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]

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

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \frac {\log \left (a+b F^x\right )}{b \log (F)} \]

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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fricas [A]  time = 0.42, size = 16, normalized size = 1.00 \[ \frac {\log \left (F^{x} b + a\right )}{b \log \relax (F)} \]

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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giac [A]  time = 0.29, size = 17, normalized size = 1.06 \[ \frac {\log \left ({\left | F^{x} b + a \right |}\right )}{b \log \relax (F)} \]

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))

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maple [A]  time = 0.00, size = 17, normalized size = 1.06 \[ \frac {\ln \left (b \,F^{x}+a \right )}{b \ln \relax (F )} \]

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 = 0.43, size = 16, normalized size = 1.00 \[ \frac {\log \left (F^{x} b + a\right )}{b \log \relax (F)} \]

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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mupad [B]  time = 3.44, size = 16, normalized size = 1.00 \[ \frac {\ln \left (a+F^x\,b\right )}{b\,\ln \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^x/(a + F^x*b),x)

[Out]

log(a + F^x*b)/(b*log(F))

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sympy [A]  time = 0.12, size = 12, normalized size = 0.75 \[ \frac {\log {\left (F^{x} + \frac {a}{b} \right )}}{b \log {\relax (F )}} \]

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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