Optimal. Leaf size=51 \[ -\frac{1}{12} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{6} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.473394, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{1}{12} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{6} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]
[Out]
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Rubi in Sympy [A] time = 43.7541, size = 44, normalized size = 0.86 \[ - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{6} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**2+1)/(x**2+2)/(x**2+3)/(x**2+4),x)
[Out]
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Mathematica [A] time = 0.0377071, size = 47, normalized size = 0.92 \[ \frac{1}{12} \left (-\tan ^{-1}\left (\frac{x}{2}\right )+2 \tan ^{-1}(x)-3 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 + x^2)*(2 + x^2)*(3 + x^2)*(4 + x^2)),x]
[Out]
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Maple [A] time = 0.016, size = 36, normalized size = 0.7 \[ -{\frac{1}{12}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{6}}-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^2+1)/(x^2+2)/(x^2+3)/(x^2+4),x)
[Out]
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Maxima [A] time = 1.52032, size = 47, normalized size = 0.92 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{12} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{6} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 4)*(x^2 + 3)*(x^2 + 2)*(x^2 + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239113, size = 73, normalized size = 1.43 \[ -\frac{1}{72} \, \sqrt{3} \sqrt{2}{\left (\sqrt{3} \sqrt{2} \arctan \left (\frac{1}{2} \, x\right ) - 2 \, \sqrt{3} \sqrt{2} \arctan \left (x\right ) - 6 \, \sqrt{2} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + 6 \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 4)*(x^2 + 3)*(x^2 + 2)*(x^2 + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.774853, size = 44, normalized size = 0.86 \[ - \frac{\operatorname{atan}{\left (\frac{x}{2} \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{6} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**2+1)/(x**2+2)/(x**2+3)/(x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.217276, size = 47, normalized size = 0.92 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{12} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{6} \, \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 + 4)*(x^2 + 3)*(x^2 + 2)*(x^2 + 1)),x, algorithm="giac")
[Out]