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

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 = 0.0466638, 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, Rules used = {1165, 628} \[ \frac{\log \left (\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\log \left (-\sqrt{2} \sqrt{d} \sqrt{e} x+d+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}} \]

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])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{d-e x^2}{d^2+e^2 x^4} \, dx &=-\frac{\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt{e}}+2 x}{-\frac{d}{e}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt{e}}-x^2} \, dx}{2 \sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt{e}}-2 x}{-\frac{d}{e}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt{e}}-x^2} \, dx}{2 \sqrt{2} \sqrt{d} \sqrt{e}}\\ &=-\frac{\log \left (d-\sqrt{2} \sqrt{d} \sqrt{e} x+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}}+\frac{\log \left (d+\sqrt{2} \sqrt{d} \sqrt{e} x+e x^2\right )}{2 \sqrt{2} \sqrt{d} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.0223228, 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 = 0.045, size = 290, normalized size = 3.2 \begin{align*}{\frac{\sqrt{2}}{8\,d}\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}\ln \left ({ \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 ({x\sqrt{2}{\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 ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ) }-{\frac{\sqrt{2}}{8\,e}\ln \left ({ \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 ({x\sqrt{2}{\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 ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{{d}^{2}}{{e}^{2}}}}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 1.29279, size = 346, normalized size = 3.84 \begin{align*} \left [\frac{\sqrt{2} \sqrt{d e} \log \left (\frac{e^{2} x^{4} + 4 \, d e x^{2} + 2 \, \sqrt{2}{\left (e x^{3} + d x\right )} \sqrt{d e} + d^{2}}{e^{2} x^{4} + d^{2}}\right )}{4 \, d e}, -\frac{\sqrt{2} \sqrt{-d e} \arctan \left (\frac{\sqrt{2} \sqrt{-d e} x}{2 \, d}\right ) - \sqrt{2} \sqrt{-d e} \arctan \left (\frac{\sqrt{2}{\left (e x^{3} - d x\right )} \sqrt{-d e}}{2 \, d^{2}}\right )}{2 \, d e}\right ] \end{align*}

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

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Sympy [A]  time = 0.330698, size = 80, normalized size = 0.89 \begin{align*} - \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} \end{align*}

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 [B]  time = 1.15793, size = 300, normalized size = 3.33 \begin{align*} \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 )} \log \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 )} \log \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}} \end{align*}

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)*log(sqrt(2)*(d^2)^(1/4)*x*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1))/d^2 - 1/8*s
qrt(2)*((d^2)^(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*e^(-6)*log(-sqrt(2)*(d^2)^(1/4)*x*e^(-1/2) + x^2 + sqrt
(d^2)*e^(-1))/d^2