3.520 \(\int \frac{1}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx\)

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]

x + 1/(d*(1 + f^(c + d*x))*Log[f]) - Log[1 + f^(c + d*x)]/(d*Log[f])

_______________________________________________________________________________________

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]

x + 1/(d*(1 + f^(c + d*x))*Log[f]) - Log[1 + f^(c + d*x)]/(d*Log[f])

_______________________________________________________________________________________

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]

Integral(1/(2*f**(c + d*x) + f**(2*c + 2*d*x) + 1), x)

_______________________________________________________________________________________

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]

x + 1/(d*(1 + f^(c + d*x))*Log[f]) - Log[1 + f^(c + d*x)]/(d*Log[f])

_______________________________________________________________________________________

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]

(x+x*exp((d*x+c)*ln(f))-1/d/ln(f)*exp((d*x+c)*ln(f)))/(exp((d*x+c)*ln(f))+1)-1/d
/ln(f)*ln(exp((d*x+c)*ln(f))+1)

_______________________________________________________________________________________

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]

-log(f^(d*x + c) + 1)/(d*log(f)) + log(f^(d*x + c))/(d*log(f)) + 1/(d*(f^(d*x +
c) + 1)*log(f))

_______________________________________________________________________________________

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]

(d*f^(d*x + c)*x*log(f) + d*x*log(f) - (f^(d*x + c) + 1)*log(f^(d*x + c) + 1) +
1)/(d*f^(d*x + c)*log(f) + d*log(f))

_______________________________________________________________________________________

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]

x + 1/(d*f**(c + d*x)*log(f) + d*log(f)) - log(f**(c + d*x) + 1)/(d*log(f))

_______________________________________________________________________________________

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]

integrate(1/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1), x)