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

Optimal. Leaf size=194 \[ \frac {23 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{30 a^6}+\frac {19 \tan ^{-1}(a x)}{60 a^6}+\frac {\cot ^{-1}(a x)^3}{6 a^6}+\frac {23 i \cot ^{-1}(a x)^2}{30 a^6}-\frac {23 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{15 a^6}-\frac {19 x}{60 a^5}+\frac {x \cot ^{-1}(a x)^2}{2 a^5}-\frac {4 x^2 \cot ^{-1}(a x)}{15 a^4}+\frac {x^3}{60 a^3}-\frac {x^3 \cot ^{-1}(a x)^2}{6 a^3}+\frac {x^4 \cot ^{-1}(a x)}{20 a^2}+\frac {1}{6} x^6 \cot ^{-1}(a x)^3+\frac {x^5 \cot ^{-1}(a x)^2}{10 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.67, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, 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 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 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 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 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 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 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]

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 = 0.69, size = 125, normalized size = 0.64 \[ \frac {10 \left (a^6 x^6+1\right ) \cot ^{-1}(a x)^3+a x \left (a^2 x^2-19\right )+2 \left (3 a^5 x^5-5 a^3 x^3+15 a x+23 i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (3 a^4 x^4-16 a^2 x^2-92 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )-19\right )+46 i \text {Li}_2\left (e^{2 i \cot ^{-1}(a x)}\right )}{60 a^6} \]

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)

________________________________________________________________________________________

fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{5} \operatorname {arccot}\left (a x\right )^{3}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {arccot}\left (a x\right )^{3}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 1.88, size = 243, normalized size = 1.25 \[ \frac {x^{6} \mathrm {arccot}\left (a x \right )^{3}}{6}+\frac {\mathrm {arccot}\left (a x \right )^{3}}{6 a^{6}}+\frac {x^{5} \mathrm {arccot}\left (a x \right )^{2}}{10 a}-\frac {x^{3} \mathrm {arccot}\left (a x \right )^{2}}{6 a^{3}}+\frac {x^{4} \mathrm {arccot}\left (a x \right )}{20 a^{2}}-\frac {4 x^{2} \mathrm {arccot}\left (a x \right )}{15 a^{4}}+\frac {x \mathrm {arccot}\left (a x \right )^{2}}{2 a^{5}}+\frac {x^{3}}{60 a^{3}}-\frac {19 x}{60 a^{5}}+\frac {23 i \polylog \left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{15 a^{6}}-\frac {19 \,\mathrm {arccot}\left (a x \right )}{60 a^{6}}+\frac {i}{3 a^{6}}-\frac {23 \,\mathrm {arccot}\left (a x \right ) \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{15 a^{6}}+\frac {23 i \polylog \left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{15 a^{6}}-\frac {23 \,\mathrm {arccot}\left (a x \right ) \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{15 a^{6}}+\frac {23 i \mathrm {arccot}\left (a x \right )^{2}}{30 a^{6}} \]

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+23/15*I/a^6*polylog(2,-(I+
a*x)/(a^2*x^2+1)^(1/2))-19/60/a^6*arccot(a*x)+23/15*I/a^6*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-23/15/a^6*arcco
t(a*x)*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+1/3*I/a^6-23/15/a^6*arccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))+23/30*I
*arccot(a*x)^2/a^6

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,{\mathrm {acot}\left (a\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^5*acot(a*x)^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {acot}^{3}{\left (a x \right )}\, dx \]

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)

________________________________________________________________________________________