3.6.32 \(\int \frac {1}{2+f^{-c-d x}+f^{c+d x}} \, dx\) [532]

Optimal. Leaf size=20 \[ -\frac {1}{d \left (1+f^{c+d x}\right ) \log (f)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2320, 32} \begin {gather*} -\frac {1}{d \log (f) \left (f^{c+d x}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(d*(1 + f^(c + d*x))*Log[f]))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2320

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]]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{d \log (f)+d f^{c+d x} \log (f)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 23, normalized size = 1.15

method	result	size
risch	\(\frac {1}{d \ln \left (f \right ) \left (f^{-d x -c}+1\right )}\)	\(23\)
norman	\(\frac {1}{d \ln \left (f \right ) \left ({\mathrm e}^{\left (-d x -c \right ) \ln \left (f \right )}+1\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+f^(-d*x-c)+f^(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 22, normalized size = 1.10 \begin {gather*} \frac {1}{d {\left (f^{-d x - c} + 1\right )} \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(d*(f^(-d*x - c) + 1)*log(f))

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{d f^{d x + c} \log \left (f\right ) + d \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(d*f^(d*x + c)*log(f) + d*log(f))

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} - \frac {1}{d f^{c + d x} \log {\left (f \right )} + d \log {\left (f \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(d*f**(c + d*x)*log(f) + d*log(f))

________________________________________________________________________________________

Giac [A]
time = 3.93, size = 22, normalized size = 1.10 \begin {gather*} -\frac {1}{{\left (f^{d x} f^{c} + 1\right )} d \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((f^(d*x)*f^c + 1)*d*log(f))

________________________________________________________________________________________

Mupad [B]
time = 3.56, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{d\,\ln \left (f\right )\,\left (f^{c+d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(d*log(f)*(f^(c + d*x) + 1))

________________________________________________________________________________________