3.27 \(\int x \cot ^{-1}(a x)^3 \, dx\)

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

[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

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Rubi [A]  time = 0.170855, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4853, 4917, 4847, 4921, 4855, 2402, 2315, 4885} \[ \frac{3 i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2}+\frac{\cot ^{-1}(a x)^3}{2 a^2}+\frac{3 i \cot ^{-1}(a x)^2}{2 a^2}-\frac{3 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^2}+\frac{1}{2} x^2 \cot ^{-1}(a x)^3+\frac{3 x \cot ^{-1}(a x)^2}{2 a} \]

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 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 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 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 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 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 2402

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 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]

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) \operatorname{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*}

Mathematica [A]  time = 0.0857338, size = 76, normalized size = 0.74 \[ \frac{3 i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x) \left (\left (a^2 x^2+1\right ) \cot ^{-1}(a x)^2+3 (a x+i) \cot ^{-1}(a x)-6 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )}{2 a^2} \]

Warning: Unable to verify antiderivative.

[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 [A]  time = 0.316, size = 162, normalized size = 1.6 \begin{align*}{\frac{{x}^{2} \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{2}}+{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{2\,{a}^{2}}}+{\frac{3\,x \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{2\,a}}+{\frac{{\frac{3\,i}{2}} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{{a}^{2}}}-3\,{\frac{{\rm arccot} \left (ax\right )}{{a}^{2}}\ln \left ( 1-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-3\,{\frac{{\rm arccot} \left (ax\right )}{{a}^{2}}\ln \left ( 1+{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{3\,i}{{a}^{2}}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{3\,i}{{a}^{2}}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{arccot}\left (a x\right )^{3}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{acot}^{3}{\left (a x \right )}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{arccot}\left (a x\right )^{3}\,{d x} \end{align*}

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)