\(\int \frac {x^2}{a^4+x^4} \, dx\) [135]

Optimal result

Integrand size = 13, antiderivative size = 109 \[ \int \frac {x^2}{a^4+x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {303, 1176, 631, 210, 1179, 642} \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\arctan \left (\frac {\sqrt {2} x}{a}+1\right )}{2 \sqrt {2} a} \]

[In]

Int[x^2/(a^4 + x^4),x]

[Out]

-1/2*ArcTan[1 - (Sqrt[2]*x)/a]/(Sqrt[2]*a) + ArcTan[1 + (Sqrt[2]*x)/a]/(2*Sqrt[2]*a) + Log[a^2 - Sqrt[2]*a*x +
x^2]/(4*Sqrt[2]*a) - Log[a^2 + Sqrt[2]*a*x + x^2]/(4*Sqrt[2]*a)

Rule 210

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])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {a^2-x^2}{a^4+x^4} \, dx\right )+\frac {1}{2} \int \frac {a^2+x^2}{a^4+x^4} \, dx \\ & = \frac {1}{4} \int \frac {1}{a^2-\sqrt {2} a x+x^2} \, dx+\frac {1}{4} \int \frac {1}{a^2+\sqrt {2} a x+x^2} \, dx+\frac {\int \frac {\sqrt {2} a+2 x}{-a^2-\sqrt {2} a x-x^2} \, dx}{4 \sqrt {2} a}+\frac {\int \frac {\sqrt {2} a-2 x}{-a^2+\sqrt {2} a x-x^2} \, dx}{4 \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}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,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 {\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt {2} x}{a}\right )+2 \arctan \left (1+\frac {\sqrt {2} x}{a}\right )+\log \left (a^2-\sqrt {2} a x+x^2\right )-\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a} \]

[In]

Integrate[x^2/(a^4 + x^4),x]

[Out]

(-2*ArcTan[1 - (Sqrt[2]*x)/a] + 2*ArcTan[1 + (Sqrt[2]*x)/a] + Log[a^2 - Sqrt[2]*a*x + x^2] - Log[a^2 + Sqrt[2]
*a*x + x^2])/(4*Sqrt[2]*a)

Maple [C] (verified)

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	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a^{4}\right )^{\frac {1}{4}}}\)	\(85\)

[In]

int(x^2/(a^4+x^4),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(1/_R*ln(x-_R),_R=RootOf(_Z^4+a^4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {1}{4} \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (i \, a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-a^{4} \left (-\frac {1}{a^{4}}\right )^{\frac {3}{4}} + x\right ) \]

[In]

integrate(x^2/(a^4+x^4),x, algorithm="fricas")

[Out]

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)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.17 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\operatorname {RootSum} {\left (256 t^{4} + 1, \left ( t \mapsto t \log {\left (64 t^{3} a + x \right )} \right )\right )}}{a} \]

[In]

integrate(x**2/(a**4+x**4),x)

[Out]

RootSum(256*_t**4 + 1, Lambda(_t, _t*log(64*_t**3*a + x)))/a

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{4 \, a} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{4 \, a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} \]

[In]

integrate(x^2/(a^4+x^4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {\sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} \]

[In]

integrate(x^2/(a^4+x^4),x, algorithm="giac")

[Out]

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*s
qrt(2)*abs(a)*log(-sqrt(2)*x*abs(a) + x^2 + abs(a)^2)/a^2

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {x^2}{a^4+x^4} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )-{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{2\,a} \]

[In]

int(x^2/(a^4 + x^4),x)

[Out]

((-1)^(1/4)*atan(((-1)^(1/4)*x)/a) - (-1)^(1/4)*atanh(((-1)^(1/4)*x)/a))/(2*a)