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 = 0.0358685, 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 \[ \frac{\log \left (a+b F^x\right )}{b \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^x/(a + b*F^x),x]

[Out]

Log[a + b*F^x]/(b*Log[F])

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Rubi in Sympy [A]  time = 11.105, size = 12, normalized size = 0.75 \[ \frac{\log{\left (F^{x} b + a \right )}}{b \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**x/(a+b*F**x),x)

[Out]

log(F**x*b + a)/(b*log(F))

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Mathematica [A]  time = 0.0045716, 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 = 0.003, size = 17, normalized size = 1.1 \[{\frac{\ln \left ( a+b{F}^{x} \right ) }{b\ln \left ( F \right ) }} \]

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 = 0.86913, size = 22, normalized size = 1.38 \[ \frac{\log \left (F^{x} b + a\right )}{b \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(F^x*b + a)/(b*log(F))

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Fricas [A]  time = 0.2675, size = 22, normalized size = 1.38 \[ \frac{\log \left (F^{x} b + a\right )}{b \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(F^x*b + a)/(b*log(F))

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Sympy [A]  time = 0.243275, size = 12, normalized size = 0.75 \[ \frac{\log{\left (F^{x} + \frac{a}{b} \right )}}{b \log{\left (F \right )}} \]

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/XCAS [A]  time = 0.233287, size = 23, normalized size = 1.44 \[ \frac{{\rm ln}\left ({\left | F^{x} b + a \right |}\right )}{b{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(F^x*b + a))/(b*ln(F))