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

Optimal. Leaf size=51 \[ -\frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x) \]

[Out]

x/(6*a^5) - x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCot[a*x])/6 - ArcTan[a*x]/(6*a^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0251326, antiderivative size = 51, 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} \[ -\frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(6*a^5) - x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCot[a*x])/6 - ArcTan[a*x]/(6*a^6)

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^5 \cot ^{-1}(a x) \, dx &=\frac{1}{6} x^6 \cot ^{-1}(a x)+\frac{1}{6} a \int \frac{x^6}{1+a^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \cot ^{-1}(a x)+\frac{1}{6} a \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{x}{6 a^5}-\frac{x^3}{18 a^3}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)-\frac{\int \frac{1}{1+a^2 x^2} \, dx}{6 a^5}\\ &=\frac{x}{6 a^5}-\frac{x^3}{18 a^3}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x)-\frac{\tan ^{-1}(a x)}{6 a^6}\\ \end{align*}

Mathematica [A]  time = 0.002514, size = 51, normalized size = 1. \[ -\frac{x^3}{18 a^3}+\frac{x}{6 a^5}-\frac{\tan ^{-1}(a x)}{6 a^6}+\frac{x^5}{30 a}+\frac{1}{6} x^6 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(6*a^5) - x^3/(18*a^3) + x^5/(30*a) + (x^6*ArcCot[a*x])/6 - ArcTan[a*x]/(6*a^6)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 42, normalized size = 0.8 \begin{align*}{\frac{x}{6\,{a}^{5}}}-{\frac{{x}^{3}}{18\,{a}^{3}}}+{\frac{{x}^{5}}{30\,a}}+{\frac{{x}^{6}{\rm arccot} \left (ax\right )}{6}}-{\frac{\arctan \left ( ax \right ) }{6\,{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x/a^5-1/18*x^3/a^3+1/30*x^5/a+1/6*x^6*arccot(a*x)-1/6*arctan(a*x)/a^6

________________________________________________________________________________________

Maxima [A]  time = 1.46764, size = 63, normalized size = 1.24 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arccot}\left (a x\right ) + \frac{1}{90} \, a{\left (\frac{3 \, a^{4} x^{5} - 5 \, a^{2} x^{3} + 15 \, x}{a^{6}} - \frac{15 \, \arctan \left (a x\right )}{a^{7}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*arccot(a*x) + 1/90*a*((3*a^4*x^5 - 5*a^2*x^3 + 15*x)/a^6 - 15*arctan(a*x)/a^7)

________________________________________________________________________________________

Fricas [A]  time = 1.84552, size = 100, normalized size = 1.96 \begin{align*} \frac{3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 15 \, a x + 15 \,{\left (a^{6} x^{6} + 1\right )} \operatorname{arccot}\left (a x\right )}{90 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90*(3*a^5*x^5 - 5*a^3*x^3 + 15*a*x + 15*(a^6*x^6 + 1)*arccot(a*x))/a^6

________________________________________________________________________________________

Sympy [A]  time = 1.59348, size = 48, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acot}{\left (a x \right )}}{6} + \frac{x^{5}}{30 a} - \frac{x^{3}}{18 a^{3}} + \frac{x}{6 a^{5}} + \frac{\operatorname{acot}{\left (a x \right )}}{6 a^{6}} & \text{for}\: a \neq 0 \\\frac{\pi x^{6}}{12} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**6*acot(a*x)/6 + x**5/(30*a) - x**3/(18*a**3) + x/(6*a**5) + acot(a*x)/(6*a**6), Ne(a, 0)), (pi*x
**6/12, True))

________________________________________________________________________________________

Giac [A]  time = 1.14345, size = 74, normalized size = 1.45 \begin{align*} \frac{1}{6} \, x^{6} \arctan \left (\frac{1}{a x}\right ) - \frac{1}{90} \, a{\left (\frac{15 \, \arctan \left (a x\right )}{a^{7}} - \frac{3 \, a^{8} x^{5} - 5 \, a^{6} x^{3} + 15 \, a^{4} x}{a^{10}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*arctan(1/(a*x)) - 1/90*a*(15*arctan(a*x)/a^7 - (3*a^8*x^5 - 5*a^6*x^3 + 15*a^4*x)/a^10)