3.8 \(\int \frac{F^x}{a+b F^x} \, dx\)

Optimal. Leaf size=16 \[ \frac{\log \left (a+b F^x\right )}{b \log (F)} \]

[Out]

Log[a + b*F^x]/(b*Log[F])

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Rubi [A]  time = 0.0205769, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 31} \[ \frac{\log \left (a+b F^x\right )}{b \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^x/(a + b*F^x),x]

[Out]

Log[a + b*F^x]/(b*Log[F])

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, 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{F^x}{a+b F^x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,F^x\right )}{\log (F)}\\ &=\frac{\log \left (a+b F^x\right )}{b \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0059151, size = 16, normalized size = 1. \[ \frac{\log \left (a+b F^x\right )}{b \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^x/(a + b*F^x),x]

[Out]

Log[a + b*F^x]/(b*Log[F])

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Maple [A]  time = 0.001, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b{F}^{x} \right ) }{b\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^x/(a+b*F^x),x)

[Out]

ln(a+b*F^x)/b/ln(F)

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Maxima [A]  time = 1.15051, size = 22, normalized size = 1.38 \begin{align*} \frac{\log \left (F^{x} b + a\right )}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^x/(a+b*F^x),x, algorithm="maxima")

[Out]

log(F^x*b + a)/(b*log(F))

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Fricas [A]  time = 1.4996, size = 36, normalized size = 2.25 \begin{align*} \frac{\log \left (F^{x} b + a\right )}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^x/(a+b*F^x),x, algorithm="fricas")

[Out]

log(F^x*b + a)/(b*log(F))

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Sympy [A]  time = 0.127843, size = 12, normalized size = 0.75 \begin{align*} \frac{\log{\left (F^{x} + \frac{a}{b} \right )}}{b \log{\left (F \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**x/(a+b*F**x),x)

[Out]

log(F**x + a/b)/(b*log(F))

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Giac [A]  time = 1.28401, size = 23, normalized size = 1.44 \begin{align*} \frac{\log \left ({\left | F^{x} b + a \right |}\right )}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^x/(a+b*F^x),x, algorithm="giac")

[Out]

log(abs(F^x*b + a))/(b*log(F))