3.6.30 \(\int \frac {x}{e^2+4 e f x^2+4 d f x^4+4 f^2 x^4} \, dx\) [530]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, 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, 1121, 632, 210} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {f} \left (2 x^2 (d+f)+e\right )}{\sqrt {d} e}\right )}{4 \sqrt {d} e \sqrt {f}} \end {gather*}

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 210

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

Rule 632

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 1121

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]

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} \text {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 )\\ &=-\text {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*}

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Mathematica [A]
time = 0.02, size = 42, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.03, size = 42, normalized size = 1.00

method	result	size
default	\(\frac {\arctan \left (\frac {2 \left (4 d f +4 f^{2}\right ) x^{2}+4 e f}{4 \sqrt {d f}\, e}\right )}{4 \sqrt {d f}\, e}\)	\(42\)
risch	\(-\frac {\ln \left (\left (2 \sqrt {-d f}-2 f \right ) x^{2}-e \right )}{8 \sqrt {-d f}\, e}+\frac {\ln \left (\left (2 \sqrt {-d f}+2 f \right ) x^{2}+e \right )}{8 \sqrt {-d f}\, e}\)	\(64\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 35, normalized size = 0.83 \begin {gather*} \frac {\arctan \left (\frac {{\left (2 \, {\left (d f + f^{2}\right )} x^{2} + f e\right )} e^{\left (-1\right )}}{\sqrt {d f}}\right ) e^{\left (-1\right )}}{4 \, \sqrt {d f}} \end {gather*}

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]

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

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Fricas [A]
time = 0.36, size = 147, normalized size = 3.50 \begin {gather*} \left [-\frac {\sqrt {-d f} e^{\left (-1\right )} \log \left (\frac {4 \, {\left (d^{2} f + 2 \, d f^{2} + f^{3}\right )} x^{4} + 4 \, {\left (d f + f^{2}\right )} x^{2} e - {\left (d - f\right )} e^{2} - 2 \, {\left (2 \, {\left (d + f\right )} x^{2} e + e^{2}\right )} \sqrt {-d f}}{4 \, {\left (d f + f^{2}\right )} x^{4} + 4 \, f x^{2} e + e^{2}}\right )}{8 \, d f}, \frac {\sqrt {d f} \arctan \left (\frac {{\left (2 \, {\left (d + f\right )} x^{2} + e\right )} \sqrt {d f} e^{\left (-1\right )}}{d}\right ) e^{\left (-1\right )}}{4 \, d f}\right ] \end {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (37) = 74\).
time = 0.38, size = 78, normalized size = 1.86 \begin {gather*} \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 {gather*}

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 = 8.13, size = 38, normalized size = 0.90 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 3.10, size = 42, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atan}\left (\frac {e\,\sqrt {f}+2\,f^{3/2}\,x^2+2\,d\,\sqrt {f}\,x^2}{\sqrt {d}\,e}\right )}{4\,\sqrt {d}\,e\,\sqrt {f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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