Optimal. Leaf size=40 \[ -\frac{\log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{1}{d \log (f) \left (f^{c+d x}+1\right )}+x \]
[Out]
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Rubi [A] time = 0.0478951, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{\log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{1}{d \log (f) \left (f^{c+d x}+1\right )}+x \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x))^(-1),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{2 f^{c + d x} + f^{2 c + 2 d x} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+2*f**(d*x+c)+f**(2*d*x+2*c)),x)
[Out]
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Mathematica [A] time = 0.0754808, size = 40, normalized size = 1. \[ -\frac{\log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{1}{d \log (f) \left (f^{c+d x}+1\right )}+x \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x))^(-1),x]
[Out]
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Maple [A] time = 0.024, size = 68, normalized size = 1.7 \[{\frac{1}{{{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}+1} \left ( x+x{{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}-{\frac{{{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}}{d\ln \left ( f \right ) }} \right ) }-{\frac{\ln \left ({{\rm e}^{ \left ( dx+c \right ) \ln \left ( f \right ) }}+1 \right ) }{d\ln \left ( f \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+2*f^(d*x+c)+f^(2*d*x+2*c)),x)
[Out]
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Maxima [A] time = 0.753029, size = 74, normalized size = 1.85 \[ -\frac{\log \left (f^{d x + c} + 1\right )}{d \log \left (f\right )} + \frac{\log \left (f^{d x + c}\right )}{d \log \left (f\right )} + \frac{1}{d{\left (f^{d x + c} + 1\right )} \log \left (f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281298, size = 80, normalized size = 2. \[ \frac{d f^{d x + c} x \log \left (f\right ) + d x \log \left (f\right ) -{\left (f^{d x + c} + 1\right )} \log \left (f^{d x + c} + 1\right ) + 1}{d f^{d x + c} \log \left (f\right ) + d \log \left (f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.121286, size = 34, normalized size = 0.85 \[ x + \frac{1}{d f^{c + d x} \log{\left (f \right )} + d \log{\left (f \right )}} - \frac{\log{\left (f^{c + d x} + 1 \right )}}{d \log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+2*f**(d*x+c)+f**(2*d*x+2*c)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{f^{2 \, d x + 2 \, c} + 2 \, f^{d x + c} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1),x, algorithm="giac")
[Out]