Integrand size = 10, antiderivative size = 80 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {x^2}{12 a^2}-\frac {x \cot ^{-1}(a x)}{2 a^3}+\frac {x^3 \cot ^{-1}(a x)}{6 a}-\frac {\cot ^{-1}(a x)^2}{4 a^4}+\frac {1}{4} x^4 \cot ^{-1}(a x)^2-\frac {\log \left (1+a^2 x^2\right )}{3 a^4} \] Output:
1/12*x^2/a^2-1/2*x*arccot(a*x)/a^3+1/6*x^3*arccot(a*x)/a-1/4*arccot(a*x)^2 /a^4+1/4*x^4*arccot(a*x)^2-1/3*ln(a^2*x^2+1)/a^4
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {a^2 x^2+2 a x \left (-3+a^2 x^2\right ) \cot ^{-1}(a x)+3 \left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^2-4 \log \left (1+a^2 x^2\right )}{12 a^4} \] Input:
Integrate[x^3*ArcCot[a*x]^2,x]
Output:
(a^2*x^2 + 2*a*x*(-3 + a^2*x^2)*ArcCot[a*x] + 3*(-1 + a^4*x^4)*ArcCot[a*x] ^2 - 4*Log[1 + a^2*x^2])/(12*a^4)
Time = 0.73 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.39, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5362, 5452, 5362, 243, 49, 2009, 5452, 5346, 240, 5420}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \cot ^{-1}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{2} a \int \frac {x^4 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\int x^2 \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{3} a \int \frac {x^3}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \int \frac {x^2}{a^2 x^2+1}dx^2+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (a^2 x^2+1\right )}\right )dx^2+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\int \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5346 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {a \int \frac {x}{a^2 x^2+1}dx+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5420 |
| \(\displaystyle \frac {1}{2} a \left (\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}}{a^2}\right )+\frac {1}{4} x^4 \cot ^{-1}(a x)^2\) |
Input:
Int[x^3*ArcCot[a*x]^2,x]
Output:
(x^4*ArcCot[a*x]^2)/4 + (a*(((x^3*ArcCot[a*x])/3 + (a*(x^2/a^2 - Log[1 + a ^2*x^2]/a^4))/6)/a^2 - (ArcCot[a*x]^2/(2*a^3) + (x*ArcCot[a*x] + Log[1 + a ^2*x^2]/(2*a))/a^2)/a^2))/2
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
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] + Simp[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]
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[-(a + b*ArcCot[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] )^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88
| method | result | size |
| parallelrisch | \(-\frac {-3 a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}-2 a^{3} x^{3} \operatorname {arccot}\left (a x \right )-a^{2} x^{2}+6 \,\operatorname {arccot}\left (a x \right ) a x +1+3 \operatorname {arccot}\left (a x \right )^{2}+4 \ln \left (a^{2} x^{2}+1\right )}{12 a^{4}}\) | \(70\) |
| parts | \(\frac {x^{4} \operatorname {arccot}\left (a x \right )^{2}}{4}+\frac {\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{3}-\operatorname {arccot}\left (a x \right ) a x +\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\frac {a^{2} x^{2}}{6}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{2}}{2 a^{4}}\) | \(76\) |
| derivativedivides | \(\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}}{4}+\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{6}-\frac {\operatorname {arccot}\left (a x \right ) a x}{2}+\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}+\frac {a^{2} x^{2}}{12}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{4}}{a^{4}}\) | \(78\) |
| default | \(\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}}{4}+\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{6}-\frac {\operatorname {arccot}\left (a x \right ) a x}{2}+\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}+\frac {a^{2} x^{2}}{12}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{4}}{a^{4}}\) | \(78\) |
