3.135 \(\int \frac {x^2}{a^4+x^4} \, dx\)

Optimal. Leaf size=109 \[ \frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{2 \sqrt {2} a} \]

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Rubi [A]  time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {297, 1162, 617, 204, 1165, 628} \[ \frac {\log \left (a^2-\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\log \left (a^2+\sqrt {2} a x+x^2\right )}{4 \sqrt {2} a}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{2 \sqrt {2} a} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a^4 + x^4),x]

[Out]

-ArcTan[1 - (Sqrt[2]*x)/a]/(2*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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 297

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 617

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 628

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 1162

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 1165

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*} \int \frac {x^2}{a^4+x^4} \, dx &=-\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 {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\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*}

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Mathematica [A]  time = 0.03, size = 79, normalized size = 0.72 \[ \frac {\log \left (a^2-\sqrt {2} a x+x^2\right )-\log \left (a^2+\sqrt {2} a x+x^2\right )-2 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{4 \sqrt {2} a} \]

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [A]  time = 0.31, size = 69, normalized size = 0.63 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} a x}{a^2+x^2}\right )}{2 \sqrt {2} a}-\frac {\tan ^{-1}\left (\frac {\frac {a}{\sqrt {2}}-\frac {x^2}{\sqrt {2} a}}{x}\right )}{2 \sqrt {2} a} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/(a^4 + x^4),x]

[Out]

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

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fricas [B]  time = 0.97, size = 199, normalized size = 1.83 \[ -\frac {1}{2} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}} \frac {1}{a^{4}}^{\frac {1}{4}} - 1\right ) - \frac {1}{2} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}} \frac {1}{a^{4}}^{\frac {1}{4}} + 1\right ) - \frac {1}{8} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}\right ) + \frac {1}{8} \, \sqrt {2} \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\sqrt {2} a^{4} \frac {1}{a^{4}}^{\frac {3}{4}} x + a^{4} \sqrt {\frac {1}{a^{4}}} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*(a^(-4))^(1/4)*arctan(-sqrt(2)*(a^(-4))^(1/4)*x + sqrt(2)*sqrt(sqrt(2)*a^4*(a^(-4))^(3/4)*x + a^4
*sqrt(a^(-4)) + x^2)*(a^(-4))^(1/4) - 1) - 1/2*sqrt(2)*(a^(-4))^(1/4)*arctan(-sqrt(2)*(a^(-4))^(1/4)*x + sqrt(
2)*sqrt(-sqrt(2)*a^4*(a^(-4))^(3/4)*x + a^4*sqrt(a^(-4)) + x^2)*(a^(-4))^(1/4) + 1) - 1/8*sqrt(2)*(a^(-4))^(1/
4)*log(sqrt(2)*a^4*(a^(-4))^(3/4)*x + a^4*sqrt(a^(-4)) + x^2) + 1/8*sqrt(2)*(a^(-4))^(1/4)*log(-sqrt(2)*a^4*(a
^(-4))^(3/4)*x + a^4*sqrt(a^(-4)) + x^2)

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giac [A]  time = 0.97, size = 114, normalized size = 1.05 \[ \frac {\sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} + \frac {\sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{8 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [C]  time = 0.27, size = 24, normalized size = 0.22

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+a^{4}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{4}\)	\(24\)
default	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a^{4}\right )^{\frac {1}{4}}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.97, size = 98, normalized size = 0.90 \[ \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{4 \, a} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{4 \, a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{8 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.10, size = 33, normalized size = 0.30 \[ \frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )-{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.14, size = 19, normalized size = 0.17 \[ \frac {\operatorname {RootSum} {\left (256 t^{4} + 1, \left (t \mapsto t \log {\left (64 t^{3} a + x \right )} \right )\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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