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

Optimal. Leaf size=90 \[ \frac{\log \left (\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\log \left (-\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0852422, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\log \left (\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\log \left (-\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.973, size = 83, normalized size = 0.92 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{d} \sqrt{e} x + d + e x^{2} \right )}}{4 \sqrt{d} \sqrt{e}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{d} \sqrt{e} x + d + e x^{2} \right )}}{4 \sqrt{d} \sqrt{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*log(-sqrt(2)*sqrt(d)*sqrt(e)*x + d + e*x**2)/(4*sqrt(d)*sqrt(e)) + sqrt
(2)*log(sqrt(2)*sqrt(d)*sqrt(e)*x + d + e*x**2)/(4*sqrt(d)*sqrt(e))

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Mathematica [A]  time = 0.0356861, size = 75, normalized size = 0.83 \[ \frac{\log \left (\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )-\log \left (\sqrt{2} \sqrt{d} \sqrt{e} x-d-e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.004, size = 290, normalized size = 3.2 \[{\frac{\sqrt{2}}{8\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ) }-{\frac{\sqrt{2}}{8\,e}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}x\sqrt{2}+\sqrt{{\frac{{d}^{2}}{{e}^{2}}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-{\frac{\sqrt{2}}{4\,e}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-{\frac{\sqrt{2}}{4\,e}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e*x^2+d)/(e^2*x^4+d^2),x)

[Out]

1/8/d*(d^2/e^2)^(1/4)*2^(1/2)*ln((x^2+(d^2/e^2)^(1/4)*x*2^(1/2)+(d^2/e^2)^(1/2))
/(x^2-(d^2/e^2)^(1/4)*x*2^(1/2)+(d^2/e^2)^(1/2)))+1/4/d*(d^2/e^2)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(d^2/e^2)^(1/4)*x+1)+1/4/d*(d^2/e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(d^2/e^2)^(1/4)*x-1)-1/8/e/(d^2/e^2)^(1/4)*2^(1/2)*ln((x^2-(d^2/e^2)^(1/4)*x*2^
(1/2)+(d^2/e^2)^(1/2))/(x^2+(d^2/e^2)^(1/4)*x*2^(1/2)+(d^2/e^2)^(1/2)))-1/4/e/(d
^2/e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(d^2/e^2)^(1/4)*x+1)-1/4/e/(d^2/e^2)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(d^2/e^2)^(1/4)*x-1)

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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(-(e*x^2 - d)/(e^2*x^4 + d^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291147, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2} \log \left (\frac{4 \, d e^{2} x^{3} + 4 \, d^{2} e x + \sqrt{2}{\left (e^{2} x^{4} + 4 \, d e x^{2} + d^{2}\right )} \sqrt{d e}}{e^{2} x^{4} + d^{2}}\right )}{4 \, \sqrt{d e}}, \frac{\sqrt{2}{\left (\arctan \left (\frac{\sqrt{2} \sqrt{-d e} x}{2 \, d}\right ) + \arctan \left (\frac{\sqrt{2}{\left (e^{2} x^{3} - d e x\right )}}{2 \, \sqrt{-d e} d}\right )\right )}}{2 \, \sqrt{-d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.28393, size = 80, normalized size = 0.89 \[ - \frac{\sqrt{2} \sqrt{\frac{1}{d e}} \log{\left (- \sqrt{2} d x \sqrt{\frac{1}{d e}} + \frac{d}{e} + x^{2} \right )}}{4} + \frac{\sqrt{2} \sqrt{\frac{1}{d e}} \log{\left (\sqrt{2} d x \sqrt{\frac{1}{d e}} + \frac{d}{e} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.276823, size = 300, normalized size = 3.33 \[ \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + 2 \, x\right )} e^{\frac{1}{2}}}{2 \,{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} -{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} - 2 \, x\right )} e^{\frac{1}{2}}}{2 \,{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-6\right )}}{4 \, d^{2}} + \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left (\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} x e^{\left (-\frac{1}{2}\right )} + x^{2} + \sqrt{d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} - \frac{\sqrt{2}{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )}{\rm ln}\left (-\sqrt{2}{\left (d^{2}\right )}^{\frac{1}{4}} x e^{\left (-\frac{1}{2}\right )} + x^{2} + \sqrt{d^{2}} e^{\left (-1\right )}\right )}{8 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*((d^2)^(1/4)*d*e^(11/2) - (d^2)^(3/4)*e^(11/2))*arctan(1/2*sqrt(2)*(
sqrt(2)*(d^2)^(1/4)*e^(-1/2) + 2*x)*e^(1/2)/(d^2)^(1/4))*e^(-6)/d^2 + 1/4*sqrt(2
)*((d^2)^(1/4)*d*e^(11/2) - (d^2)^(3/4)*e^(11/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*(
d^2)^(1/4)*e^(-1/2) - 2*x)*e^(1/2)/(d^2)^(1/4))*e^(-6)/d^2 + 1/8*sqrt(2)*((d^2)^
(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*e^(-6)*ln(sqrt(2)*(d^2)^(1/4)*x*e^(-1/2
) + x^2 + sqrt(d^2)*e^(-1))/d^2 - 1/8*sqrt(2)*((d^2)^(1/4)*d*e^(11/2) + (d^2)^(3
/4)*e^(11/2))*e^(-6)*ln(-sqrt(2)*(d^2)^(1/4)*x*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1)
)/d^2