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

Optimal. Leaf size=49 \[ \frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0354551, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.4498, size = 42, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (x \right )}}{2} + \frac{\operatorname{atanh}{\left (x \right )}}{2} + \frac{\sqrt{a x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} - \frac{\sqrt{a x^{6}} \operatorname{atanh}{\left (x \right )}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x)/2 + atanh(x)/2 + sqrt(a*x**6)*atan(x)/(2*x**3) - sqrt(a*x**6)*atanh(x)/(
2*x**3)

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Mathematica [A]  time = 0.13873, size = 0, normalized size = 0. \[ \int \left (\frac{1}{1-x^4}-\frac{\sqrt{a x^6}}{x \left (1-x^4\right )}\right ) \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.005, size = 37, normalized size = 0.8 \[{\frac{{\it Artanh} \left ( x \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( -1+x \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{a{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.777124, size = 57, normalized size = 1.16 \[ \frac{1}{2} \, \sqrt{a} \arctan \left (x\right ) - \frac{1}{4} \, \sqrt{a} \log \left (x + 1\right ) + \frac{1}{4} \, \sqrt{a} \log \left (x - 1\right ) + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(a)*arctan(x) - 1/4*sqrt(a)*log(x + 1) + 1/4*sqrt(a)*log(x - 1) + 1/2*ar
ctan(x) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Fricas [A]  time = 0.292584, size = 1, normalized size = 0.02 \[ \left [\frac{x^{3} \sqrt{-\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}} \log \left (\frac{{\left (a - 1\right )} x^{4} -{\left (a - 1\right )} x^{2} - 2 \,{\left (x^{3} - \sqrt{a x^{6}}\right )} \sqrt{-\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}}}{x^{4} + x^{2}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt{a x^{6}}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}, \frac{2 \, x^{3} \sqrt{\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}} \arctan \left (-\frac{{\left (a - 1\right )} x^{4}}{{\left (x^{3} - \sqrt{a x^{6}}\right )} \sqrt{\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt{a x^{6}}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(x^3*sqrt(-((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3)*log(((a - 1)*x^4 - (a - 1)*x^
2 - 2*(x^3 - sqrt(a*x^6))*sqrt(-((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3))/(x^4 + x^2))
+ x^3*log(x + 1) - x^3*log(x - 1) - sqrt(a*x^6)*(log(x + 1) - log(x - 1)))/x^3,
1/4*(2*x^3*sqrt(((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3)*arctan(-(a - 1)*x^4/((x^3 -
sqrt(a*x^6))*sqrt(((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3))) + x^3*log(x + 1) - x^3*lo
g(x - 1) - sqrt(a*x^6)*(log(x + 1) - log(x - 1)))/x^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{x^{5} - x}\, dx - \int \left (- \frac{\sqrt{a x^{6}}}{x^{5} - x}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(x**5 - x), x) - Integral(-sqrt(a*x**6)/(x**5 - x), x)

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GIAC/XCAS [A]  time = 0.26123, size = 65, normalized size = 1.33 \[ \frac{1}{4} \,{\left (2 \, \arctan \left (x\right ){\rm sign}\left (x\right ) -{\rm ln}\left ({\left | x + 1 \right |}\right ){\rm sign}\left (x\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ){\rm sign}\left (x\right )\right )} \sqrt{a} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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