3.75 \(\int x^3 \cot ^{-1}(a x^2) \, dx\)

Optimal. Leaf size=37 \[ -\frac{\tan ^{-1}\left (a x^2\right )}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]

[Out]

x^2/(4*a) + (x^4*ArcCot[a*x^2])/4 - ArcTan[a*x^2]/(4*a^2)

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Rubi [A]  time = 0.0193072, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5034, 275, 321, 203} \[ -\frac{\tan ^{-1}\left (a x^2\right )}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCot[a*x^2],x]

[Out]

x^2/(4*a) + (x^4*ArcCot[a*x^2])/4 - ArcTan[a*x^2]/(4*a^2)

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]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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])

Rubi steps

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

Mathematica [A]  time = 0.0056528, size = 37, normalized size = 1. \[ -\frac{\tan ^{-1}\left (a x^2\right )}{4 a^2}+\frac{x^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCot[a*x^2],x]

[Out]

x^2/(4*a) + (x^4*ArcCot[a*x^2])/4 - ArcTan[a*x^2]/(4*a^2)

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Maple [A]  time = 0.04, size = 32, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{4\,a}}+{\frac{{x}^{4}{\rm arccot} \left (a{x}^{2}\right )}{4}}-{\frac{\arctan \left ( a{x}^{2} \right ) }{4\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccot(a*x^2),x)

[Out]

1/4*x^2/a+1/4*x^4*arccot(a*x^2)-1/4*arctan(a*x^2)/a^2

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Maxima [A]  time = 1.44328, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccot}\left (a x^{2}\right ) + \frac{1}{4} \, a{\left (\frac{x^{2}}{a^{2}} - \frac{\arctan \left (a x^{2}\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccot(a*x^2),x, algorithm="maxima")

[Out]

1/4*x^4*arccot(a*x^2) + 1/4*a*(x^2/a^2 - arctan(a*x^2)/a^3)

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Fricas [A]  time = 2.10793, size = 63, normalized size = 1.7 \begin{align*} \frac{a x^{2} +{\left (a^{2} x^{4} + 1\right )} \operatorname{arccot}\left (a x^{2}\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccot(a*x^2),x, algorithm="fricas")

[Out]

1/4*(a*x^2 + (a^2*x^4 + 1)*arccot(a*x^2))/a^2

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Sympy [A]  time = 1.48478, size = 36, normalized size = 0.97 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acot}{\left (a x^{2} \right )}}{4} + \frac{x^{2}}{4 a} + \frac{\operatorname{acot}{\left (a x^{2} \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\\frac{\pi x^{4}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acot(a*x**2),x)

[Out]

Piecewise((x**4*acot(a*x**2)/4 + x**2/(4*a) + acot(a*x**2)/(4*a**2), Ne(a, 0)), (pi*x**4/8, True))

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Giac [A]  time = 1.12607, size = 49, normalized size = 1.32 \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (\frac{1}{a x^{2}}\right ) + \frac{1}{4} \, a{\left (\frac{x^{2}}{a^{2}} - \frac{\arctan \left (a x^{2}\right )}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccot(a*x^2),x, algorithm="giac")

[Out]

1/4*x^4*arctan(1/(a*x^2)) + 1/4*a*(x^2/a^2 - arctan(a*x^2)/a^3)