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

Optimal. Leaf size=61 \[ \frac{x}{2 d^2 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 \sqrt{2} d^2 \sqrt{e}} \]

[Out]

x/(2*d^2*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*Sqrt
[2]*d^2*Sqrt[e])

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Rubi [A]  time = 0.114444, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x}{2 d^2 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 \sqrt{2} d^2 \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

x/(2*d^2*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*Sqrt
[2]*d^2*Sqrt[e])

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Rubi in Sympy [A]  time = 21.3912, size = 54, normalized size = 0.89 \[ \frac{x}{2 d^{2} \sqrt{d + e x^{2}}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{4 d^{2} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*d**2*sqrt(d + e*x**2)) + sqrt(2)*atanh(sqrt(2)*sqrt(e)*x/sqrt(d + e*x**2))/
(4*d**2*sqrt(e))

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Mathematica [A]  time = 0.0516846, size = 57, normalized size = 0.93 \[ \frac{\frac{2 x}{\sqrt{d+e x^2}}+\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{e}}}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*x)/Sqrt[d + e*x^2] + (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]])/
Sqrt[e])/(4*d^2)

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Maple [B]  time = 0.03, size = 441, normalized size = 7.2 \[ -{\frac{e\sqrt{2}}{4}\ln \left ({1 \left ( 4\,d+2\,\sqrt{de} \left ( x-{\frac{\sqrt{de}}{e}} \right ) +2\,\sqrt{2}\sqrt{d}\sqrt{ \left ( x-{\frac{\sqrt{de}}{e}} \right ) ^{2}e+2\,\sqrt{de} \left ( x-{\frac{\sqrt{de}}{e}} \right ) +2\,d} \right ) \left ( x-{\frac{1}{e}\sqrt{de}} \right ) ^{-1}} \right ) \left ( \sqrt{de}+\sqrt{-de} \right ) ^{-1} \left ( -\sqrt{de}+\sqrt{-de} \right ) ^{-1}{\frac{1}{\sqrt{de}}}{\frac{1}{\sqrt{d}}}}+{\frac{e\sqrt{2}}{4}\ln \left ({1 \left ( 4\,d-2\,\sqrt{de} \left ( x+{\frac{\sqrt{de}}{e}} \right ) +2\,\sqrt{2}\sqrt{d}\sqrt{ \left ( x+{\frac{\sqrt{de}}{e}} \right ) ^{2}e-2\,\sqrt{de} \left ( x+{\frac{\sqrt{de}}{e}} \right ) +2\,d} \right ) \left ( x+{\frac{1}{e}\sqrt{de}} \right ) ^{-1}} \right ) \left ( \sqrt{de}+\sqrt{-de} \right ) ^{-1} \left ( -\sqrt{de}+\sqrt{-de} \right ) ^{-1}{\frac{1}{\sqrt{de}}}{\frac{1}{\sqrt{d}}}}-{\frac{1}{2\,d}\sqrt{ \left ( x-{\frac{1}{e}\sqrt{-de}} \right ) ^{2}e+2\,\sqrt{-de} \left ( x-{\frac{\sqrt{-de}}{e}} \right ) } \left ( \sqrt{de}+\sqrt{-de} \right ) ^{-1} \left ( -\sqrt{de}+\sqrt{-de} \right ) ^{-1} \left ( x-{\frac{1}{e}\sqrt{-de}} \right ) ^{-1}}-{\frac{1}{2\,d}\sqrt{ \left ( x+{\frac{1}{e}\sqrt{-de}} \right ) ^{2}e-2\,\sqrt{-de} \left ( x+{\frac{\sqrt{-de}}{e}} \right ) } \left ( \sqrt{de}+\sqrt{-de} \right ) ^{-1} \left ( -\sqrt{de}+\sqrt{-de} \right ) ^{-1} \left ( x+{\frac{1}{e}\sqrt{-de}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*e/((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)*2^(1/2
)/d^(1/2)*ln((4*d+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*2^(1/2)*d^(1/2)*((x-(d*e)^(1
/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)^(1/2))/(x-(d*e)^(1/2)/e))+1/4*e/
((d*e)^(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)*2^(1/2)/d^(1/
2)*ln((4*d-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*2^(1/2)*d^(1/2)*((x+(d*e)^(1/2)/e)^
2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d)^(1/2))/(x+(d*e)^(1/2)/e))-1/2/d/((d*e)^
(1/2)+(-d*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(x-1/e*(-d*e)^(1/2))*((x-1/e*(-d
*e)^(1/2))^2*e+2*(-d*e)^(1/2)*(x-1/e*(-d*e)^(1/2)))^(1/2)-1/2/d/((d*e)^(1/2)+(-d
*e)^(1/2))/(-(d*e)^(1/2)+(-d*e)^(1/2))/(x+1/e*(-d*e)^(1/2))*((x+1/e*(-d*e)^(1/2)
)^2*e-2*(-d*e)^(1/2)*(x+1/e*(-d*e)^(1/2)))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.292522, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{e x^{2} + d} \sqrt{e} x +{\left (e x^{2} + d\right )} \log \left (\frac{\sqrt{2}{\left (17 \, e^{2} x^{4} + 14 \, d e x^{2} + d^{2}\right )} \sqrt{e} + 8 \,{\left (3 \, e^{2} x^{3} + d e x\right )} \sqrt{e x^{2} + d}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )\right )}}{16 \,{\left (d^{2} e x^{2} + d^{3}\right )} \sqrt{e}}, \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{e x^{2} + d} \sqrt{-e} x +{\left (e x^{2} + d\right )} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )} \sqrt{-e}}{4 \, \sqrt{e x^{2} + d} e x}\right )\right )}}{8 \,{\left (d^{2} e x^{2} + d^{3}\right )} \sqrt{-e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*sqrt(2)*(4*sqrt(2)*sqrt(e*x^2 + d)*sqrt(e)*x + (e*x^2 + d)*log((sqrt(2)*(1
7*e^2*x^4 + 14*d*e*x^2 + d^2)*sqrt(e) + 8*(3*e^2*x^3 + d*e*x)*sqrt(e*x^2 + d))/(
e^2*x^4 - 2*d*e*x^2 + d^2)))/((d^2*e*x^2 + d^3)*sqrt(e)), 1/8*sqrt(2)*(2*sqrt(2)
*sqrt(e*x^2 + d)*sqrt(-e)*x + (e*x^2 + d)*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*sqrt(
-e)/(sqrt(e*x^2 + d)*e*x)))/((d^2*e*x^2 + d^3)*sqrt(-e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*sqrt(e*x^2 + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError