3.661 \(\int \frac{x-\sqrt{x^6}}{x \left (1-x^4\right )} \, dx\)

Optimal. Leaf size=45 \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

ArcTan[x]/2 + (Sqrt[x^6]*ArcTan[x])/(2*x^3) + ArcTanh[x]/2 - (Sqrt[x^6]*ArcTanh[
x])/(2*x^3)

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Rubi [A]  time = 0.252606, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x - Sqrt[x^6])/(x*(1 - x^4)),x]

[Out]

ArcTan[x]/2 + (Sqrt[x^6]*ArcTan[x])/(2*x^3) + ArcTanh[x]/2 - (Sqrt[x^6]*ArcTanh[
x])/(2*x^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x}{2} + \frac{\sqrt{x^{6}}}{6} + \frac{\operatorname{atan}{\left (x \right )}}{2} + \int \frac{x - \sqrt{x^{6}}}{x}\, dx + \int \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x - 1}\, dx + \int \frac{- \frac{x}{4} + \frac{\sqrt{x^{6}}}{4}}{x + 1}\, dx - \frac{\sqrt{x^{6}}}{2 x^{2}} + \frac{\sqrt{x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x-(x**6)**(1/2))/x/(-x**4+1),x)

[Out]

-x/2 + sqrt(x**6)/6 + atan(x)/2 + Integral((x - sqrt(x**6))/x, x) + Integral((-x
/4 + sqrt(x**6)/4)/(x - 1), x) + Integral((-x/4 + sqrt(x**6)/4)/(x + 1), x) - sq
rt(x**6)/(2*x**2) + sqrt(x**6)*atan(x)/(2*x**3)

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Mathematica [A]  time = 0.0916927, size = 0, normalized size = 0. \[ \int \frac{x-\sqrt{x^6}}{x \left (1-x^4\right )} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(x - Sqrt[x^6])/(x*(1 - x^4)),x]

[Out]

Integrate[(x - Sqrt[x^6])/(x*(1 - x^4)), x]

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Maple [A]  time = 0.006, size = 35, normalized size = 0.8 \[{\frac{\ln \left ( -1+x \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{{x}^{6}}}+{\frac{{\it Artanh} \left ( x \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x-(x^6)^(1/2))/x/(-x^4+1),x)

[Out]

1/4*(x^6)^(1/2)*(ln(-1+x)-ln(1+x)+2*arctan(x))/x^3+1/2*arctanh(x)+1/2*arctan(x)

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Maxima [A]  time = 0.92148, size = 3, normalized size = 0.07 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - sqrt(x^6))/((x^4 - 1)*x),x, algorithm="maxima")

[Out]

arctan(x)

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Fricas [A]  time = 0.265254, size = 3, normalized size = 0.07 \[ \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - sqrt(x^6))/((x^4 - 1)*x),x, algorithm="fricas")

[Out]

arctan(x)

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Sympy [A]  time = 0.199121, size = 2, normalized size = 0.04 \[ \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x-(x**6)**(1/2))/x/(-x**4+1),x)

[Out]

atan(x)

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GIAC/XCAS [A]  time = 0.264492, size = 42, normalized size = 0.93 \[ \frac{1}{2} \,{\left ({\rm sign}\left (x\right ) + 1\right )} \arctan \left (x\right ) - \frac{1}{4} \,{\left ({\rm sign}\left (x\right ) - 1\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \,{\left ({\rm sign}\left (x\right ) - 1\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - sqrt(x^6))/((x^4 - 1)*x),x, algorithm="giac")

[Out]

1/2*(sign(x) + 1)*arctan(x) - 1/4*(sign(x) - 1)*ln(abs(x + 1)) + 1/4*(sign(x) -
1)*ln(abs(x - 1))