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

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

[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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Rubi [A]  time = 0.02, antiderivative size = 41, 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, 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 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])

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 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^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*}

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Mathematica [A]  time = 0.00, size = 41, normalized size = 1.00 \[ \frac {\tan ^{-1}(a x)}{4 a^4}-\frac {x}{4 a^3}+\frac {1}{4} x^4 \cot ^{-1}(a x)+\frac {x^3}{12 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*x/a^3 + x^3/(12*a) + (x^4*ArcCot[a*x])/4 + ArcTan[a*x]/(4*a^4)

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fricas [A]  time = 1.01, size = 32, normalized size = 0.78 \[ \frac {a^{3} x^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )}{12 \, a^{4}} \]

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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giac [A]  time = 0.13, size = 51, normalized size = 1.24 \[ \frac {1}{12} \, {\left (\frac {3 \, x^{4} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{3} {\left (\frac {3}{a^{2} x^{2}} - 1\right )}}{a^{2}} - \frac {3 \, \arctan \left (\frac {1}{a x}\right )}{a^{5}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 34, normalized size = 0.83 \[ -\frac {x}{4 a^{3}}+\frac {x^{3}}{12 a}+\frac {x^{4} \mathrm {arccot}\left (a x \right )}{4}+\frac {\arctan \left (a x \right )}{4 a^{4}} \]

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 = 0.41, size = 38, normalized size = 0.93 \[ \frac {1}{4} \, x^{4} \operatorname {arccot}\left (a x\right ) + \frac {1}{12} \, a {\left (\frac {a^{2} x^{3} - 3 \, x}{a^{4}} + \frac {3 \, \arctan \left (a x\right )}{a^{5}}\right )} \]

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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mupad [B]  time = 0.72, size = 46, normalized size = 1.12 \[ \left \{\begin {array}{cl} \frac {\pi \,x^4}{8} & \text {\ if\ \ }a=0\\ \frac {3\,\mathrm {atan}\left (a\,x\right )-3\,a\,x+a^3\,x^3}{12\,a^4}+\frac {x^4\,\mathrm {acot}\left (a\,x\right )}{4} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(a == 0, (x^4*pi)/8, a ~= 0, (3*atan(a*x) - 3*a*x + a^3*x^3)/(12*a^4) + (x^4*acot(a*x))/4)

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sympy [A]  time = 0.68, size = 39, normalized size = 0.95 \[ \begin {cases} \frac {x^{4} \operatorname {acot}{\left (a x \right )}}{4} + \frac {x^{3}}{12 a} - \frac {x}{4 a^{3}} - \frac {\operatorname {acot}{\left (a x \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]

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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