3.9.77 \(\int \frac {\sqrt {-1+x^4}}{1+x^4} \, dx\)

Optimal. Leaf size=67 \[ \frac {1}{4} \tan ^{-1}\left (\frac {\frac {x^4}{2}-x^2-\frac {1}{2}}{x \sqrt {x^4-1}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\frac {x^4}{2}+x^2-\frac {1}{2}}{x \sqrt {x^4-1}}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {405} \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {x \left (1-x^2\right )}{\sqrt {x^4-1}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x \left (x^2+1\right )}{\sqrt {x^4-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x^4]/(1 + x^4),x]

[Out]

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

Rule 405

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*b), 4]}, Simp[(a*ArcTan[(q*
x*(a + q^2*x^2))/(a*Sqrt[a + b*x^4])])/(2*c*q), x] + Simp[(a*ArcTanh[(q*x*(a - q^2*x^2))/(a*Sqrt[a + b*x^4])])
/(2*c*q), 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]  time = 0.11, size = 108, normalized size = 1.61 \begin {gather*} -\frac {5 x \sqrt {x^4-1} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};x^4,-x^4\right )}{\left (x^4+1\right ) \left (2 x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};x^4,-x^4\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};x^4,-x^4\right )\right )-5 F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};x^4,-x^4\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[-1 + x^4]/(1 + x^4),x]

[Out]

(-5*x*Sqrt[-1 + x^4]*AppellF1[1/4, -1/2, 1, 5/4, x^4, -x^4])/((1 + x^4)*(-5*AppellF1[1/4, -1/2, 1, 5/4, x^4, -
x^4] + 2*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, x^4, -x^4] + AppellF1[5/4, 1/2, 1, 9/4, x^4, -x^4])))

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IntegrateAlgebraic [C]  time = 0.19, size = 53, normalized size = 0.79 \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]

IntegrateAlgebraic[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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fricas [A]  time = 0.56, size = 51, normalized size = 0.76 \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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - 1}}{x^{4} + 1}\,{d x} \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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maple [A]  time = 0.60, size = 101, normalized size = 1.51

method	result	size
default	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{8}\right ) \sqrt {2}}{2}\)	\(101\)
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{4}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{8}\right ) \sqrt {2}}{2}\)	\(101\)
trager	\(-\frac {\ln \left (-\frac {-4 \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}-1}\right )}{2}-\ln \left (-\frac {-4 \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}-1}\right ) \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+\RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \ln \left (-\frac {4 \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x +\sqrt {x^{4}-1}+2 x}{4 x^{2} \RootOf \left (8 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+x^{2}+1}\right )\)	\(176\)

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - 1}}{x^{4} + 1}\,{d x} \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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^4-1}}{x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - 1)^(1/2)/(x^4 + 1),x)

[Out]

int((x^4 - 1)^(1/2)/(x^4 + 1), x)

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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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