Integrand size = 10, antiderivative size = 152 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {2 x^3}{15 a}+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{5 \sqrt {2} a^{5/2}}-\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}}+\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{10 \sqrt {2} a^{5/2}} \]
2/15*x^3/a+1/5*x^5*arccot(a*x^2)-1/10*arctan(-1+x*2^(1/2)*a^(1/2))/a^(5/2) *2^(1/2)-1/10*arctan(1+x*2^(1/2)*a^(1/2))/a^(5/2)*2^(1/2)-1/20*ln(1+a*x^2- x*2^(1/2)*a^(1/2))/a^(5/2)*2^(1/2)+1/20*ln(1+a*x^2+x*2^(1/2)*a^(1/2))/a^(5 /2)*2^(1/2)
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {8 a^{3/2} x^3+12 a^{5/2} x^5 \cot ^{-1}\left (a x^2\right )+6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {a} x\right )-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {a} x\right )-3 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )+3 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{60 a^{5/2}} \]
(8*a^(3/2)*x^3 + 12*a^(5/2)*x^5*ArcCot[a*x^2] + 6*Sqrt[2]*ArcTan[1 - Sqrt[ 2]*Sqrt[a]*x] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[a]*x] - 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] + 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2 ])/(60*a^(5/2))
Time = 0.42 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5362, 843, 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.
| \(\displaystyle \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {2}{5} a \int \frac {x^6}{a^2 x^4+1}dx+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\int \frac {x^2}{a^2 x^4+1}dx}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\int \frac {a x^2+1}{a^2 x^4+1}dx}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\int \frac {1}{-\left (1-\sqrt {2} \sqrt {a} x\right )^2-1}d\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}-\frac {\int \frac {1}{-\left (\sqrt {2} \sqrt {a} x+1\right )^2-1}d\left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {-\frac {\int -\frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 \sqrt {2} a}+\frac {\int \frac {\sqrt {2} \sqrt {a} x+1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{5} a \left (\frac {x^3}{3 a^2}-\frac {\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}}{2 a}}{a^2}\right )+\frac {1}{5} x^5 \cot ^{-1}\left (a x^2\right )\) |
(x^5*ArcCot[a*x^2])/5 + (2*a*(x^3/(3*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))/a^2))/5
3.1.80.3.1 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^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && 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 = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(\frac {x^{5} \operatorname {arccot}\left (a \,x^{2}\right )}{5}+\frac {2 a \left (\frac {x^{3}}{3 a^{2}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{4} \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\right )}{5}\) | \(112\) |
| parts | \(\frac {x^{5} \operatorname {arccot}\left (a \,x^{2}\right )}{5}+\frac {2 a \left (\frac {x^{3}}{3 a^{2}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a^{4} \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\right )}{5}\) | \(112\) |
1/5*x^5*arccot(a*x^2)+2/5*a*(1/3*x^3/a^2-1/8/a^4/(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 = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {6 \, a x^{5} \operatorname {arccot}\left (a x^{2}\right ) + 4 \, x^{3} - 3 \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right ) + 3 i \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (i \, a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right ) - 3 i \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (-i \, a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right ) + 3 \, a \left (-\frac {1}{a^{10}}\right )^{\frac {1}{4}} \log \left (-a^{7} \left (-\frac {1}{a^{10}}\right )^{\frac {3}{4}} + x\right )}{30 \, a} \]
1/30*(6*a*x^5*arccot(a*x^2) + 4*x^3 - 3*a*(-1/a^10)^(1/4)*log(a^7*(-1/a^10 )^(3/4) + x) + 3*I*a*(-1/a^10)^(1/4)*log(I*a^7*(-1/a^10)^(3/4) + x) - 3*I* a*(-1/a^10)^(1/4)*log(-I*a^7*(-1/a^10)^(3/4) + x) + 3*a*(-1/a^10)^(1/4)*lo g(-a^7*(-1/a^10)^(3/4) + x))/a
Time = 9.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\begin {cases} \frac {x^{5} \operatorname {acot}{\left (a x^{2} \right )}}{5} + \frac {2 x^{3}}{15 a} - \frac {\sqrt [4]{- \frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )}}{5 a^{2}} - \frac {\log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{5 a^{3} \sqrt [4]{- \frac {1}{a^{2}}}} + \frac {\log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{10 a^{3} \sqrt [4]{- \frac {1}{a^{2}}}} - \frac {\operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{5 a^{3} \sqrt [4]{- \frac {1}{a^{2}}}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \text {otherwise} \end {cases} \]
Piecewise((x**5*acot(a*x**2)/5 + 2*x**3/(15*a) - (-1/a**2)**(1/4)*acot(a*x **2)/(5*a**2) - log(x - (-1/a**2)**(1/4))/(5*a**3*(-1/a**2)**(1/4)) + log( x**2 + sqrt(-1/a**2))/(10*a**3*(-1/a**2)**(1/4)) - atan(x/(-1/a**2)**(1/4) )/(5*a**3*(-1/a**2)**(1/4)), Ne(a, 0)), (pi*x**5/10, True))
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{5} \, x^{5} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{60} \, a {\left (\frac {8 \, x^{3}}{a^{2}} - \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )}}{a^{2}}\right )} \]
1/5*x^5*arccot(a*x^2) + 1/60*a*(8*x^3/a^2 - 3*(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)*log(a*x^2 - sqrt(2)*sqrt(a)*x + 1)/a^(3/ 2))/a^2)
Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{5} \, x^{5} \arctan \left (\frac {1}{a x^{2}}\right ) + \frac {1}{60} \, a {\left (\frac {8 \, x^{3}}{a^{2}} - \frac {6 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} {\left | a \right |}^{\frac {3}{2}}} - \frac {6 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2} {\left | a \right |}^{\frac {3}{2}}} + \frac {3 \, \sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{4}} - \frac {3 \, \sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2} {\left | a \right |}^{\frac {3}{2}}}\right )} \]
1/5*x^5*arctan(1/(a*x^2)) + 1/60*a*(8*x^3/a^2 - 6*sqrt(2)*arctan(1/2*sqrt( 2)*(2*x + sqrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/(a^2*abs(a)^(3/2)) - 6*sqrt( 2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/(a^2*abs( a)^(3/2)) + 3*sqrt(2)*sqrt(abs(a))*log(x^2 + sqrt(2)*x/sqrt(abs(a)) + 1/ab s(a))/a^4 - 3*sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/(a^2*ab s(a)^(3/2)))
Time = 0.48 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.36 \[ \int x^4 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^5\,\mathrm {acot}\left (a\,x^2\right )}{5}+\frac {2\,x^3}{15\,a}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{5\,a^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5\,a^{5/2}} \]