| risch | \(-\frac {\left (a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{16 a^{4}}+\frac {\left (3 i \pi \,a^{4} x^{4}+3 x^{4} \ln \left (-i a x +1\right ) a^{4}+2 i a^{3} x^{3}-6 i a x -3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{24 a^{4}}-\frac {i \pi \,x^{4} \ln \left (-i a x +1\right )}{8}-\frac {x^{4} \ln \left (-i a x +1\right )^{2}}{16}+\frac {\pi ^{2} x^{4}}{16}-\frac {i x^{3} \ln \left (-i a x +1\right )}{12 a}+\frac {\pi \,x^{3}}{12 a}+\frac {i x \ln \left (-i a x +1\right )}{4 a^{3}}+\frac {x^{2}}{12 a^{2}}-\frac {\pi x}{4 a^{3}}+\frac {\ln \left (-i a x +1\right )^{2}}{16 a^{4}}+\frac {\pi \arctan \left (a x \right )}{4 a^{4}}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3 a^{4}}\) | \(224\) |
Input:
int(x^3*arccot(a*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/12*(-3*a^4*x^4*arccot(a*x)^2-2*a^3*x^3*arccot(a*x)-a^2*x^2+6*arccot(a*x )*a*x+1+3*arccot(a*x)^2+4*ln(a^2*x^2+1))/a^4
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )^{2} + 2 \, {\left (a^{3} x^{3} - 3 \, a x\right )} \operatorname {arccot}\left (a x\right ) - 4 \, \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{4}} \] Input:
integrate(x^3*arccot(a*x)^2,x, algorithm="fricas")
Output:
1/12*(a^2*x^2 + 3*(a^4*x^4 - 1)*arccot(a*x)^2 + 2*(a^3*x^3 - 3*a*x)*arccot (a*x) - 4*log(a^2*x^2 + 1))/a^4
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {acot}^{2}{\left (a x \right )}}{4} + \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{6 a} + \frac {x^{2}}{12 a^{2}} - \frac {x \operatorname {acot}{\left (a x \right )}}{2 a^{3}} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{3 a^{4}} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{4 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{4}}{16} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*acot(a*x)**2,x)
Output:
Piecewise((x**4*acot(a*x)**2/4 + x**3*acot(a*x)/(6*a) + x**2/(12*a**2) - x *acot(a*x)/(2*a**3) - log(a**2*x**2 + 1)/(3*a**4) - acot(a*x)**2/(4*a**4), Ne(a, 0)), (pi**2*x**4/16, True))
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {1}{4} \, x^{4} \operatorname {arccot}\left (a x\right )^{2} + \frac {1}{6} \, a {\left (\frac {a^{2} x^{3} - 3 \, x}{a^{4}} + \frac {3 \, \arctan \left (a x\right )}{a^{5}}\right )} \operatorname {arccot}\left (a x\right ) + \frac {a^{2} x^{2} + 3 \, \arctan \left (a x\right )^{2} - 4 \, \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{4}} \] Input:
integrate(x^3*arccot(a*x)^2,x, algorithm="maxima")
Output:
1/4*x^4*arccot(a*x)^2 + 1/6*a*((a^2*x^3 - 3*x)/a^4 + 3*arctan(a*x)/a^5)*ar ccot(a*x) + 1/12*(a^2*x^2 + 3*arctan(a*x)^2 - 4*log(a^2*x^2 + 1))/a^4
\[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\int { x^{3} \operatorname {arccot}\left (a x\right )^{2} \,d x } \] Input:
integrate(x^3*arccot(a*x)^2,x, algorithm="giac")
Output:
integrate(x^3*arccot(a*x)^2, x)
Time = 1.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {x^4\,{\mathrm {acot}\left (a\,x\right )}^2}{4}-\frac {\frac {\ln \left (a^2\,x^2+1\right )}{3}-\frac {a^2\,x^2}{12}+\frac {{\mathrm {acot}\left (a\,x\right )}^2}{4}-\frac {a^3\,x^3\,\mathrm {acot}\left (a\,x\right )}{6}+\frac {a\,x\,\mathrm {acot}\left (a\,x\right )}{2}}{a^4} \] Input:
int(x^3*acot(a*x)^2,x)
Output:
(x^4*acot(a*x)^2)/4 - (log(a^2*x^2 + 1)/3 - (a^2*x^2)/12 + acot(a*x)^2/4 - (a^3*x^3*acot(a*x))/6 + (a*x*acot(a*x))/2)/a^4
Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int x^3 \cot ^{-1}(a x)^2 \, dx=\frac {3 \mathit {acot} \left (a x \right )^{2} a^{4} x^{4}-3 \mathit {acot} \left (a x \right )^{2}+2 \mathit {acot} \left (a x \right ) a^{3} x^{3}-6 \mathit {acot} \left (a x \right ) a x -4 \,\mathrm {log}\left (a^{2} x^{2}+1\right )+a^{2} x^{2}}{12 a^{4}} \] Input:
int(x^3*acot(a*x)^2,x)
Output:
(3*acot(a*x)**2*a**4*x**4 - 3*acot(a*x)**2 + 2*acot(a*x)*a**3*x**3 - 6*aco t(a*x)*a*x - 4*log(a**2*x**2 + 1) + a**2*x**2)/(12*a**4)