3.23 \(\int x^5 \cot ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=194 \[ \frac{23 i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{30 a^6}+\frac{x^3}{60 a^3}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}-\frac{19 x}{60 a^5}+\frac{19 \tan ^{-1}(a x)}{60 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{15 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{x^5 \cot ^{-1}(a x)^2}{10 a} \]

[Out]

(-19*x)/(60*a^5) + x^3/(60*a^3) - (4*x^2*ArcCot[a*x])/(15*a^4) + (x^4*ArcCot[a*x])/(20*a^2) + (((23*I)/30)*Arc
Cot[a*x]^2)/a^6 + (x*ArcCot[a*x]^2)/(2*a^5) - (x^3*ArcCot[a*x]^2)/(6*a^3) + (x^5*ArcCot[a*x]^2)/(10*a) + ArcCo
t[a*x]^3/(6*a^6) + (x^6*ArcCot[a*x]^3)/6 + (19*ArcTan[a*x])/(60*a^6) - (23*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(15
*a^6) + (((23*I)/30)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^6

________________________________________________________________________________________

Rubi [A]  time = 0.666636, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {4853, 4917, 302, 203, 321, 4921, 4855, 2402, 2315, 4847, 4885} \[ \frac{23 i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{30 a^6}+\frac{x^3}{60 a^3}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}-\frac{19 x}{60 a^5}+\frac{19 \tan ^{-1}(a x)}{60 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac{23 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{15 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{x^5 \cot ^{-1}(a x)^2}{10 a} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCot[a*x]^3,x]

[Out]

(-19*x)/(60*a^5) + x^3/(60*a^3) - (4*x^2*ArcCot[a*x])/(15*a^4) + (x^4*ArcCot[a*x])/(20*a^2) + (((23*I)/30)*Arc
Cot[a*x]^2)/a^6 + (x*ArcCot[a*x]^2)/(2*a^5) - (x^3*ArcCot[a*x]^2)/(6*a^3) + (x^5*ArcCot[a*x]^2)/(10*a) + ArcCo
t[a*x]^3/(6*a^6) + (x^6*ArcCot[a*x]^3)/6 + (19*ArcTan[a*x])/(60*a^6) - (23*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(15
*a^6) + (((23*I)/30)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^6

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 321

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 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 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 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^5 \cot ^{-1}(a x)^3 \, dx &=\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{1}{2} a \int \frac{x^6 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{\int x^4 \cot ^{-1}(a x)^2 \, dx}{2 a}-\frac{\int \frac{x^4 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a}\\ &=\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{1}{5} \int \frac{x^5 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{\int x^2 \cot ^{-1}(a x)^2 \, dx}{2 a^3}+\frac{\int \frac{x^2 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^3}\\ &=-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{\int \cot ^{-1}(a x)^2 \, dx}{2 a^5}-\frac{\int \frac{\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^5}+\frac{\int x^3 \cot ^{-1}(a x) \, dx}{5 a^2}-\frac{\int \frac{x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\int \frac{x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^2}\\ &=\frac{x^4 \cot ^{-1}(a x)}{20 a^2}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3-\frac{\int x \cot ^{-1}(a x) \, dx}{5 a^4}+\frac{\int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^4}-\frac{\int x \cot ^{-1}(a x) \, dx}{3 a^4}+\frac{\int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^4}+\frac{\int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^4}+\frac{\int \frac{x^4}{1+a^2 x^2} \, dx}{20 a}\\ &=-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3-\frac{\int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{5 a^5}-\frac{\int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{3 a^5}-\frac{\int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{a^5}-\frac{\int \frac{x^2}{1+a^2 x^2} \, dx}{10 a^3}-\frac{\int \frac{x^2}{1+a^2 x^2} \, dx}{6 a^3}+\frac{\int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx}{20 a}\\ &=-\frac{19 x}{60 a^5}+\frac{x^3}{60 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3-\frac{23 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^6}+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{20 a^5}+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{10 a^5}+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{6 a^5}-\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^5}-\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^5}-\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^5}\\ &=-\frac{19 x}{60 a^5}+\frac{x^3}{60 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{19 \tan ^{-1}(a x)}{60 a^6}-\frac{23 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^6}+\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^6}+\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^6}+\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{a^6}\\ &=-\frac{19 x}{60 a^5}+\frac{x^3}{60 a^3}-\frac{4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac{x^4 \cot ^{-1}(a x)}{20 a^2}+\frac{23 i \cot ^{-1}(a x)^2}{30 a^6}+\frac{x \cot ^{-1}(a x)^2}{2 a^5}-\frac{x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac{x^5 \cot ^{-1}(a x)^2}{10 a}+\frac{\cot ^{-1}(a x)^3}{6 a^6}+\frac{1}{6} x^6 \cot ^{-1}(a x)^3+\frac{19 \tan ^{-1}(a x)}{60 a^6}-\frac{23 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^6}+\frac{23 i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{30 a^6}\\ \end{align*}

