3.74 \(\int x^5 \cot ^{-1}(a x^2) \, dx\)

Optimal. Leaf size=41 \[ -\frac {\log \left (a^2 x^4+1\right )}{12 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]

[Out]

1/12*x^4/a+1/6*x^6*arccot(a*x^2)-1/12*ln(a^2*x^4+1)/a^3

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Rubi [A]  time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5034, 266, 43} \[ -\frac {\log \left (a^2 x^4+1\right )}{12 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCot[a*x^2],x]

[Out]

x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5034

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

Rubi steps

\begin {align*} \int x^5 \cot ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac {1}{3} a \int \frac {x^7}{1+a^2 x^4} \, dx\\ &=\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac {1}{12} a \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^4\right )\\ &=\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac {1}{12} a \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^4\right )\\ &=\frac {x^4}{12 a}+\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right )-\frac {\log \left (1+a^2 x^4\right )}{12 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 1.00 \[ -\frac {\log \left (a^2 x^4+1\right )}{12 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCot[a*x^2],x]

[Out]

x^4/(12*a) + (x^6*ArcCot[a*x^2])/6 - Log[1 + a^2*x^4]/(12*a^3)

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fricas [A]  time = 1.56, size = 39, normalized size = 0.95 \[ \frac {2 \, a^{3} x^{6} \operatorname {arccot}\left (a x^{2}\right ) + a^{2} x^{4} - \log \left (a^{2} x^{4} + 1\right )}{12 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x^2),x, algorithm="fricas")

[Out]

1/12*(2*a^3*x^6*arccot(a*x^2) + a^2*x^4 - log(a^2*x^4 + 1))/a^3

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giac [A]  time = 0.13, size = 40, normalized size = 0.98 \[ \frac {1}{6} \, x^{6} \arctan \left (\frac {1}{a x^{2}}\right ) + \frac {1}{12} \, {\left (\frac {x^{4}}{a^{2}} - \frac {\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x^2),x, algorithm="giac")

[Out]

1/6*x^6*arctan(1/(a*x^2)) + 1/12*(x^4/a^2 - log(a^2*x^4 + 1)/a^4)*a

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maple [A]  time = 0.04, size = 36, normalized size = 0.88 \[ \frac {x^{4}}{12 a}+\frac {x^{6} \mathrm {arccot}\left (a \,x^{2}\right )}{6}-\frac {\ln \left (a^{2} x^{4}+1\right )}{12 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccot(a*x^2),x)

[Out]

1/12*x^4/a+1/6*x^6*arccot(a*x^2)-1/12*ln(a^2*x^4+1)/a^3

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maxima [A]  time = 0.32, size = 38, normalized size = 0.93 \[ \frac {1}{6} \, x^{6} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{12} \, {\left (\frac {x^{4}}{a^{2}} - \frac {\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccot(a*x^2),x, algorithm="maxima")

[Out]

1/6*x^6*arccot(a*x^2) + 1/12*(x^4/a^2 - log(a^2*x^4 + 1)/a^4)*a

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mupad [B]  time = 0.65, size = 35, normalized size = 0.85 \[ \frac {x^6\,\mathrm {acot}\left (a\,x^2\right )}{6}-\frac {\ln \left (a^2\,x^4+1\right )}{12\,a^3}+\frac {x^4}{12\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*acot(a*x^2),x)

[Out]

(x^6*acot(a*x^2))/6 - log(a^2*x^4 + 1)/(12*a^3) + x^4/(12*a)

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sympy [A]  time = 1.78, size = 39, normalized size = 0.95 \[ \begin {cases} \frac {x^{6} \operatorname {acot}{\left (a x^{2} \right )}}{6} + \frac {x^{4}}{12 a} - \frac {\log {\left (a^{2} x^{4} + 1 \right )}}{12 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi x^{6}}{12} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acot(a*x**2),x)

[Out]

Piecewise((x**6*acot(a*x**2)/6 + x**4/(12*a) - log(a**2*x**4 + 1)/(12*a**3), Ne(a, 0)), (pi*x**6/12, True))

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