3.197 \(\int \frac{1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx\)

Optimal. Leaf size=72 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2} \sqrt{e}}+\frac{x}{4 d^2 \left (d+e x^2\right )} \]

[Out]

x/(4*d^2*(d + e*x^2)) + ArcTan[(Sqrt[e]*x)/Sqrt[d]]/(2*d^(5/2)*Sqrt[e]) + ArcTan
h[(Sqrt[e]*x)/Sqrt[d]]/(4*d^(5/2)*Sqrt[e])

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Rubi [A]  time = 0.141373, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2} \sqrt{e}}+\frac{x}{4 d^2 \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

x/(4*d^2*(d + e*x^2)) + ArcTan[(Sqrt[e]*x)/Sqrt[d]]/(2*d^(5/2)*Sqrt[e]) + ArcTan
h[(Sqrt[e]*x)/Sqrt[d]]/(4*d^(5/2)*Sqrt[e])

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Rubi in Sympy [A]  time = 35.0936, size = 63, normalized size = 0.88 \[ \frac{x}{4 d^{2} \left (d + e x^{2}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 d^{\frac{5}{2}} \sqrt{e}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{4 d^{\frac{5}{2}} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(4*d**2*(d + e*x**2)) + atan(sqrt(e)*x/sqrt(d))/(2*d**(5/2)*sqrt(e)) + atanh(s
qrt(e)*x/sqrt(d))/(4*d**(5/2)*sqrt(e))

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Mathematica [A]  time = 0.0590362, size = 65, normalized size = 0.9 \[ \frac{\frac{\sqrt{d} x}{d+e x^2}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}}{4 d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d]*x)/(d + e*x^2) + (2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + ArcTanh[(Sq
rt[e]*x)/Sqrt[d]]/Sqrt[e])/(4*d^(5/2))

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Maple [A]  time = 0.014, size = 55, normalized size = 0.8 \[{\frac{1}{4\,{d}^{2}}{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{x}{4\,{d}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{1}{2\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/d^2/(d*e)^(1/2)*arctanh(x*e/(d*e)^(1/2))+1/4*x/d^2/(e*x^2+d)+1/2/d^2/(d*e)^(
1/2)*arctan(x*e/(d*e)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290197, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (e x^{2} + d\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) +{\left (e x^{2} + d\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} + d\right )} \sqrt{d e}}{e x^{2} - d}\right ) + 2 \, \sqrt{d e} x}{8 \,{\left (d^{2} e x^{2} + d^{3}\right )} \sqrt{d e}}, \frac{{\left (e x^{2} + d\right )} \arctan \left (\frac{\sqrt{-d e} x}{d}\right ) +{\left (e x^{2} + d\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) + \sqrt{-d e} x}{4 \,{\left (d^{2} e x^{2} + d^{3}\right )} \sqrt{-d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(4*(e*x^2 + d)*arctan(sqrt(d*e)*x/d) + (e*x^2 + d)*log((2*d*e*x + (e*x^2 +
d)*sqrt(d*e))/(e*x^2 - d)) + 2*sqrt(d*e)*x)/((d^2*e*x^2 + d^3)*sqrt(d*e)), 1/4*(
(e*x^2 + d)*arctan(sqrt(-d*e)*x/d) + (e*x^2 + d)*log((2*d*e*x + (e*x^2 - d)*sqrt
(-d*e))/(e*x^2 + d)) + sqrt(-d*e)*x)/((d^2*e*x^2 + d^3)*sqrt(-d*e))]

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Sympy [A]  time = 2.37143, size = 226, normalized size = 3.14 \[ \frac{x}{4 d^{3} + 4 d^{2} e x^{2}} - \frac{\sqrt{\frac{1}{d^{5} e}} \log{\left (- \frac{d^{8} e \left (\frac{1}{d^{5} e}\right )^{\frac{3}{2}}}{10} - \frac{9 d^{3} \sqrt{\frac{1}{d^{5} e}}}{10} + x \right )}}{8} + \frac{\sqrt{\frac{1}{d^{5} e}} \log{\left (\frac{d^{8} e \left (\frac{1}{d^{5} e}\right )^{\frac{3}{2}}}{10} + \frac{9 d^{3} \sqrt{\frac{1}{d^{5} e}}}{10} + x \right )}}{8} - \frac{\sqrt{- \frac{1}{d^{5} e}} \log{\left (- \frac{4 d^{8} e \left (- \frac{1}{d^{5} e}\right )^{\frac{3}{2}}}{5} - \frac{9 d^{3} \sqrt{- \frac{1}{d^{5} e}}}{5} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{5} e}} \log{\left (\frac{4 d^{8} e \left (- \frac{1}{d^{5} e}\right )^{\frac{3}{2}}}{5} + \frac{9 d^{3} \sqrt{- \frac{1}{d^{5} e}}}{5} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(4*d**3 + 4*d**2*e*x**2) - sqrt(1/(d**5*e))*log(-d**8*e*(1/(d**5*e))**(3/2)/10
- 9*d**3*sqrt(1/(d**5*e))/10 + x)/8 + sqrt(1/(d**5*e))*log(d**8*e*(1/(d**5*e))*
*(3/2)/10 + 9*d**3*sqrt(1/(d**5*e))/10 + x)/8 - sqrt(-1/(d**5*e))*log(-4*d**8*e*
(-1/(d**5*e))**(3/2)/5 - 9*d**3*sqrt(-1/(d**5*e))/5 + x)/4 + sqrt(-1/(d**5*e))*l
og(4*d**8*e*(-1/(d**5*e))**(3/2)/5 + 9*d**3*sqrt(-1/(d**5*e))/5 + x)/4

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError