3.365 \(\int \frac{e-4 f x^3}{e^2+4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx\)

Optimal. Leaf size=38 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]

[Out]

ArcTan[(2*Sqrt[d]*Sqrt[f]*x)/(e + 2*f*x^3)]/(2*Sqrt[d]*Sqrt[f])

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Rubi [A]  time = 0.10521, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Int[(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6),x]

[Out]

ArcTan[(2*Sqrt[d]*Sqrt[f]*x)/(e + 2*f*x^3)]/(2*Sqrt[d]*Sqrt[f])

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Rubi in Sympy [A]  time = 46.3091, size = 36, normalized size = 0.95 \[ \frac{\operatorname{atan}{\left (\frac{4 \sqrt{d} \sqrt{f} x}{2 e + 4 f x^{3}} \right )}}{2 \sqrt{d} \sqrt{f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-4*f*x**3+e)/(4*f**2*x**6+4*e*f*x**3+4*d*f*x**2+e**2),x)

[Out]

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

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Mathematica [C]  time = 0.0929538, size = 87, normalized size = 2.29 \[ -\frac{\text{RootSum}\left [4 \text{$\#$1}^6 f^2+4 \text{$\#$1}^3 e f+4 \text{$\#$1}^2 d f+e^2\&,\frac{4 \text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{6 \text{$\#$1}^5 f+3 \text{$\#$1}^2 e+2 \text{$\#$1} d}\&\right ]}{4 f} \]

Antiderivative was successfully verified.

[In]  Integrate[(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6),x]

[Out]

-RootSum[e^2 + 4*d*f*#1^2 + 4*e*f*#1^3 + 4*f^2*#1^6 & , (-(e*Log[x - #1]) + 4*f*
Log[x - #1]*#1^3)/(2*d*#1 + 3*e*#1^2 + 6*f*#1^5) & ]/(4*f)

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Maple [C]  time = 0.012, size = 70, normalized size = 1.8 \[{\frac{1}{4\,f}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{f}^{2}{{\it \_Z}}^{6}+4\,ef{{\it \_Z}}^{3}+4\,df{{\it \_Z}}^{2}+{e}^{2} \right ) }{\frac{ \left ( -4\,{{\it \_R}}^{3}f+e \right ) \ln \left ( x-{\it \_R} \right ) }{6\,f{{\it \_R}}^{5}+3\,e{{\it \_R}}^{2}+2\,d{\it \_R}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*f*x^3+e)/(4*f^2*x^6+4*e*f*x^3+4*d*f*x^2+e^2),x)

[Out]

1/4/f*sum((-4*_R^3*f+e)/(6*_R^5*f+3*_R^2*e+2*_R*d)*ln(x-_R),_R=RootOf(4*_Z^6*f^2
+4*_Z^3*e*f+4*_Z^2*d*f+e^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{4 \, f x^{3} - e}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2} + e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*f*x^3 - e)/(4*f^2*x^6 + 4*e*f*x^3 + 4*d*f*x^2 + e^2),x, algorithm="maxima")

[Out]

-integrate((4*f*x^3 - e)/(4*f^2*x^6 + 4*e*f*x^3 + 4*d*f*x^2 + e^2), x)

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Fricas [A]  time = 0.286313, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{8 \, d f^{2} x^{4} + 4 \, d e f x -{\left (4 \, f^{2} x^{6} + 4 \, e f x^{3} - 4 \, d f x^{2} + e^{2}\right )} \sqrt{-d f}}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2} + e^{2}}\right )}{4 \, \sqrt{-d f}}, -\frac{\arctan \left (\frac{\sqrt{d f} x^{2}}{d}\right ) - \arctan \left (\frac{2 \, f^{2} x^{5} + e f x^{2} + 2 \, d f x}{\sqrt{d f} e}\right )}{2 \, \sqrt{d f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*f*x^3 - e)/(4*f^2*x^6 + 4*e*f*x^3 + 4*d*f*x^2 + e^2),x, algorithm="fricas")

[Out]

[1/4*log(-(8*d*f^2*x^4 + 4*d*e*f*x - (4*f^2*x^6 + 4*e*f*x^3 - 4*d*f*x^2 + e^2)*s
qrt(-d*f))/(4*f^2*x^6 + 4*e*f*x^3 + 4*d*f*x^2 + e^2))/sqrt(-d*f), -1/2*(arctan(s
qrt(d*f)*x^2/d) - arctan((2*f^2*x^5 + e*f*x^2 + 2*d*f*x)/(sqrt(d*f)*e)))/sqrt(d*
f)]

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Sympy [A]  time = 3.05482, size = 70, normalized size = 1.84 \[ \frac{\sqrt{- \frac{1}{d f}} \log{\left (- d x \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} - \frac{\sqrt{- \frac{1}{d f}} \log{\left (d x \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*f*x**3+e)/(4*f**2*x**6+4*e*f*x**3+4*d*f*x**2+e**2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{4 \, f x^{3} - e}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2} + e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(4*f*x^3 - e)/(4*f^2*x^6 + 4*e*f*x^3 + 4*d*f*x^2 + e^2),x, algorithm="giac")

[Out]

integrate(-(4*f*x^3 - e)/(4*f^2*x^6 + 4*e*f*x^3 + 4*d*f*x^2 + e^2), x)