3.1.65 \(\int \frac {\sqrt {1-x^4}}{1+x^4} \, dx\) [65]

Optimal. Leaf size=49 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x \left (1+x^2\right )}{\sqrt {1-x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x \left (1-x^2\right )}{\sqrt {1-x^4}}\right ) \]

[Out]

1/2*arctan(x*(x^2+1)/(-x^4+1)^(1/2))+1/2*arctanh(x*(-x^2+1)/(-x^4+1)^(1/2))

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Rubi [A]
time = 0.00, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {414} \begin {gather*} \frac {1}{2} \text {ArcTan}\left (\frac {x \left (x^2+1\right )}{\sqrt {1-x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x \left (1-x^2\right )}{\sqrt {1-x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^4]/(1 + x^4),x]

[Out]

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

Rule 414

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^4}}{1+x^4} \, dx &=\frac {1}{2} \tan ^{-1}\left (\frac {x \left (1+x^2\right )}{\sqrt {1-x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {x \left (1-x^2\right )}{\sqrt {1-x^4}}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 57, normalized size = 1.16 \begin {gather*} \left (\frac {1}{4}-\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {1-x^4}}\right )-\left (\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {1-x^4}}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^4]/(1 + x^4),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(41)=82\).
time = 0.42, size = 113, normalized size = 2.31

method	result	size
default	\(\frac {\left (-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}}{x}\right ) \sqrt {2}}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}}{x}\right ) \sqrt {2}}{4}-\frac {\sqrt {2}\, \ln \left (\frac {1+\frac {-x^{4}+1}{2 x^{2}}-\frac {\sqrt {-x^{4}+1}}{x}}{1+\frac {-x^{4}+1}{2 x^{2}}+\frac {\sqrt {-x^{4}+1}}{x}}\right )}{8}\right ) \sqrt {2}}{2}\)	\(113\)
elliptic	\(\frac {\left (-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}}{x}\right ) \sqrt {2}}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}}{x}\right ) \sqrt {2}}{4}-\frac {\sqrt {2}\, \ln \left (\frac {1+\frac {-x^{4}+1}{2 x^{2}}-\frac {\sqrt {-x^{4}+1}}{x}}{1+\frac {-x^{4}+1}{2 x^{2}}+\frac {\sqrt {-x^{4}+1}}{x}}\right )}{8}\right ) \sqrt {2}}{2}\)	\(113\)
trager	\(\frac {\ln \left (-\frac {4 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +\sqrt {-x^{4}+1}}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-x^{2}-1}\right )}{2}-\ln \left (-\frac {4 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +\sqrt {-x^{4}+1}}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-x^{2}-1}\right ) \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+\RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (-\frac {-4 \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +\sqrt {-x^{4}+1}+2 x}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-x^{2}+1}\right )\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+1)^(1/2)/(x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/4*arctan(1+(-x^4+1)^(1/2)/x)*2^(1/2)-1/4*arctan(-1+(-x^4+1)^(1/2)/x)*2^(1/2)-1/8*2^(1/2)*ln((1+1/2*(-x
^4+1)/x^2-(-x^4+1)^(1/2)/x)/(1+1/2*(-x^4+1)/x^2+(-x^4+1)^(1/2)/x)))*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)^(1/2)/(x^4+1),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^4 + 1)/(x^4 + 1), x)

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Fricas [A]
time = 1.05, size = 56, normalized size = 1.14 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1} x}{x^{2} - 1}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{4} - 2 \, x^{2} - 2 \, \sqrt {-x^{4} + 1} x - 1}{x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)^(1/2)/(x^4+1),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{x^{4} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+1)**(1/2)/(x**4+1),x)

[Out]

Integral(sqrt(-(x - 1)*(x + 1)*(x**2 + 1))/(x**4 + 1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)^(1/2)/(x^4+1),x, algorithm="giac")

[Out]

integrate(sqrt(-x^4 + 1)/(x^4 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {1-x^4}}{x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^4)^(1/2)/(x^4 + 1),x)

[Out]

int((1 - x^4)^(1/2)/(x^4 + 1), x)

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