Integrand size = 10, antiderivative size = 135 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=-\frac {\cot ^{-1}\left (a x^2\right )}{x}+\frac {\sqrt {a} \arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}-\frac {\sqrt {a} \arctan \left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}+\frac {\sqrt {a} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}}-\frac {\sqrt {a} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}} \]
-arccot(a*x^2)/x-1/2*arctan(-1+x*2^(1/2)*a^(1/2))*a^(1/2)*2^(1/2)-1/2*arct an(1+x*2^(1/2)*a^(1/2))*a^(1/2)*2^(1/2)+1/4*ln(1+a*x^2-x*2^(1/2)*a^(1/2))* a^(1/2)*2^(1/2)-1/4*ln(1+a*x^2+x*2^(1/2)*a^(1/2))*a^(1/2)*2^(1/2)
Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=-\frac {\cot ^{-1}\left (a x^2\right )}{x}+\frac {\sqrt {a} \left (2 \arctan \left (1-\sqrt {2} \sqrt {a} x\right )-2 \arctan \left (1+\sqrt {2} \sqrt {a} x\right )+\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )\right )}{2 \sqrt {2}} \]
-(ArcCot[a*x^2]/x) + (Sqrt[a]*(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])
Time = 0.37 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5362, 755, 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.
| \(\displaystyle \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -2 a \int \frac {1}{a^2 x^4+1}dx-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle -2 a \left (\frac {1}{2} \int \frac {1-a x^2}{a^2 x^4+1}dx+\frac {1}{2} \int \frac {a x^2+1}{a^2 x^4+1}dx\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -2 a \left (\frac {1}{2} \int \frac {1-a x^2}{a^2 x^4+1}dx+\frac {1}{2} \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -2 a \left (\frac {1}{2} \int \frac {1-a x^2}{a^2 x^4+1}dx+\frac {1}{2} \left (\frac {\int \frac {1}{-\left (1-\sqrt {2} \sqrt {a} x\right )^2-1}d\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}-\frac {\int \frac {1}{-\left (\sqrt {2} \sqrt {a} x+1\right )^2-1}d\left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -2 a \left (\frac {1}{2} \int \frac {1-a x^2}{a^2 x^4+1}dx+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -2 a \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{\sqrt {a} \left (x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {a} \left (x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}\right )}dx}{2 \sqrt {2} \sqrt {a}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -2 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {a} x}{x^2-\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 \sqrt {2} a}+\frac {\int \frac {\sqrt {2} \sqrt {a} x+1}{x^2+\frac {\sqrt {2} x}{\sqrt {a}}+\frac {1}{a}}dx}{2 a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}\right )+\frac {1}{2} \left (\frac {\log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}\right )\right )-\frac {\cot ^{-1}\left (a x^2\right )}{x}\) |
-(ArcCot[a*x^2]/x) - 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 + (-1/2*Log[1 - S qrt[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)
3.1.83.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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_)^(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 = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {\operatorname {arccot}\left (a \,x^{2}\right )}{x}-\frac {a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4}\) | \(98\) |
| parts | \(-\frac {\operatorname {arccot}\left (a \,x^{2}\right )}{x}-\frac {a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4}\) | \(98\) |
-arccot(a*x^2)/x-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 = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=-\frac {\left (-a^{2}\right )^{\frac {1}{4}} x \log \left (a x + \left (-a^{2}\right )^{\frac {1}{4}}\right ) + i \, \left (-a^{2}\right )^{\frac {1}{4}} x \log \left (a x + i \, \left (-a^{2}\right )^{\frac {1}{4}}\right ) - i \, \left (-a^{2}\right )^{\frac {1}{4}} x \log \left (a x - i \, \left (-a^{2}\right )^{\frac {1}{4}}\right ) - \left (-a^{2}\right )^{\frac {1}{4}} x \log \left (a x - \left (-a^{2}\right )^{\frac {1}{4}}\right ) + 2 \, \operatorname {arccot}\left (a x^{2}\right )}{2 \, x} \]
-1/2*((-a^2)^(1/4)*x*log(a*x + (-a^2)^(1/4)) + I*(-a^2)^(1/4)*x*log(a*x + I*(-a^2)^(1/4)) - I*(-a^2)^(1/4)*x*log(a*x - I*(-a^2)^(1/4)) - (-a^2)^(1/4 )*x*log(a*x - (-a^2)^(1/4)) + 2*arccot(a*x^2))/x
Time = 6.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=\begin {cases} a^{2} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \operatorname {acot}{\left (a x^{2} \right )} + a \sqrt [4]{- \frac {1}{a^{2}}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )} - \frac {a \sqrt [4]{- \frac {1}{a^{2}}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{2} - a \sqrt [4]{- \frac {1}{a^{2}}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{x} & \text {for}\: a \neq 0 \\- \frac {\pi }{2 x} & \text {otherwise} \end {cases} \]
Piecewise((a**2*(-1/a**2)**(3/4)*acot(a*x**2) + a*(-1/a**2)**(1/4)*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)) - acot(a*x**2)/x, Ne(a, 0)), (-pi/ (2*x), True))
Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=-\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}}\right )} - \frac {\operatorname {arccot}\left (a x^{2}\right )}{x} \]
-1/4*a*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*a*x + sqrt(2)*sqrt(a))/sqrt(a))/sq rt(a) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*a*x - sqrt(2)*sqrt(a))/sqrt(a))/sq rt(a) + sqrt(2)*log(a*x^2 + sqrt(2)*sqrt(a)*x + 1)/sqrt(a) - sqrt(2)*log(a *x^2 - sqrt(2)*sqrt(a)*x + 1)/sqrt(a)) - arccot(a*x^2)/x
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=-\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} + \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} - \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}}\right )} - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{x} \]
-1/4*a*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(a)))*sqrt(abs (a)))/sqrt(abs(a)) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs( a)))*sqrt(abs(a)))/sqrt(abs(a)) + sqrt(2)*log(x^2 + sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/sqrt(abs(a)) - sqrt(2)*log(x^2 - sqrt(2)*x/sqrt(abs(a)) + 1/a bs(a))/sqrt(abs(a))) - arctan(1/(a*x^2))/x
Time = 0.77 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.33 \[ \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx=-\frac {\mathrm {acot}\left (a\,x^2\right )}{x}+{\left (-1\right )}^{1/4}\,\sqrt {a}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i}+{\left (-1\right )}^{1/4}\,\sqrt {a}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i} \]