3.2 \(\int x^4 \cot ^{-1}(a x) \, dx\)

Optimal. Leaf size=49 \[ -\frac{x^2}{10 a^3}+\frac{\log \left (a^2 x^2+1\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x) \]

[Out]

-x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCot[a*x])/5 + Log[1 + a^2*x^2]/(10*a^5)

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Rubi [A]  time = 0.0363023, antiderivative size = 49, 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, 266, 43} \[ -\frac{x^2}{10 a^3}+\frac{\log \left (a^2 x^2+1\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCot[a*x],x]

[Out]

-x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCot[a*x])/5 + Log[1 + a^2*x^2]/(10*a^5)

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 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 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])

Rubi steps

\begin{align*} \int x^4 \cot ^{-1}(a x) \, dx &=\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{1}{5} a \int \frac{x^5}{1+a^2 x^2} \, dx\\ &=\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{1}{10} a \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{1}{10} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{10 a^3}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x)+\frac{\log \left (1+a^2 x^2\right )}{10 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0123587, size = 49, normalized size = 1. \[ -\frac{x^2}{10 a^3}+\frac{\log \left (a^2 x^2+1\right )}{10 a^5}+\frac{x^4}{20 a}+\frac{1}{5} x^5 \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCot[a*x],x]

[Out]

-x^2/(10*a^3) + x^4/(20*a) + (x^5*ArcCot[a*x])/5 + Log[1 + a^2*x^2]/(10*a^5)

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Maple [A]  time = 0.04, size = 42, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{10\,{a}^{3}}}+{\frac{{x}^{4}}{20\,a}}+{\frac{{x}^{5}{\rm arccot} \left (ax\right )}{5}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{10\,{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccot(a*x),x)

[Out]

-1/10*x^2/a^3+1/20*x^4/a+1/5*x^5*arccot(a*x)+1/10*ln(a^2*x^2+1)/a^5

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Maxima [A]  time = 0.97502, size = 62, normalized size = 1.27 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arccot}\left (a x\right ) + \frac{1}{20} \, a{\left (\frac{a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac{2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x),x, algorithm="maxima")

[Out]

1/5*x^5*arccot(a*x) + 1/20*a*((a^2*x^4 - 2*x^2)/a^4 + 2*log(a^2*x^2 + 1)/a^6)

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Fricas [A]  time = 1.85585, size = 104, normalized size = 2.12 \begin{align*} \frac{4 \, a^{5} x^{5} \operatorname{arccot}\left (a x\right ) + a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} + 1\right )}{20 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x),x, algorithm="fricas")

[Out]

1/20*(4*a^5*x^5*arccot(a*x) + a^4*x^4 - 2*a^2*x^2 + 2*log(a^2*x^2 + 1))/a^5

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Sympy [A]  time = 1.19608, size = 46, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acot}{\left (a x \right )}}{5} + \frac{x^{4}}{20 a} - \frac{x^{2}}{10 a^{3}} + \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi x^{5}}{10} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acot(a*x),x)

[Out]

Piecewise((x**5*acot(a*x)/5 + x**4/(20*a) - x**2/(10*a**3) + log(a**2*x**2 + 1)/(10*a**5), Ne(a, 0)), (pi*x**5
/10, True))

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Giac [A]  time = 1.08115, size = 68, normalized size = 1.39 \begin{align*} \frac{1}{5} \, x^{5} \arctan \left (\frac{1}{a x}\right ) + \frac{1}{20} \, a{\left (\frac{a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac{2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccot(a*x),x, algorithm="giac")

[Out]

1/5*x^5*arctan(1/(a*x)) + 1/20*a*((a^2*x^4 - 2*x^2)/a^4 + 2*log(a^2*x^2 + 1)/a^6)