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

Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^2 (d+f)+e\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

[Out]

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

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Rubi [A]  time = 0.0661598, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6, 1107, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^2 (d+f)+e\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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{x}{e^2+4 e f x^2+4 d f x^4+4 f^2 x^4} \, dx &=\int \frac{x}{e^2+4 e f x^2+4 \left (d f+f^2\right ) x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{e^2+4 e f x+4 \left (d f+f^2\right ) x^2} \, dx,x,x^2\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{-16 d e^2 f-x^2} \, dx,x,4 f \left (e+2 (d+f) x^2\right )\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (e+2 (d+f) x^2\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}}\\ \end{align*}

Mathematica [A]  time = 0.0212751, size = 42, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^2 (d+f)+e\right )}{\sqrt{d} e}\right )}{4 \sqrt{d} e \sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 42, normalized size = 1. \begin{align*}{\frac{1}{4\,e}\arctan \left ({\frac{2\, \left ( 4\,df+4\,{f}^{2} \right ){x}^{2}+4\,fe}{4\,e}{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*d*f*x^4+4*f^2*x^4+4*e*f*x^2+e^2),x)

[Out]

1/4/e/(d*f)^(1/2)*arctan(1/4*(2*(4*d*f+4*f^2)*x^2+4*f*e)/e/(d*f)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43143, size = 340, normalized size = 8.1 \begin{align*} \left [-\frac{\sqrt{-d f} \log \left (\frac{4 \,{\left (d^{2} f + 2 \, d f^{2} + f^{3}\right )} x^{4} - d e^{2} + e^{2} f + 4 \,{\left (d e f + e f^{2}\right )} x^{2} - 2 \,{\left (2 \,{\left (d e + e f\right )} x^{2} + e^{2}\right )} \sqrt{-d f}}{4 \,{\left (d f + f^{2}\right )} x^{4} + 4 \, e f x^{2} + e^{2}}\right )}{8 \, d e f}, \frac{\sqrt{d f} \arctan \left (\frac{{\left (2 \,{\left (d + f\right )} x^{2} + e\right )} \sqrt{d f}}{d e}\right )}{4 \, d e f}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*d*f*x^4+4*f^2*x^4+4*e*f*x^2+e^2),x, algorithm="fricas")

[Out]

[-1/8*sqrt(-d*f)*log((4*(d^2*f + 2*d*f^2 + f^3)*x^4 - d*e^2 + e^2*f + 4*(d*e*f + e*f^2)*x^2 - 2*(2*(d*e + e*f)
*x^2 + e^2)*sqrt(-d*f))/(4*(d*f + f^2)*x^4 + 4*e*f*x^2 + e^2))/(d*e*f), 1/4*sqrt(d*f)*arctan((2*(d + f)*x^2 +
e)*sqrt(d*f)/(d*e))/(d*e*f)]

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Sympy [B]  time = 0.526625, size = 78, normalized size = 1.86 \begin{align*} \frac{- \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{2} + \frac{- d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{8} + \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{2} + \frac{d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{8}}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*d*f*x**4+4*f**2*x**4+4*e*f*x**2+e**2),x)

[Out]

(-sqrt(-1/(d*f))*log(x**2 + (-d*e*sqrt(-1/(d*f)) + e)/(2*d + 2*f))/8 + sqrt(-1/(d*f))*log(x**2 + (d*e*sqrt(-1/
(d*f)) + e)/(2*d + 2*f))/8)/e

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Giac [A]  time = 1.30321, size = 51, normalized size = 1.21 \begin{align*} \frac{\arctan \left (\frac{{\left (2 \, d f x^{2} + 2 \, f^{2} x^{2} + f e\right )} e^{\left (-1\right )}}{\sqrt{d f}}\right ) e^{\left (-1\right )}}{4 \, \sqrt{d f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*d*f*x^4+4*f^2*x^4+4*e*f*x^2+e^2),x, algorithm="giac")

[Out]

1/4*arctan((2*d*f*x^2 + 2*f^2*x^2 + f*e)*e^(-1)/sqrt(d*f))*e^(-1)/sqrt(d*f)