∫ x2 cot−1(ax)3 dx [26]

3.1.26 \(\int x^2 \cot ^{-1}(a x)^3 \, dx\) [26]

Optimal. Leaf size=157 \[ \frac {x \cot ^{-1}(a x)}{a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \cot ^{-1}(a x)^2}{2 a}-\frac {i \cot ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^3+\frac {\cot ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {\log \left (1+a^2 x^2\right )}{2 a^3}-\frac {i \cot ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3}+\frac {\text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^3} \]

[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 = 0.21, 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 = {4947, 5037, 4931, 266, 5005, 5041, 4965, 5115, 6745} \begin {gather*} \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} \end {gather*}

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 266

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 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]

Rule 5115

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar

cCot[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 6745

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 = 0.25, size = 149, normalized size = 0.95 \begin {gather*} \frac {-i \pi ^3+24 a x \cot ^{-1}(a x)+12 \cot ^{-1}(a x)^2+12 a^2 x^2 \cot ^{-1}(a x)^2+8 i \cot ^{-1}(a x)^3+8 a^3 x^3 \cot ^{-1}(a x)^3+24 \cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )-24 \log \left (\frac {1}{a \sqrt {1+\frac {1}{a^2 x^2}} x}\right )+24 i \cot ^{-1}(a x) \text {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )+12 \text {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )}{24 a^3} \end {gather*}

Antiderivative was successfully veriﬁed.

[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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.96, size = 1665, normalized size = 10.61


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1665\)
default	\(\text {Expression too large to display}\)	\(1665\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccot(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

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

+a*x)/(a^2*x^2+1)^(1/2))-arccot(a*x)^2*ln((I+a*x)^2/(a^2*x^2+1)-1)+1/24*I*arccot(a*x)*(24+3*I*csgn(I*((I+a*x)^

2/(a^2*x^2+1)-1)^2)^3*arccot(a*x)*Pi*a*x+3*I*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*ar

ccot(a*x)*Pi*a*x-8*arccot(a*x)^2+6*csgn(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^3*arccot(a*x)*Pi+

3*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^3*arccot(a*x)*Pi-6*csgn(I*(I+a*x)^2/(a^2*x^2+1))^3*arccot(a*x)*Pi-12*csg

n(I*(I+a*x)^2/(a^2*x^2+1)/((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*arccot(a*x)*Pi-3*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*arccot(a*x)*Pi-24*I*arccot(a*x)*ln(2)-24*I*a*x+12*arccot(a*x)*Pi-12*I*arccot(a*x)-3

*csgn(I*(I+a*x)^2/(a^2*x^2+1)-I)^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)*arccot(a*x)*Pi

-6*csgn(I*(I+a*x)^2/(a^2*x^2+1)-I)*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*arccot(a*x)*

Pi+6*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))*arccot(a*x)*Pi+

6*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)*arccot(a*x)*

Pi+3*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)*arccot(a*x)*Pi-6*csgn(I*((I+a*x)^

2/(a^2*x^2+1)-1))*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*arccot(a*x)*Pi-6*csgn(I*(I+a*x)/(a^2*x^2+1)^(1/2))^2*c

sgn(I*(I+a*x)^2/(a^2*x^2+1))*arccot(a*x)*Pi+12*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)*Pi-6*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))*cs

gn(I/((I+a*x)^2/(a^2*x^2+1)-1)^2)*arccot(a*x)*Pi+3*I*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)*arccot(a*x)*Pi*a*x-6*I*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*((I+a*x)^2/(a^2*x^2+1)-1)^2)^2*

arccot(a*x)*Pi*a*x+3*I*csgn(I*(I+a*x)^2/(a^2*x^2+1)-I)^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)*arccot(a*x)*Pi*a*x+6*I*csgn(I*(I+a*x)^2/(a^2*x^2+1)-I)*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*arccot(a*x)*Pi*a*x)-ln((I+a*x)/(a^2*x^2+1)^(1/2)-1)-ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+arccot(a*x

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

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

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

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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^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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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^2*arccot(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^2*arccot(a*x)^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {acot}^{3}{\left (a x \right )}\, dx \end {gather*}

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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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^2*arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^2*arccot(a*x)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \end {gather*}

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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