I)^2*csgn(I*(I+a*x
)^2/(a^2*x^2+1)-I)*Pi-1/8*I/a^3*arccot(a*x)^2*csgn(-I*(I+a*x)^4/(a^2*x^2+1)^2+2*I*(I+a*x)^2/(a^2*x^2+1)-I)*csg
n(I*(I+a*x)^2/(a^2*x^2+1)-I)^2*Pi-1/4*I/a^3*arccot(a*x)^2*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(I+a*x)^2
/(a^2*x^2+1))*Pi+1/2*I/a^3*arccot(a*x)^2*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(I+a*x)^2/(a^2*x^2+1))^2*Pi+
1/4*I/a^3*arccot(a*x)^2*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
)^2*Pi+1/4*I/a^3*arccot(a*x)^2*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*csgn(I/((I+a*x)^2/(
a^2*x^2+1)-1)^2)*Pi-1/4*I/a^3*arccot(a*x)^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*((I+a*x)^2/(a^2*x^2+1
)-1))*Pi+1/8*I/a^3*arccot(a*x)^2*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1))^2*Pi+1/
2*arccot(a*x)^2/a^3-1/4*I/a^3*arccot(a*x)^2*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)*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)^2)*Pi
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{24} \, x^{3} \arctan \left (1, a x\right )^{3} - \frac {1}{32} \, x^{3} \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {28 \, a^{2} x^{4} \arctan \left (1, a x\right )^{3} + 4 \, a^{2} x^{4} \arctan \left (1, a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 4 \, a x^{3} \arctan \left (1, a x\right )^{2} + 28 \, x^{2} \arctan \left (1, a x\right )^{3} + {\left (3 \, a^{2} x^{4} \arctan \left (1, a x\right ) - a x^{3} + 3 \, x^{2} \arctan \left (1, a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{32 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*arccot(a*x)^3,x, algorithm="maxima")
[Out]
1/24*x^3*arctan2(1, a*x)^3 - 1/32*x^3*arctan2(1, a*x)*log(a^2*x^2 + 1)^2 + integrate(1/32*(28*a^2*x^4*arctan2(
1, a*x)^3 + 4*a^2*x^4*arctan2(1, a*x)*log(a^2*x^2 + 1) + 4*a*x^3*arctan2(1, a*x)^2 + 28*x^2*arctan2(1, a*x)^3
+ (3*a^2*x^4*arctan2(1, a*x) - a*x^3 + 3*x^2*arctan2(1, a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^2 + 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*acot(a*x)^3,x)
[Out]
int(x^2*acot(a*x)^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*acot(a*x)**3,x)
[Out]
Integral(x**2*acot(a*x)**3, x)
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