3.35 \(\int \frac{d+e x^2}{d^2+f x^2+e^2 x^4} \, dx\)

Optimal. Leaf size=82 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-f}+2 e x}{\sqrt{2 d e+f}}\right )}{\sqrt{2 d e+f}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-f}-2 e x}{\sqrt{2 d e+f}}\right )}{\sqrt{2 d e+f}} \]

[Out]

-(ArcTan[(Sqrt[2*d*e - f] - 2*e*x)/Sqrt[2*d*e + f]]/Sqrt[2*d*e + f]) + ArcTan[(S
qrt[2*d*e - f] + 2*e*x)/Sqrt[2*d*e + f]]/Sqrt[2*d*e + f]

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Rubi [A]  time = 0.204049, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-f}+2 e x}{\sqrt{2 d e+f}}\right )}{\sqrt{2 d e+f}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-f}-2 e x}{\sqrt{2 d e+f}}\right )}{\sqrt{2 d e+f}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(Sqrt[2*d*e - f] - 2*e*x)/Sqrt[2*d*e + f]]/Sqrt[2*d*e + f]) + ArcTan[(S
qrt[2*d*e - f] + 2*e*x)/Sqrt[2*d*e + f]]/Sqrt[2*d*e + f]

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Rubi in Sympy [A]  time = 17.7761, size = 73, normalized size = 0.89 \[ \frac{\operatorname{atan}{\left (\frac{2 e x - \sqrt{2 d e - f}}{\sqrt{2 d e + f}} \right )}}{\sqrt{2 d e + f}} + \frac{\operatorname{atan}{\left (\frac{2 e x + \sqrt{2 d e - f}}{\sqrt{2 d e + f}} \right )}}{\sqrt{2 d e + f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan((2*e*x - sqrt(2*d*e - f))/sqrt(2*d*e + f))/sqrt(2*d*e + f) + atan((2*e*x +
sqrt(2*d*e - f))/sqrt(2*d*e + f))/sqrt(2*d*e + f)

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Mathematica [B]  time = 0.183775, size = 181, normalized size = 2.21 \[ \frac{\frac{\left (\sqrt{f^2-4 d^2 e^2}+2 d e-f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}\right )}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}+\frac{\left (\sqrt{f^2-4 d^2 e^2}-2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]

Antiderivative was successfully verified.

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

[Out]

(((2*d*e - f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f - Sqrt[-4*d^2
*e^2 + f^2]]])/Sqrt[f - Sqrt[-4*d^2*e^2 + f^2]] + ((-2*d*e + f + Sqrt[-4*d^2*e^2
+ f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[f + Sqrt[-
4*d^2*e^2 + f^2]])/(Sqrt[2]*Sqrt[-4*d^2*e^2 + f^2])

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Maple [A]  time = 0.037, size = 71, normalized size = 0.9 \[ -{1\arctan \left ({1 \left ( -2\,ex+\sqrt{2\,de-f} \right ){\frac{1}{\sqrt{2\,de+f}}}} \right ){\frac{1}{\sqrt{2\,de+f}}}}+{1\arctan \left ({1 \left ( 2\,ex+\sqrt{2\,de-f} \right ){\frac{1}{\sqrt{2\,de+f}}}} \right ){\frac{1}{\sqrt{2\,de+f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.283345, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (\frac{2 \,{\left (2 \, d e^{2} + e f\right )} x^{3} - 2 \,{\left (2 \, d^{2} e + d f\right )} x +{\left (e^{2} x^{4} -{\left (4 \, d e + f\right )} x^{2} + d^{2}\right )} \sqrt{-2 \, d e - f}}{e^{2} x^{4} + f x^{2} + d^{2}}\right )}{2 \, \sqrt{-2 \, d e - f}}, \frac{\arctan \left (\frac{e x}{\sqrt{2 \, d e + f}}\right ) + \arctan \left (\frac{e^{2} x^{3} +{\left (d e + f\right )} x}{\sqrt{2 \, d e + f} d}\right )}{\sqrt{2 \, d e + f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.37044, size = 122, normalized size = 1.49 \[ - \frac{\sqrt{- \frac{1}{2 d e + f}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (- 2 d e \sqrt{- \frac{1}{2 d e + f}} - f \sqrt{- \frac{1}{2 d e + f}}\right )}{e} \right )}}{2} + \frac{\sqrt{- \frac{1}{2 d e + f}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (2 d e \sqrt{- \frac{1}{2 d e + f}} + f \sqrt{- \frac{1}{2 d e + f}}\right )}{e} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(2*d*e + f))*log(-d/e + x**2 + x*(-2*d*e*sqrt(-1/(2*d*e + f)) - f*sqrt(
-1/(2*d*e + f)))/e)/2 + sqrt(-1/(2*d*e + f))*log(-d/e + x**2 + x*(2*d*e*sqrt(-1/
(2*d*e + f)) + f*sqrt(-1/(2*d*e + f)))/e)/2

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GIAC/XCAS [A]  time = 0.452962, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done