3.1.13 \(\int x^4 \cot ^{-1}(a x)^2 \, dx\) [13]

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

[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 = 0.15, 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 = {4947, 5037, 308, 209, 327, 5041, 4965, 2449, 2352} \begin {gather*} \frac {3 \text {ArcTan}(a x)}{10 a^5}+\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{5 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} \end {gather*}

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 209

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

Rule 308

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 327

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

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 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4965

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(
Log[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 5037

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 5041

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) \text {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 = 0.38, size = 95, normalized size = 0.70 \begin {gather*} \frac {a x \left (-9+a^2 x^2\right )+6 \left (i+a^5 x^5\right ) \cot ^{-1}(a x)^2+3 \cot ^{-1}(a x) \left (-3-2 a^2 x^2+a^4 x^4-4 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+6 i \text {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{30 a^5} \end {gather*}

Antiderivative was successfully verified.

[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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Maple [A]
time = 0.16, size = 208, normalized size = 1.54

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \mathrm {arccot}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \mathrm {arccot}\left (a x \right )}{10}-\frac {\mathrm {arccot}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{5}+\frac {a^{3} x^{3}}{30}-\frac {3 a x}{10}+\frac {3 \arctan \left (a x \right )}{10}-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{10}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{10}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{10}+\frac {i \ln \left (a x -i\right )^{2}}{20}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{10}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{10}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{10}-\frac {i \ln \left (a x +i\right )^{2}}{20}}{a^{5}}\)	\(208\)
default	\(\frac {\frac {a^{5} x^{5} \mathrm {arccot}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \mathrm {arccot}\left (a x \right )}{10}-\frac {\mathrm {arccot}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{5}+\frac {a^{3} x^{3}}{30}-\frac {3 a x}{10}+\frac {3 \arctan \left (a x \right )}{10}-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{10}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{10}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{10}+\frac {i \ln \left (a x -i\right )^{2}}{20}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{10}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{10}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{10}-\frac {i \ln \left (a x +i\right )^{2}}{20}}{a^{5}}\)	\(208\)
risch	\(\frac {i \ln \left (i a x +1\right ) x^{4}}{20 a}-\frac {i \ln \left (i a x +1\right ) x^{2}}{10 a^{3}}-\frac {i \ln \left (-i a x +1\right ) x^{4}}{20 a}+\frac {i \ln \left (-i a x +1\right ) x^{2}}{10 a^{3}}+\frac {i \pi \ln \left (i a x +1\right ) x^{5}}{10}-\frac {i \pi \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{10 a^{5}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{5 a^{5}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{5 a^{5}}-\frac {\ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {\ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {3 x}{10 a^{4}}+\frac {x^{3}}{30 a^{2}}+\frac {3 \arctan \left (a x \right )}{10 a^{5}}+\frac {\pi ^{2} x^{5}}{20}-\frac {137 \pi }{300 a^{5}}+\frac {413 i}{2250 a^{5}}+\frac {i \ln \left (i a x +1\right )^{2}}{20 a^{5}}-\frac {i \ln \left (-i a x +1\right )^{2}}{20 a^{5}}+\frac {i \pi ^{2}}{20 a^{5}}+\frac {i \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{5 a^{5}}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {\pi \ln \left (a^{2} x^{2}+1\right )}{10 a^{5}}-\frac {\pi \,x^{2}}{10 a^{3}}+\frac {\pi \,x^{4}}{20 a}\)	\(349\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccot(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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