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

Optimal result

Integrand size = 8, antiderivative size = 51 \[ \int x^5 \cot ^{-1}(a x) \, dx=\frac {x}{6 a^5}-\frac {x^3}{18 a^3}+\frac {x^5}{30 a}+\frac {1}{6} x^6 \cot ^{-1}(a x)-\frac {\arctan (a x)}{6 a^6} \]

[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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 308, 209} \[ \int x^5 \cot ^{-1}(a x) \, dx=-\frac {\arctan (a x)}{6 a^6}+\frac {x}{6 a^5}-\frac {x^3}{18 a^3}+\frac {1}{6} x^6 \cot ^{-1}(a x)+\frac {x^5}{30 a} \]

[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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 308

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 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \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 {\arctan (a x)}{6 a^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int x^5 \cot ^{-1}(a x) \, dx=\frac {x}{6 a^5}-\frac {x^3}{18 a^3}+\frac {x^5}{30 a}+\frac {1}{6} x^6 \cot ^{-1}(a x)-\frac {\arctan (a x)}{6 a^6} \]

[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] (verified)

Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\frac {a^{6} x^{6} \operatorname {arccot}\left (a x \right )}{6}+\frac {a^{5} x^{5}}{30}-\frac {a^{3} x^{3}}{18}+\frac {a x}{6}-\frac {\arctan \left (a x \right )}{6}}{a^{6}}\)	\(44\)
default	\(\frac {\frac {a^{6} x^{6} \operatorname {arccot}\left (a x \right )}{6}+\frac {a^{5} x^{5}}{30}-\frac {a^{3} x^{3}}{18}+\frac {a x}{6}-\frac {\arctan \left (a x \right )}{6}}{a^{6}}\)	\(44\)
parallelrisch	\(\frac {15 a^{6} x^{6} \operatorname {arccot}\left (a x \right )+3 a^{5} x^{5}-5 a^{3} x^{3}+15 a x +15 \,\operatorname {arccot}\left (a x \right )}{90 a^{6}}\)	\(45\)
parts	\(\frac {x^{6} \operatorname {arccot}\left (a x \right )}{6}+\frac {a \left (\frac {\frac {1}{5} a^{4} x^{5}-\frac {1}{3} a^{2} x^{3}+x}{a^{6}}-\frac {\arctan \left (a x \right )}{a^{7}}\right )}{6}\)	\(46\)
risch	\(\frac {i x^{6} \ln \left (i a x +1\right )}{12}-\frac {i x^{6} \ln \left (-i a x +1\right )}{12}+\frac {x^{6} \pi }{12}+\frac {x^{5}}{30 a}-\frac {x^{3}}{18 a^{3}}+\frac {x}{6 a^{5}}-\frac {\arctan \left (a x \right )}{6 a^{6}}\)	\(67\)

[In]

int(x^5*arccot(a*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80 \[ \int x^5 \cot ^{-1}(a x) \, dx=\frac {3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 15 \, a x + 15 \, {\left (a^{6} x^{6} + 1\right )} \operatorname {arccot}\left (a x\right )}{90 \, a^{6}} \]

[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] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.94 \[ \int x^5 \cot ^{-1}(a x) \, dx=\begin {cases} \frac {x^{6} \operatorname {acot}{\left (a x \right )}}{6} + \frac {x^{5}}{30 a} - \frac {x^{3}}{18 a^{3}} + \frac {x}{6 a^{5}} + \frac {\operatorname {acot}{\left (a x \right )}}{6 a^{6}} & \text {for}\: a \neq 0 \\\frac {\pi x^{6}}{12} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int x^5 \cot ^{-1}(a x) \, dx=\frac {1}{6} \, x^{6} \operatorname {arccot}\left (a x\right ) + \frac {1}{90} \, a {\left (\frac {3 \, a^{4} x^{5} - 5 \, a^{2} x^{3} + 15 \, x}{a^{6}} - \frac {15 \, \arctan \left (a x\right )}{a^{7}}\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.16 \[ \int x^5 \cot ^{-1}(a x) \, dx=\frac {1}{90} \, {\left (\frac {15 \, x^{6} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{5} {\left (\frac {5}{a^{2} x^{2}} - \frac {15}{a^{4} x^{4}} - 3\right )}}{a^{2}} + \frac {15 \, \arctan \left (\frac {1}{a x}\right )}{a^{7}}\right )} a \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int x^5 \cot ^{-1}(a x) \, dx=\left \{\begin {array}{cl} \frac {\pi \,x^6}{12} & \text {\ if\ \ }a=0\\ \frac {x^6\,\mathrm {acot}\left (a\,x\right )}{6}-\frac {\frac {\mathrm {atan}\left (a\,x\right )}{6}-\frac {a\,x}{6}+\frac {a^3\,x^3}{18}-\frac {a^5\,x^5}{30}}{a^6} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

[In]

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

[Out]

piecewise(a == 0, (x^6*pi)/12, a ~= 0, - (atan(a*x)/6 - (a*x)/6 + (a^3*x^3)/18 - (a^5*x^5)/30)/a^6 + (x^6*acot
(a*x))/6)