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

Optimal. Leaf size=75 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]
/(Sqrt[2]*Sqrt[d]*Sqrt[e])

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Rubi [A]  time = 0.0498835, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1162, 617, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)/(d^2 + e^2*x^4),x]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]]
/(Sqrt[2]*Sqrt[d]*Sqrt[e])

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{d+e x^2}{d^2+e^2 x^4} \, dx &=\frac{\int \frac{1}{\frac{d}{e}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt{e}}+x^2} \, dx}{2 e}+\frac{\int \frac{1}{\frac{d}{e}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt{e}}+x^2} \, dx}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.0321727, size = 60, normalized size = 0.8 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}+1\right )-\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{2} \sqrt{d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-ArcTan[1 - (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]] + ArcTan[1 + (Sqrt[2]*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[2]*Sqrt[d]*Sqrt[e])

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Maple [B]  time = 0.07, size = 290, normalized size = 3.9 \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 = 1.36301, size = 343, normalized size = 4.57 \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 = 0.176202, size = 87, normalized size = 1.16 \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 = 1.16751, size = 300, normalized size = 4. \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