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

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

[Out]

-x/(4*a^3) + x^3/(12*a) + (x^4*ArcCot[a*x])/4 + ArcTan[a*x]/(4*a^4)

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Rubi [A]  time = 0.0208234, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4853, 302, 203} \[ -\frac{x}{4 a^3}+\frac{\tan ^{-1}(a x)}{4 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(4*a^3) + x^3/(12*a) + (x^4*ArcCot[a*x])/4 + ArcTan[a*x]/(4*a^4)

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 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 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}(a x) \, dx &=\frac{1}{4} x^4 \cot ^{-1}(a x)+\frac{1}{4} a \int \frac{x^4}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} x^4 \cot ^{-1}(a x)+\frac{1}{4} a \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{x}{4 a^3}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \cot ^{-1}(a x)+\frac{\int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}\\ &=-\frac{x}{4 a^3}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \cot ^{-1}(a x)+\frac{\tan ^{-1}(a x)}{4 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0022742, size = 41, normalized size = 1. \[ -\frac{x}{4 a^3}+\frac{\tan ^{-1}(a x)}{4 a^4}+\frac{x^3}{12 a}+\frac{1}{4} x^4 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(4*a^3) + x^3/(12*a) + (x^4*ArcCot[a*x])/4 + ArcTan[a*x]/(4*a^4)

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Maple [A]  time = 0.042, size = 34, normalized size = 0.8 \begin{align*} -{\frac{x}{4\,{a}^{3}}}+{\frac{{x}^{3}}{12\,a}}+{\frac{{x}^{4}{\rm arccot} \left (ax\right )}{4}}+{\frac{\arctan \left ( ax \right ) }{4\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x/a^3+1/12*x^3/a+1/4*x^4*arccot(a*x)+1/4*arctan(a*x)/a^4

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Maxima [A]  time = 1.45369, size = 51, normalized size = 1.24 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{arccot}\left (a x\right ) + \frac{1}{12} \, a{\left (\frac{a^{2} x^{3} - 3 \, x}{a^{4}} + \frac{3 \, \arctan \left (a x\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arccot(a*x) + 1/12*a*((a^2*x^3 - 3*x)/a^4 + 3*arctan(a*x)/a^5)

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Fricas [A]  time = 1.85826, size = 78, normalized size = 1.9 \begin{align*} \frac{a^{3} x^{3} - 3 \, a x + 3 \,{\left (a^{4} x^{4} - 1\right )} \operatorname{arccot}\left (a x\right )}{12 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.883826, size = 39, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acot}{\left (a x \right )}}{4} + \frac{x^{3}}{12 a} - \frac{x}{4 a^{3}} - \frac{\operatorname{acot}{\left (a x \right )}}{4 a^{4}} & \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),x)

[Out]

Piecewise((x**4*acot(a*x)/4 + x**3/(12*a) - x/(4*a**3) - acot(a*x)/(4*a**4), Ne(a, 0)), (pi*x**4/8, True))

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Giac [A]  time = 1.11311, size = 61, normalized size = 1.49 \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (\frac{1}{a x}\right ) + \frac{1}{12} \, a{\left (\frac{3 \, \arctan \left (a x\right )}{a^{5}} + \frac{a^{4} x^{3} - 3 \, a^{2} x}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arctan(1/(a*x)) + 1/12*a*(3*arctan(a*x)/a^5 + (a^4*x^3 - 3*a^2*x)/a^6)