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])

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Rubi [A]  time = 0.0265826, 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, Rules used = {2282, 44} \[ -\frac{\log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac{1}{d \log (f) \left (f^{c+d x}+1\right )}+x \]

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])

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0338202, size = 37, normalized size = 0.92 \[ \frac{\frac{1}{f^{c+d x}+1}-\log \left (f^{c+d x}+1\right )+d x \log (f)}{d \log (f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 68, normalized size = 1.7 \begin{align*}{\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 ) }} \end{align*}

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
)

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Maxima [A]  time = 0.989898, size = 74, normalized size = 1.85 \begin{align*} -\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 )} \end{align*}

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, 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))

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Fricas [A]  time = 1.60264, size = 159, normalized size = 3.98 \begin{align*} \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 )} \end{align*}

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, 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*l
og(f))

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Sympy [A]  time = 0.116324, size = 34, normalized size = 0.85 \begin{align*} 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 )}} \end{align*}

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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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(1/(1+2*f^(d*x+c)+f^(2*d*x+2*c)),x, algorithm="giac")

[Out]

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