3.1.15 \(\int x^2 \cot ^{-1}(a x)^2 \, dx\) [15]

Optimal. Leaf size=111 \[ \frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\text {ArcTan}(a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^3} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4947, 5037, 327, 209, 5041, 4965, 2449, 2352} \begin {gather*} -\frac {\text {ArcTan}(a x)}{3 a^3}-\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{3 a^3}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac {x}{3 a^2}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {x^2 \cot ^{-1}(a x)}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.21, size = 76, normalized size = 0.68 \begin {gather*} \frac {a x+\left (-i+a^3 x^3\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (1+a^2 x^2+2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )-i \text {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{3 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (93 ) = 186\).
time = 0.14, size = 188, normalized size = 1.69

method	result	size
derivativedivides	\(\frac {\frac {a^{3} x^{3} \mathrm {arccot}\left (a x \right )^{2}}{3}+\frac {\mathrm {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}-\frac {i \ln \left (a x -i\right )^{2}}{12}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6}+\frac {i \ln \left (a x +i\right )^{2}}{12}}{a^{3}}\)	\(188\)
default	\(\frac {\frac {a^{3} x^{3} \mathrm {arccot}\left (a x \right )^{2}}{3}+\frac {\mathrm {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\mathrm {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{6}-\frac {i \ln \left (a x -i\right )^{2}}{12}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{6}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{6}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{6}+\frac {i \ln \left (a x +i\right )^{2}}{12}}{a^{3}}\)	\(188\)
risch	\(-\frac {i \ln \left (-i a x +1\right ) x^{2}}{6 a}+\frac {i \ln \left (i a x +1\right ) x^{2}}{6 a}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a^{3}}-\frac {i \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a^{3}}-\frac {i \pi \ln \left (-i a x +1\right ) x^{3}}{6}+\frac {i \pi \ln \left (i a x +1\right ) x^{3}}{6}+\frac {x}{3 a^{2}}+\frac {\pi ^{2} x^{3}}{12}+\frac {11 \pi }{18 a^{3}}-\frac {\ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (-i a x +1\right )^{2} x^{3}}{12}-\frac {\pi \ln \left (a^{2} x^{2}+1\right )}{6 a^{3}}-\frac {\arctan \left (a x \right )}{3 a^{3}}-\frac {i \ln \left (i a x +1\right )^{2}}{12 a^{3}}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}+\frac {\pi \,x^{2}}{6 a}-\frac {17 i}{54 a^{3}}+\frac {i \ln \left (-i a x +1\right )^{2}}{12 a^{3}}-\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{6 a^{3}}-\frac {i \pi ^{2}}{12 a^{3}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{3 a^{3}}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(1/3*a^3*x^3*arccot(a*x)^2+1/3*arccot(a*x)*a^2*x^2-1/3*arccot(a*x)*ln(a^2*x^2+1)+1/3*a*x-1/3*arctan(a*x)
+1/6*I*ln(a*x-I)*ln(a^2*x^2+1)-1/6*I*dilog(-1/2*I*(I+a*x))-1/6*I*ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/12*I*ln(a*x-I)
^2-1/6*I*ln(I+a*x)*ln(a^2*x^2+1)+1/6*I*dilog(1/2*I*(a*x-I))+1/6*I*ln(I+a*x)*ln(1/2*I*(a*x-I))+1/12*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^2*arccot(a*x)^2,x, algorithm="maxima")

[Out]

1/12*x^3*arctan2(1, a*x)^2 - 1/48*x^3*log(a^2*x^2 + 1)^2 + integrate(1/48*(36*a^2*x^4*arctan2(1, a*x)^2 + 4*a^
2*x^4*log(a^2*x^2 + 1) + 8*a*x^3*arctan2(1, a*x) + 36*x^2*arctan2(1, a*x)^2 + 3*(a^2*x^4 + x^2)*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^2*arccot(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^2*arccot(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(x^2*arccot(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*acot(a*x)^2, x)

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