Integrand size = 6, antiderivative size = 132 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x \cot ^{-1}\left (a x^2\right )-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}} \]
x*arccot(a*x^2)+1/2*arctan(-1+x*2^(1/2)*a^(1/2))*2^(1/2)/a^(1/2)+1/2*arcta n(1+x*2^(1/2)*a^(1/2))*2^(1/2)/a^(1/2)+1/4*ln(1+a*x^2-x*2^(1/2)*a^(1/2))*2 ^(1/2)/a^(1/2)-1/4*ln(1+a*x^2+x*2^(1/2)*a^(1/2))*2^(1/2)/a^(1/2)
Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x \cot ^{-1}\left (a x^2\right )+\frac {-2 \arctan \left (1-\sqrt {2} \sqrt {a} x\right )+2 \arctan \left (1+\sqrt {2} \sqrt {a} x\right )+\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}} \]
x*ArcCot[a*x^2] + (-2*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] + 2*ArcTan[1 + Sqrt[2] *Sqrt[a]*x] + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Log[1 + Sqrt[2]*Sqrt[a] *x + a*x^2])/(2*Sqrt[2]*Sqrt[a])
Time = 0.37 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {5346, 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.
| \(\displaystyle \int \cot ^{-1}\left (a x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5346 |
| \(\displaystyle 2 a \int \frac {x^2}{a^2 x^4+1}dx+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 2 a \left (\frac {\int \frac {a x^2+1}{a^2 x^4+1}dx}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 a \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 a \left (\frac {\frac {\int \frac {1}{-\left (1-\sqrt {2} \sqrt {a} x\right )^2-1}d\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}-\frac {\int \frac {1}{-\left (\sqrt {2} \sqrt {a} x+1\right )^2-1}d\left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 a \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\int \frac {1-a x^2}{a^2 x^4+1}dx}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 a \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {-\frac {\int -\frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 a \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 \sqrt {2} a}+\frac {\int \frac {\sqrt {2} \sqrt {a} x+1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 a \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}}{2 a}-\frac {\frac {\log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}}{2 a}\right )+x \cot ^{-1}\left (a x^2\right )\) |
x*ArcCot[a*x^2] + 2*a*((-(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))
3.1.82.3.1 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[((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_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])
Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(x \,\operatorname {arccot}\left (a \,x^{2}\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\) | \(97\) |
| parts | \(x \,\operatorname {arccot}\left (a \,x^{2}\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\) | \(97\) |
x*arccot(a*x^2)+1/4/a/(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 = 96, normalized size of antiderivative = 0.73 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{2} \, \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{2} i \, \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (i \, a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} + x\right ) + \frac {1}{2} i \, \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (-i \, a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{2} \, \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (-a \left (-\frac {1}{a^{2}}\right )^{\frac {3}{4}} + x\right ) \]
x*arccot(a*x^2) + 1/2*(-1/a^2)^(1/4)*log(a*(-1/a^2)^(3/4) + x) - 1/2*I*(-1 /a^2)^(1/4)*log(I*a*(-1/a^2)^(3/4) + x) + 1/2*I*(-1/a^2)^(1/4)*log(-I*a*(- 1/a^2)^(3/4) + x) - 1/2*(-1/a^2)^(1/4)*log(-a*(-1/a^2)^(3/4) + x)
Time = 2.98 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=\begin {cases} x \operatorname {acot}{\left (a x^{2} \right )} + \sqrt [4]{- \frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )} + \frac {\log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{a \sqrt [4]{- \frac {1}{a^{2}}}} - \frac {\log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{2 a \sqrt [4]{- \frac {1}{a^{2}}}} + \frac {\operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{a \sqrt [4]{- \frac {1}{a^{2}}}} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} \]
Piecewise((x*acot(a*x**2) + (-1/a**2)**(1/4)*acot(a*x**2) + log(x - (-1/a* *2)**(1/4))/(a*(-1/a**2)**(1/4)) - log(x**2 + sqrt(-1/a**2))/(2*a*(-1/a**2 )**(1/4)) + atan(x/(-1/a**2)**(1/4))/(a*(-1/a**2)**(1/4)), Ne(a, 0)), (pi* x/2, True))
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.91 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )} + x \operatorname {arccot}\left (a x^{2}\right ) \]
1/4*a*(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)) + x*arccot(a*x^2)
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.09 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} - \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}} + \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}}\right )} + x \arctan \left (\frac {1}{a x^{2}}\right ) \]
1/4*a*(2*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(a )))*sqrt(abs(a)))/a^2 + 2*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x - s qrt(2)/sqrt(abs(a)))*sqrt(abs(a)))/a^2 - sqrt(2)*sqrt(abs(a))*log(x^2 + sq rt(2)*x/sqrt(abs(a)) + 1/abs(a))/a^2 + sqrt(2)*sqrt(abs(a))*log(x^2 - sqrt (2)*x/sqrt(abs(a)) + 1/abs(a))/a^2) + x*arctan(1/(a*x^2))
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.32 \[ \int \cot ^{-1}\left (a x^2\right ) \, dx=x\,\mathrm {acot}\left (a\,x^2\right )+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}} \]