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)

________________________________________________________________________________________

Rubi [A]  time = 0.267543, 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, Rules used = {6688, 2185, 2184, 2190, 2279, 2391, 2191, 2282, 36, 29, 31} \[ -\frac{\text{PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac{x \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{x}{d \log (f) \left (f^{c+d x}+1\right )}-\frac{x}{d \log (f)}+\frac{x^2}{2} \]

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)

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 2185

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2184

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[((c + d*x)^m*(F^(g*(e + f*x)))^n)/(a + b*(F^(g*(e + f*x)))^n)
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2191

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1))/(b*f*g*n*(p +
1)*Log[F]), x] - Dist[(d*m)/(b*f*g*n*(p + 1)*Log[F]), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx &=\int \frac{x}{\left (1+f^{c+d x}\right )^2} \, dx\\ &=-\int \frac{f^{c+d x} x}{\left (1+f^{c+d x}\right )^2} \, dx+\int \frac{x}{1+f^{c+d x}} \, dx\\ &=\frac{x^2}{2}+\frac{x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac{\int \frac{1}{1+f^{c+d x}} \, dx}{d \log (f)}-\int \frac{f^{c+d x} x}{1+f^{c+d x}} \, dx\\ &=\frac{x^2}{2}+\frac{x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac{x \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\int \log \left (1+f^{c+d x}\right ) \, dx}{d \log (f)}\\ &=\frac{x^2}{2}+\frac{x}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac{x \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,f^{c+d x}\right )}{d^2 \log ^2(f)}\\ &=\frac{x^2}{2}-\frac{x}{d \log (f)}+\frac{x}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac{\log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac{x \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac{\text{Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}\\ \end{align*}

Mathematica [A]  time = 0.176088, 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)

________________________________________________________________________________________

Maple [A]  time = 0.059, size = 143, normalized size = 1.5 \begin{align*}{\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{\ln \left ({f}^{dx}{f}^{c}+1 \right ) x}{d\ln \left ( f \right ) }}-{\frac{{\it polylog} \left ( 2,-{f}^{dx}{f}^{c} \right ) }{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}-{\frac{\ln \left ({f}^{dx}{f}^{c} \right ) }{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}+{\frac{\ln \left ({f}^{dx}{f}^{c}+1 \right ) }{ \left ( \ln \left ( f \right ) \right ) ^{2}{d}^{2}}}-{\frac{c\ln \left ({f}^{dx}{f}^{c} \right ) }{\ln \left ( f \right ){d}^{2}}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.00549, size = 154, normalized size = 1.6 \begin{align*} \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}} \end{align*}

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, 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)

________________________________________________________________________________________

Fricas [A]  time = 1.54127, size = 356, normalized size = 3.71 \begin{align*} \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 )}} \end{align*}

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, 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)*log(f)^2 + d^2*log(f)^2)

________________________________________________________________________________________

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

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*log(f))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{f^{2 \, d x + 2 \, c} + 2 \, f^{d x + c} + 1}\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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