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