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

Optimal. Leaf size=96 \[ -\frac{\text{PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac{x \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{x}{d \log (f) \left (f^{c+d x}+1\right )}-\frac{x}{d \log (f)}+\frac{x^2}{2} \]

[Out]

x^2/2 - x/(d*Log[f]) + x/(d*(1 + f^(c + d*x))*Log[f]) + Log[1 + f^(c + d*x)]/(d^
2*Log[f]^2) - (x*Log[1 + f^(c + d*x)])/(d*Log[f]) - PolyLog[2, -f^(c + d*x)]/(d^
2*Log[f]^2)

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Rubi [A]  time = 0.369568, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44 \[ -\frac{\text{PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac{x \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{x}{d \log (f) \left (f^{c+d x}+1\right )}-\frac{x}{d \log (f)}+\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x)),x]

[Out]

x^2/2 - x/(d*Log[f]) + x/(d*(1 + f^(c + d*x))*Log[f]) + Log[1 + f^(c + d*x)]/(d^
2*Log[f]^2) - (x*Log[1 + f^(c + d*x)])/(d*Log[f]) - PolyLog[2, -f^(c + d*x)]/(d^
2*Log[f]^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{2 f^{c + d x} + f^{2 c + 2 d x} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(1+2*f**(d*x+c)+f**(2*d*x+2*c)),x)

[Out]

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

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Mathematica [A]  time = 0.249826, size = 88, normalized size = 0.92 \[ -\frac{\text{PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}+\frac{1}{2} x \left (\frac{2}{d \log (f) f^{c+d x}+d \log (f)}+x\right )-\frac{x \left (\log \left (f^{c+d x}+1\right )+1\right )}{d \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x)),x]

[Out]

(x*(x + 2/(d*Log[f] + d*f^(c + d*x)*Log[f])))/2 + Log[1 + f^(c + d*x)]/(d^2*Log[
f]^2) - (x*(1 + Log[1 + f^(c + d*x)]))/(d*Log[f]) - PolyLog[2, -f^(c + d*x)]/(d^
2*Log[f]^2)

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Maple [A]  time = 0.044, size = 134, normalized size = 1.4 \[{\frac{x}{d \left ( 1+{f}^{dx+c} \right ) \ln \left ( f \right ) }}+{\frac{{x}^{2}}{2}}+{\frac{cx}{d}}+{\frac{{c}^{2}}{2\,{d}^{2}}}-{\frac{x\ln \left ( 1+{f}^{dx+c} \right ) }{d\ln \left ( f \right ) }}-{\frac{{\it polylog} \left ( 2,-{f}^{dx+c} \right ) }{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}+{\frac{\ln \left ( 1+{f}^{dx+c} \right ) }{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}-{\frac{\ln \left ({f}^{dx+c} \right ) }{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}-{\frac{c\ln \left ({f}^{dx+c} \right ) }{{d}^{2}\ln \left ( f \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(1+2*f^(d*x+c)+f^(2*d*x+2*c)),x)

[Out]

x/d/(1+f^(d*x+c))/ln(f)+1/2*x^2+1/d*c*x+1/2/d^2*c^2-x*ln(1+f^(d*x+c))/d/ln(f)-po
lylog(2,-f^(d*x+c))/d^2/ln(f)^2+ln(1+f^(d*x+c))/d^2/ln(f)^2-1/d^2/ln(f)^2*ln(f^(
d*x+c))-1/d^2/ln(f)*c*ln(f^(d*x+c))

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Maxima [A]  time = 0.802529, size = 154, normalized size = 1.6 \[ \frac{x}{d f^{d x} f^{c} \log \left (f\right ) + d \log \left (f\right )} + \frac{\log \left (f^{d x}\right )^{2}}{2 \, d^{2} \log \left (f\right )^{2}} - \frac{\log \left (f^{d x} f^{c} + 1\right ) \log \left (f^{d x}\right ) +{\rm Li}_2\left (-f^{d x} f^{c}\right )}{d^{2} \log \left (f\right )^{2}} + \frac{\log \left (f^{d x} f^{c} + 1\right )}{d^{2} \log \left (f\right )^{2}} - \frac{\log \left (f^{d x}\right )}{d^{2} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1),x, algorithm="maxima")

[Out]

x/(d*f^(d*x)*f^c*log(f) + d*log(f)) + 1/2*log(f^(d*x))^2/(d^2*log(f)^2) - (log(f
^(d*x)*f^c + 1)*log(f^(d*x)) + dilog(-f^(d*x)*f^c))/(d^2*log(f)^2) + log(f^(d*x)
*f^c + 1)/(d^2*log(f)^2) - log(f^(d*x))/(d^2*log(f)^2)

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Fricas [A]  time = 0.267916, size = 193, normalized size = 2.01 \[ \frac{{\left (d^{2} x^{2} - c^{2}\right )} \log \left (f\right )^{2} +{\left ({\left (d^{2} x^{2} - c^{2}\right )} \log \left (f\right )^{2} - 2 \,{\left (d x + c\right )} \log \left (f\right )\right )} f^{d x + c} - 2 \,{\left (f^{d x + c} + 1\right )}{\rm Li}_2\left (-f^{d x + c}\right ) - 2 \,{\left (d x \log \left (f\right ) +{\left (d x \log \left (f\right ) - 1\right )} f^{d x + c} - 1\right )} \log \left (f^{d x + c} + 1\right ) - 2 \, c \log \left (f\right )}{2 \,{\left (d^{2} f^{d x + c} \log \left (f\right )^{2} + d^{2} \log \left (f\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1),x, algorithm="fricas")

[Out]

1/2*((d^2*x^2 - c^2)*log(f)^2 + ((d^2*x^2 - c^2)*log(f)^2 - 2*(d*x + c)*log(f))*
f^(d*x + c) - 2*(f^(d*x + c) + 1)*dilog(-f^(d*x + c)) - 2*(d*x*log(f) + (d*x*log
(f) - 1)*f^(d*x + c) - 1)*log(f^(d*x + c) + 1) - 2*c*log(f))/(d^2*f^(d*x + c)*lo
g(f)^2 + d^2*log(f)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x}{d f^{c + d x} \log{\left (f \right )} + d \log{\left (f \right )}} + \frac{\int \frac{d x \log{\left (f \right )}}{e^{c \log{\left (f \right )}} e^{d x \log{\left (f \right )}} + 1}\, dx + \int \left (- \frac{1}{e^{c \log{\left (f \right )}} e^{d x \log{\left (f \right )}} + 1}\right )\, dx}{d \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(1+2*f**(d*x+c)+f**(2*d*x+2*c)),x)

[Out]

x/(d*f**(c + d*x)*log(f) + d*log(f)) + (Integral(d*x*log(f)/(exp(c*log(f))*exp(d
*x*log(f)) + 1), x) + Integral(-1/(exp(c*log(f))*exp(d*x*log(f)) + 1), x))/(d*lo
g(f))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{f^{2 \, d x + 2 \, c} + 2 \, f^{d x + c} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(f^(2*d*x + 2*c) + 2*f^(d*x + c) + 1),x, algorithm="giac")

[Out]

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