3.880 \(\int \frac{x^2}{\sqrt{-1+x^4} \left (1+x^4\right )} \, dx\)

Optimal. Leaf size=49 \[ -\frac{1}{4} \tan ^{-1}\left (\frac{x^2+1}{x \sqrt{x^4-1}}\right )-\frac{1}{4} \tanh ^{-1}\left (\frac{1-x^2}{x \sqrt{x^4-1}}\right ) \]

[Out]

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

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Rubi [C]  time = 0.240622, antiderivative size = 47, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \left (\frac{1}{8}+\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) x}{\sqrt{x^4-1}}\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) \tan ^{-1}\left (\frac{(1+i) x}{\sqrt{x^4-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[-1 + x^4]*(1 + x^4)),x]

[Out]

(-1/8 - I/8)*ArcTan[((1 + I)*x)/Sqrt[-1 + x^4]] + (1/8 + I/8)*ArcTanh[((1 + I)*x
)/Sqrt[-1 + x^4]]

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Rubi in Sympy [A]  time = 72.0828, size = 197, normalized size = 4.02 \[ \frac{\left (\frac{1}{4} - \frac{i}{4}\right ) \sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{x^{4} - 1}} + \frac{\left (\frac{1}{4} + \frac{i}{4}\right ) \sqrt{- x^{4} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{x^{4} - 1}} + \frac{\left (\frac{1}{4} - \frac{i}{4}\right ) \sqrt{x^{4} - 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{- x^{4} + 1}} + \frac{\left (\frac{1}{4} + \frac{i}{4}\right ) \sqrt{x^{4} - 1} F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{\sqrt{- x^{4} + 1}} - \frac{i \sqrt{x^{4} - 1} \Pi \left (- i; \operatorname{asin}{\left (x \right )}\middle | -1\right )}{2 \sqrt{- x^{2} + 1} \sqrt{x^{2} + 1}} - \frac{\left (1 - i\right )^{2} \sqrt{x^{4} - 1} \Pi \left (i; \operatorname{asin}{\left (x \right )}\middle | -1\right )}{4 \sqrt{- x^{2} + 1} \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/4 - I/4)*sqrt(-x**4 + 1)*elliptic_f(asin(x), -1)/sqrt(x**4 - 1) + (1/4 + I/4)
*sqrt(-x**4 + 1)*elliptic_f(asin(x), -1)/sqrt(x**4 - 1) + (1/4 - I/4)*sqrt(x**4
- 1)*elliptic_f(asin(x), -1)/sqrt(-x**4 + 1) + (1/4 + I/4)*sqrt(x**4 - 1)*ellipt
ic_f(asin(x), -1)/sqrt(-x**4 + 1) - I*sqrt(x**4 - 1)*elliptic_pi(-I, asin(x), -1
)/(2*sqrt(-x**2 + 1)*sqrt(x**2 + 1)) - (1 - I)**2*sqrt(x**4 - 1)*elliptic_pi(I,
asin(x), -1)/(4*sqrt(-x**2 + 1)*sqrt(x**2 + 1))

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Mathematica [C]  time = 0.187742, size = 114, normalized size = 2.33 \[ -\frac{7 x^3 F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};x^4,-x^4\right )}{3 \sqrt{x^4-1} \left (x^4+1\right ) \left (2 x^4 \left (2 F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};x^4,-x^4\right )-F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};x^4,-x^4\right )\right )-7 F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};x^4,-x^4\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2/(Sqrt[-1 + x^4]*(1 + x^4)),x]

[Out]

(-7*x^3*AppellF1[3/4, 1/2, 1, 7/4, x^4, -x^4])/(3*Sqrt[-1 + x^4]*(1 + x^4)*(-7*A
ppellF1[3/4, 1/2, 1, 7/4, x^4, -x^4] + 2*x^4*(2*AppellF1[7/4, 1/2, 2, 11/4, x^4,
-x^4] - AppellF1[7/4, 3/2, 1, 11/4, x^4, -x^4])))

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Maple [B]  time = 0.026, size = 88, normalized size = 1.8 \[{\frac{1}{8}\arctan \left ({\frac{1}{x}\sqrt{{x}^{4}-1}}+1 \right ) }-{\frac{1}{8}\arctan \left ( -{\frac{1}{x}\sqrt{{x}^{4}-1}}+1 \right ) }+{\frac{1}{16}\ln \left ({1 \left ({\frac{{x}^{4}-1}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt{{x}^{4}-1}}+1 \right ) \left ({\frac{{x}^{4}-1}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt{{x}^{4}-1}}+1 \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*arctan((x^4-1)^(1/2)/x+1)-1/8*arctan(-(x^4-1)^(1/2)/x+1)+1/16*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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (x^{4} + 1\right )} \sqrt{x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^4 + 1)*sqrt(x^4 - 1)),x, algorithm="maxima")

[Out]

integrate(x^2/((x^4 + 1)*sqrt(x^4 - 1)), x)

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Fricas [A]  time = 0.321909, size = 92, normalized size = 1.88 \[ \frac{1}{4} \, \arctan \left (\frac{x^{3} + \sqrt{x^{4} - 1} x^{2} - x}{x^{3} + x + \sqrt{x^{4} - 1}}\right ) + \frac{1}{8} \, \log \left (\frac{x^{4} + 2 \, x^{2} + 2 \, \sqrt{x^{4} - 1} x - 1}{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^4 + 1)*sqrt(x^4 - 1)),x, algorithm="fricas")

[Out]

1/4*arctan((x^3 + sqrt(x^4 - 1)*x^2 - x)/(x^3 + x + sqrt(x^4 - 1))) + 1/8*log((x
^4 + 2*x^2 + 2*sqrt(x^4 - 1)*x - 1)/(x^4 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(sqrt((x - 1)*(x + 1)*(x**2 + 1))*(x**4 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (x^{4} + 1\right )} \sqrt{x^{4} - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^4 + 1)*sqrt(x^4 - 1)),x, algorithm="giac")

[Out]

integrate(x^2/((x^4 + 1)*sqrt(x^4 - 1)), x)