Integrand size = 10, antiderivative size = 150 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\frac {2 a}{3 x}-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}-\frac {a^{3/2} \arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{3 \sqrt {2}}+\frac {a^{3/2} \arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{3 \sqrt {2}}+\frac {a^{3/2} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{6 \sqrt {2}}-\frac {a^{3/2} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{6 \sqrt {2}} \]
2/3*a/x-1/3*arccot(a*x^2)/x^3+1/6*a^(3/2)*arctan(-1+x*2^(1/2)*a^(1/2))*2^( 1/2)+1/6*a^(3/2)*arctan(1+x*2^(1/2)*a^(1/2))*2^(1/2)+1/12*a^(3/2)*ln(1+a*x ^2-x*2^(1/2)*a^(1/2))*2^(1/2)-1/12*a^(3/2)*ln(1+a*x^2+x*2^(1/2)*a^(1/2))*2 ^(1/2)
Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\frac {-4 \cot ^{-1}\left (a x^2\right )+a x^2 \left (8-2 \sqrt {2} \sqrt {a} x \arctan \left (1-\sqrt {2} \sqrt {a} x\right )+2 \sqrt {2} \sqrt {a} x \arctan \left (1+\sqrt {2} \sqrt {a} x\right )+\sqrt {2} \sqrt {a} x \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\sqrt {2} \sqrt {a} x \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )\right )}{12 x^3} \]
(-4*ArcCot[a*x^2] + a*x^2*(8 - 2*Sqrt[2]*Sqrt[a]*x*ArcTan[1 - Sqrt[2]*Sqrt [a]*x] + 2*Sqrt[2]*Sqrt[a]*x*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] + Sqrt[2]*Sqrt[ a]*x*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Sqrt[2]*Sqrt[a]*x*Log[1 + Sqrt[2 ]*Sqrt[a]*x + a*x^2]))/(12*x^3)
Time = 0.40 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5362, 847, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {2}{3} a \int \frac {1}{x^2 \left (a^2 x^4+1\right )}dx-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {2}{3} a \left (a^2 \left (-\int \frac {x^2}{a^2 x^4+1}dx\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\int \frac {a x^2+1}{a^2 x^4+1}dx}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\int \frac {1}{-\left (1-\sqrt {2} \sqrt {a} x\right )^2-1}d\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}-\frac {\int \frac {1}{-\left (\sqrt {2} \sqrt {a} x+1\right )^2-1}d\left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {-\frac {\int -\frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 \sqrt {2} a}+\frac {\int \frac {\sqrt {2} \sqrt {a} x+1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2}{3} a \left (-\left (a^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}}{2 a}\right )\right )-\frac {1}{x}\right )-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}\) |
-1/3*ArcCot[a*x^2]/x^3 - (2*a*(-x^(-1) - a^2*((-(ArcTan[1 - Sqrt[2]*Sqrt[a ]*x]/(Sqrt[2]*Sqrt[a])) + ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(Sqrt[2]*Sqrt[a])) /(2*a) - (-1/2*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(Sqrt[2]*Sqrt[a]) + Log[ 1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(2*Sqrt[2]*Sqrt[a]))/(2*a))))/3
3.1.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
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]
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(-\frac {\operatorname {arccot}\left (a \,x^{2}\right )}{3 x^{3}}-\frac {2 a \left (-\frac {1}{x}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\right )}{3}\) | \(106\) |
| parts | \(-\frac {\operatorname {arccot}\left (a \,x^{2}\right )}{3 x^{3}}-\frac {2 a \left (-\frac {1}{x}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\right )}{3}\) | \(106\) |
