3.200 \(\int \frac{\sqrt{d+e x^2}}{d^2-e^2 x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0762615, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.4736, size = 36, normalized size = 0.95 \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{2 d \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0248995, size = 38, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.034, size = 986, normalized size = 26. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.29583, size = 1, normalized size = 0.03 \[ \left [\frac{\sqrt{2} \log \left (\frac{\sqrt{2}{\left (17 \, e^{2} x^{4} + 14 \, d e x^{2} + d^{2}\right )} \sqrt{e} + 8 \,{\left (3 \, e^{2} x^{3} + d e x\right )} \sqrt{e x^{2} + d}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{8 \, d \sqrt{e}}, \frac{\sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )} \sqrt{-e}}{4 \, \sqrt{e x^{2} + d} e x}\right )}{4 \, d \sqrt{-e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*sqrt(2)*log((sqrt(2)*(17*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*sqrt(e)), 1/4*sqrt(2)
*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*sqrt(-e)/(sqrt(e*x^2 + d)*e*x))/(d*sqrt(-e))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.326614, size = 177, normalized size = 4.66 \[ -\frac{{\left (\sqrt{2} i \arctan \left (\frac{e^{\frac{1}{2}}}{\sqrt{-\frac{d e + \sqrt{d^{2}} e}{d}}}\right ) e^{\frac{1}{2}} - \sqrt{2} i \arctan \left (\frac{e^{\frac{1}{2}}}{\sqrt{-\frac{d e - \sqrt{d^{2}} e}{d}}}\right ) e^{\frac{1}{2}}\right )} e^{\left (-1\right )}{\rm sign}\left (x\right )}{4 \,{\left | d \right |}} + \frac{\sqrt{2} i \arctan \left (\frac{\sqrt{\frac{d}{x^{2}} + e}}{\sqrt{-\frac{d e{\rm sign}\left (x\right ) + \sqrt{d^{2}} e}{d{\rm sign}\left (x\right )}}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \,{\left | d \right |}{\left |{\rm sign}\left (x\right ) \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(sqrt(2)*i*arctan(e^(1/2)/sqrt(-(d*e + sqrt(d^2)*e)/d))*e^(1/2) - sqrt(2)*i
*arctan(e^(1/2)/sqrt(-(d*e - sqrt(d^2)*e)/d))*e^(1/2))*e^(-1)*sign(x)/abs(d) + 1
/2*sqrt(2)*i*arctan(sqrt(d/x^2 + e)/sqrt(-(d*e*sign(x) + sqrt(d^2)*e)/(d*sign(x)
)))*e^(-1/2)/(abs(d)*abs(sign(x)))