Mathematica [A]  time = 0.600621, size = 125, normalized size = 0.64 \[ \frac{46 i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+a x \left (a^2 x^2-19\right )+10 \left (a^6 x^6+1\right ) \cot ^{-1}(a x)^3+2 \left (3 a^5 x^5-5 a^3 x^3+15 a x+23 i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (3 a^4 x^4-16 a^2 x^2-92 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )-19\right )}{60 a^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5*ArcCot[a*x]^3,x]

[Out]

(a*x*(-19 + a^2*x^2) + 2*(23*I + 15*a*x - 5*a^3*x^3 + 3*a^5*x^5)*ArcCot[a*x]^2 + 10*(1 + a^6*x^6)*ArcCot[a*x]^
3 + ArcCot[a*x]*(-19 - 16*a^2*x^2 + 3*a^4*x^4 - 92*Log[1 - E^((2*I)*ArcCot[a*x])]) + (46*I)*PolyLog[2, E^((2*I
)*ArcCot[a*x])])/(60*a^6)

________________________________________________________________________________________

Maple [A]  time = 0.531, size = 243, normalized size = 1.3 \begin{align*}{\frac{{x}^{6} \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{6}}+{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{6\,{a}^{6}}}+{\frac{{x}^{5} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{10\,a}}-{\frac{{x}^{3} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{6\,{a}^{3}}}+{\frac{{x}^{4}{\rm arccot} \left (ax\right )}{20\,{a}^{2}}}-{\frac{4\,{x}^{2}{\rm arccot} \left (ax\right )}{15\,{a}^{4}}}+{\frac{x \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{2\,{a}^{5}}}+{\frac{{x}^{3}}{60\,{a}^{3}}}-{\frac{19\,x}{60\,{a}^{5}}}+{\frac{{\frac{i}{3}}}{{a}^{6}}}-{\frac{19\,{\rm arccot} \left (ax\right )}{60\,{a}^{6}}}+{\frac{{\frac{23\,i}{15}}}{{a}^{6}}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{23\,{\rm arccot} \left (ax\right )}{15\,{a}^{6}}\ln \left ( 1+{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{23\,i}{15}}}{{a}^{6}}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{23\,{\rm arccot} \left (ax\right )}{15\,{a}^{6}}\ln \left ( 1-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{{\frac{23\,i}{30}} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccot(a*x)^3,x)

[Out]

1/6*x^6*arccot(a*x)^3+1/6*arccot(a*x)^3/a^6+1/10*x^5*arccot(a*x)^2/a-1/6*x^3*arccot(a*x)^2/a^3+1/20*x^4*arccot
(a*x)/a^2-4/15*x^2*arccot(a*x)/a^4+1/2*x*arccot(a*x)^2/a^5+1/60*x^3/a^3-19/60*x/a^5+1/3*I/a^6-19/60/a^6*arccot
(a*x)+23/15*I/a^6*polylog(2,-(a*x+I)/(a^2*x^2+1)^(1/2))-23/15/a^6*arccot(a*x)*ln(1+(a*x+I)/(a^2*x^2+1)^(1/2))+
23/15*I/a^6*polylog(2,(a*x+I)/(a^2*x^2+1)^(1/2))-23/15/a^6*arccot(a*x)*ln(1-(a*x+I)/(a^2*x^2+1)^(1/2))+23/30*I
*arccot(a*x)^2/a^6

________________________________________________________________________________________

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^5*arccot(a*x)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{5} \operatorname{arccot}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^5*arccot(a*x)^3, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{acot}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acot(a*x)**3,x)

[Out]

Integral(x**5*acot(a*x)**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{arccot}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^5*arccot(a*x)^3, x)