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

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

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Rubi [A]  time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.03, size = 60, normalized size = 0.80 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2}{d^2+e^2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.50, size = 137, normalized size = 1.83 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.17, size = 222, normalized size = 2.96 \begin {gather*} \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 {gather*}

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

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maple [B]  time = 0.01, size = 290, normalized size = 3.87 \begin {gather*} \frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )}{4 d}+\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )}{4 d}+\frac {\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}\right )}{8 d}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}{x^{2}+\left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {d^{2}}{e^{2}}}}\right )}{8 \left (\frac {d^{2}}{e^{2}}\right )^{\frac {1}{4}} e} \end {gather*}

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 [A]  time = 2.48, size = 74, normalized size = 0.99 \begin {gather*} \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, e x + \sqrt {2} \sqrt {d} \sqrt {e}\right )}}{2 \, \sqrt {d e}}\right )}{2 \, \sqrt {d e}} + \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, e x - \sqrt {2} \sqrt {d} \sqrt {e}\right )}}{2 \, \sqrt {d e}}\right )}{2 \, \sqrt {d e}} \end {gather*}

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]

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

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mupad [B]  time = 4.41, size = 57, normalized size = 0.76 \begin {gather*} \frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,e^{3/2}\,x^3}{2\,d^{3/2}}+\frac {\sqrt {2}\,\sqrt {e}\,x}{2\,\sqrt {d}}\right )\right )}{4\,\sqrt {d}\,\sqrt {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)/(d^2 + e^2*x^4),x)

[Out]

(2^(1/2)*(2*atan((2^(1/2)*e^(1/2)*x)/(2*d^(1/2))) + 2*atan((2^(1/2)*e^(3/2)*x^3)/(2*d^(3/2)) + (2^(1/2)*e^(1/2
)*x)/(2*d^(1/2)))))/(4*d^(1/2)*e^(1/2))

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sympy [A]  time = 0.22, size = 87, normalized size = 1.16 \begin {gather*} - \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 {gather*}

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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