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

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

[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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Rubi [A]  time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.00 \[ -\frac {x^2}{10 a^3}+\frac {\log \left (a^2 x^2+1\right )}{10 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {x^4}{20 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*x^2/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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fricas [A]  time = 0.54, size = 45, normalized size = 0.92 \[ \frac {4 \, a^{5} x^{5} \operatorname {arccot}\left (a x\right ) + a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} + 1\right )}{20 \, a^{5}} \]

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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giac [A]  time = 0.12, size = 74, normalized size = 1.51 \[ \frac {1}{20} \, {\left (\frac {4 \, x^{5} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{4} {\left (\frac {2}{a^{2} x^{2}} - \frac {3}{a^{4} x^{4}} - 1\right )}}{a^{2}} + \frac {2 \, \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{6}} - \frac {2 \, \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{6}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*x^5*arctan(1/(a*x))/a - x^4*(2/(a^2*x^2) - 3/(a^4*x^4) - 1)/a^2 + 2*log(1/(a^2*x^2) + 1)/a^6 - 2*log(1
/(a^2*x^2))/a^6)*a

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maple [A]  time = 0.04, size = 42, normalized size = 0.86 \[ -\frac {x^{2}}{10 a^{3}}+\frac {x^{4}}{20 a}+\frac {x^{5} \mathrm {arccot}\left (a x \right )}{5}+\frac {\ln \left (a^{2} x^{2}+1\right )}{10 a^{5}} \]

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 = 0.31, size = 46, normalized size = 0.94 \[ \frac {1}{5} \, x^{5} \operatorname {arccot}\left (a x\right ) + \frac {1}{20} \, a {\left (\frac {a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \]

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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mupad [B]  time = 0.70, size = 56, normalized size = 1.14 \[ \left \{\begin {array}{cl} \frac {\pi \,x^5}{10} & \text {\ if\ \ }a=0\\ \frac {2\,\ln \left (a^2\,x^2+1\right )-2\,a^2\,x^2+a^4\,x^4}{20\,a^5}+\frac {x^5\,\mathrm {acot}\left (a\,x\right )}{5} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(a == 0, (x^5*pi)/10, a ~= 0, (2*log(a^2*x^2 + 1) - 2*a^2*x^2 + a^4*x^4)/(20*a^5) + (x^5*acot(a*x))/5
)

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sympy [A]  time = 0.87, size = 46, normalized size = 0.94 \[ \begin {cases} \frac {x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {x^{4}}{20 a} - \frac {x^{2}}{10 a^{3}} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \text {otherwise} \end {cases} \]

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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