Integrand size = 8, antiderivative size = 53 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {x \cot ^{-1}(a x)}{a}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {\log \left (1+a^2 x^2\right )}{2 a^2} \] Output:
x*arccot(a*x)/a+1/2*arccot(a*x)^2/a^2+1/2*x^2*arccot(a*x)^2+1/2*ln(a^2*x^2 +1)/a^2
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {2 a x \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2+\log \left (1+a^2 x^2\right )}{2 a^2} \] Input:
Integrate[x*ArcCot[a*x]^2,x]
Output:
(2*a*x*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 + Log[1 + a^2*x^2])/(2*a^ 2)
Time = 0.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5362, 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.
\(\displaystyle \int x \cot ^{-1}(a x)^2 \, dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle a \int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)^2\) |
\(\Big \downarrow \) 5452 |
\(\displaystyle a \left (\frac {\int \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2\) |
\(\Big \downarrow \) 5346 |
\(\displaystyle a \left (\frac {a \int \frac {x}{a^2 x^2+1}dx+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2\) |
\(\Big \downarrow \) 240 |
\(\displaystyle a \left (\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2\) |
\(\Big \downarrow \) 5420 |
\(\displaystyle a \left (\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)^2\) |
Input:
Int[x*ArcCot[a*x]^2,x]
Output:
(x^2*ArcCot[a*x]^2)/2 + a*(ArcCot[a*x]^2/(2*a^3) + (x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a))/a^2)
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
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 = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}+2 \,\operatorname {arccot}\left (a x \right ) a x +\operatorname {arccot}\left (a x \right )^{2}+\ln \left (a^{2} x^{2}+1\right )}{2 a^{2}}\) | \(44\) |
parts | \(\frac {x^{2} \operatorname {arccot}\left (a x \right )^{2}}{2}+\frac {-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{a^{2}}\) | \(55\) |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}}{2}-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{a^{2}}\) | \(57\) |
default | \(\frac {\frac {a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}}{2}-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{a^{2}}\) | \(57\) |
risch | \(-\frac {\left (a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{8 a^{2}}+\frac {\left (i \pi \,a^{2} x^{2}+x^{2} \ln \left (-i a x +1\right ) a^{2}+2 i a x +\ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{4 a^{2}}-\frac {i \pi \,x^{2} \ln \left (-i a x +1\right )}{4}-\frac {x^{2} \ln \left (-i a x +1\right )^{2}}{8}+\frac {\pi ^{2} x^{2}}{8}-\frac {i x \ln \left (-i a x +1\right )}{2 a}+\frac {\pi x}{2 a}-\frac {\ln \left (-i a x +1\right )^{2}}{8 a^{2}}-\frac {\pi \arctan \left (a x \right )}{2 a^{2}}+\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a^{2}}\) | \(178\) |
Input:
int(x*arccot(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/2*(a^2*x^2*arccot(a*x)^2+2*arccot(a*x)*a*x+arccot(a*x)^2+ln(a^2*x^2+1))/ a^2
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {2 \, a x \operatorname {arccot}\left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )^{2} + \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{2}} \] Input:
integrate(x*arccot(a*x)^2,x, algorithm="fricas")
Output:
1/2*(2*a*x*arccot(a*x) + (a^2*x^2 + 1)*arccot(a*x)^2 + log(a^2*x^2 + 1))/a ^2
Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int x \cot ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {acot}^{2}{\left (a x \right )}}{2} + \frac {x \operatorname {acot}{\left (a x \right )}}{a} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a^{2}} + \frac {\operatorname {acot}^{2}{\left (a x \right )}}{2 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{2}}{8} & \text {otherwise} \end {cases} \] Input:
integrate(x*acot(a*x)**2,x)
Output:
Piecewise((x**2*acot(a*x)**2/2 + x*acot(a*x)/a + log(a**2*x**2 + 1)/(2*a** 2) + acot(a*x)**2/(2*a**2), Ne(a, 0)), (pi**2*x**2/8, True))
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {arccot}\left (a x\right )^{2} + a {\left (\frac {x}{a^{2}} - \frac {\arctan \left (a x\right )}{a^{3}}\right )} \operatorname {arccot}\left (a x\right ) - \frac {\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{2}} \] Input:
integrate(x*arccot(a*x)^2,x, algorithm="maxima")
Output:
1/2*x^2*arccot(a*x)^2 + a*(x/a^2 - arctan(a*x)/a^3)*arccot(a*x) - 1/2*(arc tan(a*x)^2 - log(a^2*x^2 + 1))/a^2
\[ \int x \cot ^{-1}(a x)^2 \, dx=\int { x \operatorname {arccot}\left (a x\right )^{2} \,d x } \] Input:
integrate(x*arccot(a*x)^2,x, algorithm="giac")
Output:
integrate(x*arccot(a*x)^2, x)
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {x^2\,{\mathrm {acot}\left (a\,x\right )}^2}{2}+\frac {\frac {{\mathrm {acot}\left (a\,x\right )}^2}{2}+a\,x\,\mathrm {acot}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2+1\right )}{2}}{a^2} \] Input:
int(x*acot(a*x)^2,x)
Output:
(x^2*acot(a*x)^2)/2 + (log(a^2*x^2 + 1)/2 + acot(a*x)^2/2 + a*x*acot(a*x)) /a^2
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.81 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {\mathit {acot} \left (a x \right )^{2} a^{2} x^{2}+\mathit {acot} \left (a x \right )^{2}+2 \mathit {acot} \left (a x \right ) a x +\mathrm {log}\left (a^{2} x^{2}+1\right )}{2 a^{2}} \] Input:
int(x*acot(a*x)^2,x)
Output:
(acot(a*x)**2*a**2*x**2 + acot(a*x)**2 + 2*acot(a*x)*a*x + log(a**2*x**2 + 1))/(2*a**2)