-1/3*arccot(a*x^2)/x^3-2/3*a*(-1/x-1/8/(1/a^2)^(1/4)*2^(1/2)*(ln((x^2-(1/a ^2)^(1/4)*x*2^(1/2)+(1/a^2)^(1/2))/(x^2+(1/a^2)^(1/4)*x*2^(1/2)+(1/a^2)^(1 /2)))+2*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)+2*arctan(2^(1/2)/(1/a^2)^(1/4)*x -1)))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\frac {\left (-a^{6}\right )^{\frac {1}{4}} x^{3} \log \left (a^{5} x + \left (-a^{6}\right )^{\frac {3}{4}}\right ) - i \, \left (-a^{6}\right )^{\frac {1}{4}} x^{3} \log \left (a^{5} x + i \, \left (-a^{6}\right )^{\frac {3}{4}}\right ) + i \, \left (-a^{6}\right )^{\frac {1}{4}} x^{3} \log \left (a^{5} x - i \, \left (-a^{6}\right )^{\frac {3}{4}}\right ) - \left (-a^{6}\right )^{\frac {1}{4}} x^{3} \log \left (a^{5} x - \left (-a^{6}\right )^{\frac {3}{4}}\right ) + 4 \, a x^{2} - 2 \, \operatorname {arccot}\left (a x^{2}\right )}{6 \, x^{3}} \]
1/6*((-a^6)^(1/4)*x^3*log(a^5*x + (-a^6)^(3/4)) - I*(-a^6)^(1/4)*x^3*log(a ^5*x + I*(-a^6)^(3/4)) + I*(-a^6)^(1/4)*x^3*log(a^5*x - I*(-a^6)^(3/4)) - (-a^6)^(1/4)*x^3*log(a^5*x - (-a^6)^(3/4)) + 4*a*x^2 - 2*arccot(a*x^2))/x^ 3
Time = 12.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\begin {cases} \frac {a^{2} \sqrt [4]{- \frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )}}{3} + \frac {a \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{3 \sqrt [4]{- \frac {1}{a^{2}}}} - \frac {a \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{6 \sqrt [4]{- \frac {1}{a^{2}}}} + \frac {a \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{3 \sqrt [4]{- \frac {1}{a^{2}}}} + \frac {2 a}{3 x} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\- \frac {\pi }{6 x^{3}} & \text {otherwise} \end {cases} \]
Piecewise((a**2*(-1/a**2)**(1/4)*acot(a*x**2)/3 + a*log(x - (-1/a**2)**(1/ 4))/(3*(-1/a**2)**(1/4)) - a*log(x**2 + sqrt(-1/a**2))/(6*(-1/a**2)**(1/4) ) + a*atan(x/(-1/a**2)**(1/4))/(3*(-1/a**2)**(1/4)) + 2*a/(3*x) - acot(a*x **2)/(3*x**3), Ne(a, 0)), (-pi/(6*x**3), True))
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\frac {1}{12} \, {\left (a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )} + \frac {8}{x}\right )} a - \frac {\operatorname {arccot}\left (a x^{2}\right )}{3 \, x^{3}} \]
1/12*(a^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*a*x + sqrt(2)*sqrt(a))/sqrt(a)) /a^(3/2) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*a*x - sqrt(2)*sqrt(a))/sqrt(a)) /a^(3/2) - sqrt(2)*log(a*x^2 + sqrt(2)*sqrt(a)*x + 1)/a^(3/2) + sqrt(2)*lo g(a*x^2 - sqrt(2)*sqrt(a)*x + 1)/a^(3/2)) + 8/x)*a - 1/3*arccot(a*x^2)/x^3
Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\frac {1}{12} \, {\left (\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{{\left | a \right |}^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{{\left | a \right |}^{\frac {3}{2}}} - \sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right ) + \frac {\sqrt {2} a^{2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{{\left | a \right |}^{\frac {3}{2}}} + \frac {8}{x}\right )} a - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{3 \, x^{3}} \]
1/12*(2*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(a)))*sqrt(a bs(a)))/abs(a)^(3/2) + 2*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqr t(abs(a)))*sqrt(abs(a)))/abs(a)^(3/2) - sqrt(2)*sqrt(abs(a))*log(x^2 + sqr t(2)*x/sqrt(abs(a)) + 1/abs(a)) + sqrt(2)*a^2*log(x^2 - sqrt(2)*x/sqrt(abs (a)) + 1/abs(a))/abs(a)^(3/2) + 8/x)*a - 1/3*arctan(1/(a*x^2))/x^3
Time = 0.84 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.35 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^4} \, dx=\frac {2\,a}{3\,x}-\frac {\mathrm {acot}\left (a\,x^2\right )}{3\,x^3}+\frac {{\left (-1\right )}^{1/4}\,a^{3/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{3}+\frac {{\left (-1\right )}^{1/4}\,a^{3/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \]