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

Optimal. Leaf size=78 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}-2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}+2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}} \]

[Out]

ArcTanh[(Sqrt[b + 2*d*e] - 2*e*x)/Sqrt[b - 2*d*e]]/Sqrt[b - 2*d*e] - ArcTanh[(Sq
rt[b + 2*d*e] + 2*e*x)/Sqrt[b - 2*d*e]]/Sqrt[b - 2*d*e]

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Rubi [A]  time = 0.174408, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}-2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}+2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(Sqrt[b + 2*d*e] - 2*e*x)/Sqrt[b - 2*d*e]]/Sqrt[b - 2*d*e] - ArcTanh[(Sq
rt[b + 2*d*e] + 2*e*x)/Sqrt[b - 2*d*e]]/Sqrt[b - 2*d*e]

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Rubi in Sympy [A]  time = 18.1295, size = 75, normalized size = 0.96 \[ - \frac{\operatorname{atanh}{\left (\frac{2 e x - \sqrt{b + 2 d e}}{\sqrt{b - 2 d e}} \right )}}{\sqrt{b - 2 d e}} - \frac{\operatorname{atanh}{\left (\frac{2 e x + \sqrt{b + 2 d e}}{\sqrt{b - 2 d e}} \right )}}{\sqrt{b - 2 d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.181215, size = 189, normalized size = 2.42 \[ \frac{\frac{\left (\sqrt{b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}\right )}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}+\frac{\left (\sqrt{b^2-4 d^2 e^2}-b-2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}\right )}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 75, normalized size = 1. \[{1\arctan \left ({1 \left ( 2\,ex+\sqrt{2\,de+b} \right ){\frac{1}{\sqrt{2\,de-b}}}} \right ){\frac{1}{\sqrt{2\,de-b}}}}-{1\arctan \left ({1 \left ( -2\,ex+\sqrt{2\,de+b} \right ){\frac{1}{\sqrt{2\,de-b}}}} \right ){\frac{1}{\sqrt{2\,de-b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.293069, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (\frac{2 \,{\left (2 \, d e^{2} - b e\right )} x^{3} - 2 \,{\left (2 \, d^{2} e - b d\right )} x +{\left (e^{2} x^{4} -{\left (4 \, d e - b\right )} x^{2} + d^{2}\right )} \sqrt{-2 \, d e + b}}{e^{2} x^{4} - b x^{2} + d^{2}}\right )}{2 \, \sqrt{-2 \, d e + b}}, \frac{\arctan \left (\frac{e x}{\sqrt{2 \, d e - b}}\right ) + \arctan \left (\frac{e^{2} x^{3} +{\left (d e - b\right )} x}{\sqrt{2 \, d e - b} d}\right )}{\sqrt{2 \, d e - b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.48727, size = 110, normalized size = 1.41 \[ \frac{\sqrt{\frac{1}{b - 2 d e}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (- b \sqrt{\frac{1}{b - 2 d e}} + 2 d e \sqrt{\frac{1}{b - 2 d e}}\right )}{e} \right )}}{2} - \frac{\sqrt{\frac{1}{b - 2 d e}} \log{\left (- \frac{d}{e} + x^{2} + \frac{x \left (b \sqrt{\frac{1}{b - 2 d e}} - 2 d e \sqrt{\frac{1}{b - 2 d e}}\right )}{e} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.438578, 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 - b*x^2 + d^2),x, algorithm="giac")

[Out]

Done