3.540 \(\int \frac{1}{a+b f^{-c-d x}+c f^{c+d x}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 \tanh ^{-1}\left (\frac{a+2 c f^{c+d x}}{\sqrt{a^2-4 b c}}\right )}{d \log (f) \sqrt{a^2-4 b c}} \]

[Out]

(-2*ArcTanh[(a + 2*c*f^(c + d*x))/Sqrt[a^2 - 4*b*c]])/(Sqrt[a^2 - 4*b*c]*d*Log[f
])

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Rubi [A]  time = 0.123476, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{2 \tanh ^{-1}\left (\frac{a+2 c f^{c+d x}}{\sqrt{a^2-4 b c}}\right )}{d \log (f) \sqrt{a^2-4 b c}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*f^(-c - d*x) + c*f^(c + d*x))^(-1),x]

[Out]

(-2*ArcTanh[(a + 2*c*f^(c + d*x))/Sqrt[a^2 - 4*b*c]])/(Sqrt[a^2 - 4*b*c]*d*Log[f
])

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Rubi in Sympy [A]  time = 14.2553, size = 60, normalized size = 1.28 \[ - \frac{2 f^{- c - d x} f^{c + d x} \operatorname{atanh}{\left (\frac{a + 2 c f^{c + d x}}{\sqrt{a^{2} - 4 b c}} \right )}}{d \sqrt{a^{2} - 4 b c} \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*f**(-d*x-c)+c*f**(d*x+c)),x)

[Out]

-2*f**(-c - d*x)*f**(c + d*x)*atanh((a + 2*c*f**(c + d*x))/sqrt(a**2 - 4*b*c))/(
d*sqrt(a**2 - 4*b*c)*log(f))

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Mathematica [A]  time = 0.073438, size = 51, normalized size = 1.09 \[ \frac{2 \tan ^{-1}\left (\frac{a+2 c f^{c+d x}}{\sqrt{4 b c-a^2}}\right )}{d \log (f) \sqrt{4 b c-a^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*f^(-c - d*x) + c*f^(c + d*x))^(-1),x]

[Out]

(2*ArcTan[(a + 2*c*f^(c + d*x))/Sqrt[-a^2 + 4*b*c]])/(Sqrt[-a^2 + 4*b*c]*d*Log[f
])

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Maple [B]  time = 0.044, size = 135, normalized size = 2.9 \[{\frac{1}{d\ln \left ( f \right ) }\ln \left ({f}^{-dx-c}+{\frac{1}{2\,b} \left ( a\sqrt{{a}^{2}-4\,cb}+{a}^{2}-4\,cb \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}}-{\frac{1}{d\ln \left ( f \right ) }\ln \left ({f}^{-dx-c}+{\frac{1}{2\,b} \left ( a\sqrt{{a}^{2}-4\,cb}-{a}^{2}+4\,cb \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}} \right ){\frac{1}{\sqrt{{a}^{2}-4\,cb}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*f^(-d*x-c)+c*f^(d*x+c)),x)

[Out]

1/(a^2-4*b*c)^(1/2)/d/ln(f)*ln(f^(-d*x-c)+1/2*(a*(a^2-4*b*c)^(1/2)+a^2-4*c*b)/b/
(a^2-4*b*c)^(1/2))-1/(a^2-4*b*c)^(1/2)/d/ln(f)*ln(f^(-d*x-c)+1/2*(a*(a^2-4*b*c)^
(1/2)-a^2+4*c*b)/b/(a^2-4*b*c)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*f^(d*x + c) + b*f^(-d*x - c) + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278909, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{2 \, \sqrt{a^{2} - 4 \, b c} c^{2} f^{2 \, d x + 2 \, c} - a^{3} + 4 \, a b c - 2 \,{\left (a^{2} c - 4 \, b c^{2} - \sqrt{a^{2} - 4 \, b c} a c\right )} f^{d x + c} +{\left (a^{2} - 2 \, b c\right )} \sqrt{a^{2} - 4 \, b c}}{c f^{2 \, d x + 2 \, c} + a f^{d x + c} + b}\right )}{\sqrt{a^{2} - 4 \, b c} d \log \left (f\right )}, \frac{2 \, \arctan \left (-\frac{2 \, \sqrt{-a^{2} + 4 \, b c} c f^{d x + c} + \sqrt{-a^{2} + 4 \, b c} a}{a^{2} - 4 \, b c}\right )}{\sqrt{-a^{2} + 4 \, b c} d \log \left (f\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*f^(d*x + c) + b*f^(-d*x - c) + a),x, algorithm="fricas")

[Out]

[log((2*sqrt(a^2 - 4*b*c)*c^2*f^(2*d*x + 2*c) - a^3 + 4*a*b*c - 2*(a^2*c - 4*b*c
^2 - sqrt(a^2 - 4*b*c)*a*c)*f^(d*x + c) + (a^2 - 2*b*c)*sqrt(a^2 - 4*b*c))/(c*f^
(2*d*x + 2*c) + a*f^(d*x + c) + b))/(sqrt(a^2 - 4*b*c)*d*log(f)), 2*arctan(-(2*s
qrt(-a^2 + 4*b*c)*c*f^(d*x + c) + sqrt(-a^2 + 4*b*c)*a)/(a^2 - 4*b*c))/(sqrt(-a^
2 + 4*b*c)*d*log(f))]

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Sympy [A]  time = 0.487013, size = 66, normalized size = 1.4 \[ \operatorname{RootSum}{\left (z^{2} \left (a^{2} d^{2} \log{\left (f \right )}^{2} - 4 b c d^{2} \log{\left (f \right )}^{2}\right ) - 1, \left ( i \mapsto i \log{\left (f^{c + d x} + \frac{- i a^{2} d \log{\left (f \right )} + 4 i b c d \log{\left (f \right )} + a}{2 c} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*f**(-d*x-c)+c*f**(d*x+c)),x)

[Out]

RootSum(_z**2*(a**2*d**2*log(f)**2 - 4*b*c*d**2*log(f)**2) - 1, Lambda(_i, _i*lo
g(f**(c + d*x) + (-_i*a**2*d*log(f) + 4*_i*b*c*d*log(f) + a)/(2*c))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{c f^{d x + c} + b f^{-d x - c} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*f^(d*x + c) + b*f^(-d*x - c) + a),x, algorithm="giac")

[Out]

integrate(1/(c*f^(d*x + c) + b*f^(-d*x - c) + a), x)