Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0367356, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^4+1}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + x^4]/(1 - x^4),x]
[Out]
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Rubi in Sympy [A] time = 2.58519, size = 46, normalized size = 0.87 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}} \right )}}{4} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**4+1)**(1/2)/(-x**4+1),x)
[Out]
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Mathematica [C] time = 0.134014, size = 108, normalized size = 2.04 \[ -\frac{5 x \sqrt{x^4+1} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-x^4,x^4\right )}{\left (x^4-1\right ) \left (2 x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-x^4,x^4\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-x^4,x^4\right )\right )+5 F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-x^4,x^4\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[1 + x^4]/(1 - x^4),x]
[Out]
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Maple [C] time = 0.04, size = 365, normalized size = 6.9 \[ -{\frac{{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{i}{2}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,-i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{\frac{i}{2}}{\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{i}{2}}{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{ \left ( -1 \right ) ^{{\frac{3}{4}}}{\it EllipticPi} \left ( \sqrt [4]{-1}x,i,\sqrt{-i}- \left ( -1 \right ) ^{{\frac{3}{4}}} \right ) \sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^4+1)^(1/2)/(-x^4+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{x^{4} + 1}}{x^{4} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(x^4 + 1)/(x^4 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25386, size = 81, normalized size = 1.53 \[ \frac{1}{8} \, \sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2} x}{\sqrt{x^{4} + 1}}\right ) + \log \left (\frac{\sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} + 4 \, \sqrt{x^{4} + 1} x}{x^{4} - 2 \, x^{2} + 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(x^4 + 1)/(x^4 - 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{x^{4} + 1}}{x^{4} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**4+1)**(1/2)/(-x**4+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{x^{4} + 1}}{x^{4} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(x^4 + 1)/(x^4 - 1),x, algorithm="giac")
[Out]