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

Optimal. Leaf size=148 \[ -\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{a^4}-\frac {\tan ^{-1}(a x)}{4 a^4}-\frac {\cot ^{-1}(a x)^3}{4 a^4}-\frac {i \cot ^{-1}(a x)^2}{a^4}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^4}+\frac {x}{4 a^3}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {x^3 \cot ^{-1}(a x)^2}{4 a} \]

[Out]

1/4*x/a^3+1/4*x^2*arccot(a*x)/a^2-I*arccot(a*x)^2/a^4-3/4*x*arccot(a*x)^2/a^3+1/4*x^3*arccot(a*x)^2/a-1/4*arcc
ot(a*x)^3/a^4+1/4*x^4*arccot(a*x)^3-1/4*arctan(a*x)/a^4+2*arccot(a*x)*ln(2/(1+I*a*x))/a^4-I*polylog(2,1-2/(1+I
*a*x))/a^4

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Rubi [A]  time = 0.39, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4853, 4917, 321, 203, 4921, 4855, 2402, 2315, 4847, 4885} \[ -\frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}+\frac {x}{4 a^3}-\frac {\tan ^{-1}(a x)}{4 a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}-\frac {\cot ^{-1}(a x)^3}{4 a^4}-\frac {i \cot ^{-1}(a x)^2}{a^4}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {x^3 \cot ^{-1}(a x)^2}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(4*a^3) + (x^2*ArcCot[a*x])/(4*a^2) - (I*ArcCot[a*x]^2)/a^4 - (3*x*ArcCot[a*x]^2)/(4*a^3) + (x^3*ArcCot[a*x]
^2)/(4*a) - ArcCot[a*x]^3/(4*a^4) + (x^4*ArcCot[a*x]^3)/4 - ArcTan[a*x]/(4*a^4) + (2*ArcCot[a*x]*Log[2/(1 + I*
a*x)])/a^4 - (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4

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 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^3 \cot ^{-1}(a x)^3 \, dx &=\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {1}{4} (3 a) \int \frac {x^4 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {3 \int x^2 \cot ^{-1}(a x)^2 \, dx}{4 a}-\frac {3 \int \frac {x^2 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}\\ &=\frac {x^3 \cot ^{-1}(a x)^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {1}{2} \int \frac {x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {3 \int \cot ^{-1}(a x)^2 \, dx}{4 a^3}+\frac {3 \int \frac {\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}\\ &=-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {\int x \cot ^{-1}(a x) \, dx}{2 a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {3 \int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}\\ &=\frac {x^2 \cot ^{-1}(a x)}{4 a^2}-\frac {i \cot ^{-1}(a x)^2}{a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {\int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac {3 \int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{4 a}\\ &=\frac {x}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}-\frac {i \cot ^{-1}(a x)^2}{a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}-\frac {\int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}+\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}+\frac {3 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}\\ &=\frac {x}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}-\frac {i \cot ^{-1}(a x)^2}{a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3-\frac {\tan ^{-1}(a x)}{4 a^4}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}-\frac {i \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}\\ &=\frac {x}{4 a^3}+\frac {x^2 \cot ^{-1}(a x)}{4 a^2}-\frac {i \cot ^{-1}(a x)^2}{a^4}-\frac {3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac {x^3 \cot ^{-1}(a x)^2}{4 a}-\frac {\cot ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^3-\frac {\tan ^{-1}(a x)}{4 a^4}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}-\frac {i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^4}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 96, normalized size = 0.65 \[ \frac {\left (a^4 x^4-1\right ) \cot ^{-1}(a x)^3+\left (a^3 x^3-3 a x-4 i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (a^2 x^2+8 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )+1\right )-4 i \text {Li}_2\left (e^{2 i \cot ^{-1}(a x)}\right )+a x}{4 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x + (-4*I - 3*a*x + a^3*x^3)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x]^3 + ArcCot[a*x]*(1 + a^2*x^2 + 8*Lo
g[1 - E^((2*I)*ArcCot[a*x])]) - (4*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(4*a^4)

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {arccot}\left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arccot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arccot(a*x)^3, x)

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maple [A]  time = 1.55, size = 209, normalized size = 1.41 \[ \frac {x^{4} \mathrm {arccot}\left (a x \right )^{3}}{4}-\frac {\mathrm {arccot}\left (a x \right )^{3}}{4 a^{4}}+\frac {x^{3} \mathrm {arccot}\left (a x \right )^{2}}{4 a}-\frac {3 x \mathrm {arccot}\left (a x \right )^{2}}{4 a^{3}}+\frac {x^{2} \mathrm {arccot}\left (a x \right )}{4 a^{2}}-\frac {i \mathrm {arccot}\left (a x \right )^{2}}{a^{4}}+\frac {\mathrm {arccot}\left (a x \right )}{4 a^{4}}+\frac {x}{4 a^{3}}-\frac {i}{4 a^{4}}+\frac {2 \,\mathrm {arccot}\left (a x \right ) \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{4}}-\frac {2 i \polylog \left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{4}}+\frac {2 \,\mathrm {arccot}\left (a x \right ) \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{4}}-\frac {2 i \polylog \left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arccot(a*x)^3-1/4*arccot(a*x)^3/a^4+1/4*x^3*arccot(a*x)^2/a-3/4*x*arccot(a*x)^2/a^3+1/4*x^2*arccot(a*x
)/a^2-I*arccot(a*x)^2/a^4+1/4/a^4*arccot(a*x)+1/4*x/a^3-1/4*I/a^4+2/a^4*arccot(a*x)*ln(1+(I+a*x)/(a^2*x^2+1)^(
1/2))-2*I/a^4*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))+2/a^4*arccot(a*x)*ln(1-(I+a*x)/(a^2*x^2+1)^(1/2))-2*I/a^4*
polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*acot(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acot(a*x)**3, x)

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