Integrand size = 13, antiderivative size = 109 \[ \int \frac {x^2}{a^4+x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a} \]
-1/4*arctan(1-x*2^(1/2)/a)/a*2^(1/2)+1/4*arctan(1+x*2^(1/2)/a)/a*2^(1/2)+1 /8*ln(a^2+x^2-a*x*2^(1/2))/a*2^(1/2)-1/8*ln(a^2+x^2+a*x*2^(1/2))/a*2^(1/2)
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt {2} x}{a}\right )+2 \arctan \left (1+\frac {\sqrt {2} x}{a}\right )+\log \left (a^2-\sqrt {2} a x+x^2\right )-\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a} \]
(-2*ArcTan[1 - (Sqrt[2]*x)/a] + 2*ArcTan[1 + (Sqrt[2]*x)/a] + Log[a^2 - Sq rt[2]*a*x + x^2] - Log[a^2 + Sqrt[2]*a*x + x^2])/(4*Sqrt[2]*a)
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {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.
\(\displaystyle \int \frac {x^2}{a^4+x^4} \, dx\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {1}{2} \int \frac {a^2+x^2}{a^4+x^4}dx-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{a^2-\sqrt {2} x a+x^2}dx+\frac {1}{2} \int \frac {1}{a^2+\sqrt {2} x a+x^2}dx\right )-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} x}{a}\right )^2-1}d\left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} x}{a}+1\right )^2-1}d\left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}\right )-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )-\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4}dx\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} a-2 x}{a^2-\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}+\frac {\int -\frac {\sqrt {2} \left (a+\sqrt {2} x\right )}{a^2+\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} a-2 x}{a^2-\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}-\frac {\int \frac {\sqrt {2} \left (a+\sqrt {2} x\right )}{a^2+\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} a-2 x}{a^2-\sqrt {2} x a+x^2}dx}{2 \sqrt {2} a}-\frac {\int \frac {a+\sqrt {2} x}{a^2+\sqrt {2} x a+x^2}dx}{2 a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{2 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{2 \sqrt {2} a}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{\sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{\sqrt {2} a}\right )\) |
(-(ArcTan[1 - (Sqrt[2]*x)/a]/(Sqrt[2]*a)) + ArcTan[1 + (Sqrt[2]*x)/a]/(Sqr t[2]*a))/2 + (Log[a^2 - Sqrt[2]*a*x + x^2]/(2*Sqrt[2]*a) - Log[a^2 + Sqrt[ 2]*a*x + x^2]/(2*Sqrt[2]*a))/2
3.2.35.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.22
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+a^{4}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{4}\) | \(24\) |
default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a^{4}\right )^{\frac {1}{4}}}\) | \(85\) |
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (i \, a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) \]
1/4*(-1/a^4)^(1/4)*log(a^4*(-1/a^4)^(3/4) + x) - 1/4*I*(-1/a^4)^(1/4)*log( I*a^4*(-1/a^4)^(3/4) + x) + 1/4*I*(-1/a^4)^(1/4)*log(-I*a^4*(-1/a^4)^(3/4) + x) - 1/4*(-1/a^4)^(1/4)*log(-a^4*(-1/a^4)^(3/4) + x)
Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.17 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\operatorname {RootSum} {\left (256 t^{4} + 1, \left ( t \mapsto t \log {\left (64 t^{3} a + x \right )} \right )\right )}}{a} \]
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{4 \, a} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{4 \, a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} \]
1/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a + 2*x)/a)/a + 1/4*sqrt(2)*arctan (-1/2*sqrt(2)*(sqrt(2)*a - 2*x)/a)/a - 1/8*sqrt(2)*log(sqrt(2)*a*x + a^2 + x^2)/a + 1/8*sqrt(2)*log(-sqrt(2)*a*x + a^2 + x^2)/a
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} \]
1/4*sqrt(2)*abs(a)*arctan(1/2*sqrt(2)*(sqrt(2)*abs(a) + 2*x)/abs(a))/a^2 + 1/4*sqrt(2)*abs(a)*arctan(-1/2*sqrt(2)*(sqrt(2)*abs(a) - 2*x)/abs(a))/a^2 - 1/8*sqrt(2)*abs(a)*log(sqrt(2)*x*abs(a) + x^2 + abs(a)^2)/a^2 + 1/8*sqr t(2)*abs(a)*log(-sqrt(2)*x*abs(a) + x^2 + abs(a)^2)/a^2
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )-{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{2\,a} \]
Time = 0.00 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {\sqrt {2}\, a -2 x}{\sqrt {2}\, a}\right )+2 \mathit {atan} \left (\frac {\sqrt {2}\, a +2 x}{\sqrt {2}\, a}\right )+\mathrm {log}\left (-\sqrt {2}\, a x +a^{2}+x^{2}\right )-\mathrm {log}\left (\sqrt {2}\, a x +a^{2}+x^{2}\right )\right )}{8 a} \]