3.13 \(\int x^4 \cot ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=135 \[ \frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{5 a^5}+\frac {3 \tan ^{-1}(a x)}{10 a^5}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{5 a^5}-\frac {3 x}{10 a^4}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^3}{30 a^2}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {x^4 \cot ^{-1}(a x)}{10 a} \]

[Out]

-3/10*x/a^4+1/30*x^3/a^2-1/5*x^2*arccot(a*x)/a^3+1/10*x^4*arccot(a*x)/a+1/5*I*arccot(a*x)^2/a^5+1/5*x^5*arccot
(a*x)^2+3/10*arctan(a*x)/a^5-2/5*arccot(a*x)*ln(2/(1+I*a*x))/a^5+1/5*I*polylog(2,1-2/(1+I*a*x))/a^5

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Rubi [A]  time = 0.21, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {4853, 4917, 302, 203, 321, 4921, 4855, 2402, 2315} \[ \frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5}+\frac {x^3}{30 a^2}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}-\frac {3 x}{10 a^4}+\frac {3 \tan ^{-1}(a x)}{10 a^5}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {x^4 \cot ^{-1}(a x)}{10 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCot[a*x]^2,x]

[Out]

(-3*x)/(10*a^4) + x^3/(30*a^2) - (x^2*ArcCot[a*x])/(5*a^3) + (x^4*ArcCot[a*x])/(10*a) + ((I/5)*ArcCot[a*x]^2)/
a^5 + (x^5*ArcCot[a*x]^2)/5 + (3*ArcTan[a*x])/(10*a^5) - (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(5*a^5) + ((I/5)*P
olyLog[2, 1 - 2/(1 + I*a*x)])/a^5

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

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Mathematica [A]  time = 0.55, size = 95, normalized size = 0.70 \[ \frac {6 \left (a^5 x^5+i\right ) \cot ^{-1}(a x)^2+a x \left (a^2 x^2-9\right )+3 \cot ^{-1}(a x) \left (a^4 x^4-2 a^2 x^2-4 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )-3\right )+6 i \text {Li}_2\left (e^{2 i \cot ^{-1}(a x)}\right )}{30 a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*ArcCot[a*x]^2,x]

[Out]

(a*x*(-9 + a^2*x^2) + 6*(I + a^5*x^5)*ArcCot[a*x]^2 + 3*ArcCot[a*x]*(-3 - 2*a^2*x^2 + a^4*x^4 - 4*Log[1 - E^((
2*I)*ArcCot[a*x])]) + (6*I)*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(30*a^5)

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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {arccot}\left (a x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^4*arccot(a*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {arccot}\left (a x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^4*arccot(a*x)^2, x)

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maple [B]  time = 0.14, size = 233, normalized size = 1.73 \[ \frac {x^{5} \mathrm {arccot}\left (a x \right )^{2}}{5}+\frac {x^{4} \mathrm {arccot}\left (a x \right )}{10 a}-\frac {x^{2} \mathrm {arccot}\left (a x \right )}{5 a^{3}}+\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{5 a^{5}}+\frac {x^{3}}{30 a^{2}}-\frac {3 x}{10 a^{4}}+\frac {3 \arctan \left (a x \right )}{10 a^{5}}-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{10 a^{5}}+\frac {i \ln \left (a x -i\right )^{2}}{20 a^{5}}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{10 a^{5}}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{10 a^{5}}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{10 a^{5}}-\frac {i \ln \left (a x +i\right )^{2}}{20 a^{5}}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{10 a^{5}}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{10 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccot(a*x)^2,x)

[Out]

1/5*x^5*arccot(a*x)^2+1/10*x^4*arccot(a*x)/a-1/5*x^2*arccot(a*x)/a^3+1/5/a^5*arccot(a*x)*ln(a^2*x^2+1)+1/30*x^
3/a^2-3/10*x/a^4+3/10*arctan(a*x)/a^5-1/10*I/a^5*ln(a*x-I)*ln(a^2*x^2+1)+1/20*I/a^5*ln(a*x-I)^2+1/10*I/a^5*dil
og(-1/2*I*(I+a*x))+1/10*I/a^5*ln(a*x-I)*ln(-1/2*I*(I+a*x))+1/10*I/a^5*ln(I+a*x)*ln(a^2*x^2+1)-1/20*I/a^5*ln(I+
a*x)^2-1/10*I/a^5*dilog(1/2*I*(a*x-I))-1/10*I/a^5*ln(I+a*x)*ln(1/2*I*(a*x-I))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{20} \, x^{5} \arctan \left (1, a x\right )^{2} - \frac {1}{80} \, x^{5} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {60 \, a^{2} x^{6} \arctan \left (1, a x\right )^{2} + 4 \, a^{2} x^{6} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a x^{5} \arctan \left (1, a x\right ) + 60 \, x^{4} \arctan \left (1, a x\right )^{2} + 5 \, {\left (a^{2} x^{6} + x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{80 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x)^2,x, algorithm="maxima")

[Out]

1/20*x^5*arctan2(1, a*x)^2 - 1/80*x^5*log(a^2*x^2 + 1)^2 + integrate(1/80*(60*a^2*x^6*arctan2(1, a*x)^2 + 4*a^
2*x^6*log(a^2*x^2 + 1) + 8*a*x^5*arctan2(1, a*x) + 60*x^4*arctan2(1, a*x)^2 + 5*(a^2*x^6 + x^4)*log(a^2*x^2 +
1)^2)/(a^2*x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acot(a*x)^2,x)

[Out]

int(x^4*acot(a*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acot(a*x)**2,x)

[Out]

Integral(x**4*acot(a*x)**2, x)

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