3.368 \(\int \frac{e-2 f (-1+n) x^n}{e^2-4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx\)

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

[Out]

ArcTanh[(2*Sqrt[d]*Sqrt[f]*x)/(e + 2*f*x^n)]/(2*Sqrt[d]*Sqrt[f])

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Rubi [A]  time = 0.153447, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^n}\right )}{2 \sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Int[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)),x]

[Out]

ArcTanh[(2*Sqrt[d]*Sqrt[f]*x)/(e + 2*f*x^n)]/(2*Sqrt[d]*Sqrt[f])

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Rubi in Sympy [A]  time = 74.1072, size = 44, normalized size = 1.16 \[ \frac{\operatorname{atanh}{\left (\frac{2 \sqrt{d} \sqrt{f} x \left (n - 1\right )}{e \left (n - 1\right ) + 2 f x^{n} \left (n - 1\right )} \right )}}{2 \sqrt{d} \sqrt{f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e-2*f*(-1+n)*x**n)/(e**2-4*d*f*x**2+4*e*f*x**n+4*f**2*x**(2*n)),x)

[Out]

atanh(2*sqrt(d)*sqrt(f)*x*(n - 1)/(e*(n - 1) + 2*f*x**n*(n - 1)))/(2*sqrt(d)*sqr
t(f))

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Mathematica [A]  time = 0.141701, size = 0, normalized size = 0. \[ \int \frac{e-2 f (-1+n) x^n}{e^2-4 d f x^2+4 e f x^n+4 f^2 x^{2 n}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)),x]

[Out]

Integrate[(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)),
x]

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Maple [B]  time = 0.074, size = 72, normalized size = 1.9 \[{\frac{1}{4}\ln \left ({x}^{n}+{\frac{1}{2\,f} \left ( 2\,dfx+e\sqrt{df} \right ){\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}}-{\frac{1}{4}\ln \left ({x}^{n}+{\frac{1}{2\,f} \left ( -2\,dfx+e\sqrt{df} \right ){\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e-2*f*(-1+n)*x^n)/(e^2-4*d*f*x^2+4*e*f*x^n+4*f^2*x^(2*n)),x)

[Out]

1/4/(d*f)^(1/2)*ln(x^n+1/2*(2*d*f*x+e*(d*f)^(1/2))/(d*f)^(1/2)/f)-1/4/(d*f)^(1/2
)*ln(x^n+1/2*(-2*d*f*x+e*(d*f)^(1/2))/(d*f)^(1/2)/f)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, f{\left (n - 1\right )} x^{n} - e}{4 \, d f x^{2} - 4 \, f^{2} x^{2 \, n} - 4 \, e f x^{n} - e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*f*(n - 1)*x^n - e)/(4*d*f*x^2 - 4*f^2*x^(2*n) - 4*e*f*x^n - e^2),x, algorithm="maxima")

[Out]

integrate((2*f*(n - 1)*x^n - e)/(4*d*f*x^2 - 4*f^2*x^(2*n) - 4*e*f*x^n - e^2), x
)

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Fricas [A]  time = 0.307275, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{4 \, d e f x + 4 \, \sqrt{d f} f^{2} x^{2 \, n} + 4 \,{\left (2 \, d f^{2} x + \sqrt{d f} e f\right )} x^{n} +{\left (4 \, d f x^{2} + e^{2}\right )} \sqrt{d f}}{4 \, d f x^{2} - 4 \, f^{2} x^{2 \, n} - 4 \, e f x^{n} - e^{2}}\right )}{4 \, \sqrt{d f}}, \frac{\arctan \left (\frac{2 \, \sqrt{-d f} f x^{n} + \sqrt{-d f} e}{2 \, d f x}\right )}{2 \, \sqrt{-d f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*f*(n - 1)*x^n - e)/(4*d*f*x^2 - 4*f^2*x^(2*n) - 4*e*f*x^n - e^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e-2*f*(-1+n)*x**n)/(e**2-4*d*f*x**2+4*e*f*x**n+4*f**2*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, f{\left (n - 1\right )} x^{n} - e}{4 \, d f x^{2} - 4 \, f^{2} x^{2 \, n} - 4 \, e f x^{n} - e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*f*(n - 1)*x^n - e)/(4*d*f*x^2 - 4*f^2*x^(2*n) - 4*e*f*x^n - e^2),x, algorithm="giac")

[Out]

integrate((2*f*(n - 1)*x^n - e)/(4*d*f*x^2 - 4*f^2*x^(2*n) - 4*e*f*x^n - e^2), x
)