3.4 \(\int a^x \, dx\)

Optimal. Leaf size=8 \[ \frac{a^x}{\log (a)} \]

[Out]

a^x/Log[a]

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Rubi [A]  time = 0.002335, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 3, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2194} \[ \frac{a^x}{\log (a)} \]

Antiderivative was successfully verified.

[In]

Int[a^x,x]

[Out]

a^x/Log[a]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int a^x \, dx &=\frac{a^x}{\log (a)}\\ \end{align*}

Mathematica [A]  time = 0.0005207, size = 8, normalized size = 1. \[ \frac{a^x}{\log (a)} \]

Antiderivative was successfully verified.

[In]

Integrate[a^x,x]

[Out]

a^x/Log[a]

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Maple [A]  time = 0., size = 9, normalized size = 1.1 \begin{align*}{\frac{{a}^{x}}{\ln \left ( a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x,x)

[Out]

a^x/ln(a)

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Maxima [A]  time = 0.9221, size = 11, normalized size = 1.38 \begin{align*} \frac{a^{x}}{\log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x,x, algorithm="maxima")

[Out]

a^x/log(a)

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Fricas [A]  time = 1.89986, size = 16, normalized size = 2. \begin{align*} \frac{a^{x}}{\log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x,x, algorithm="fricas")

[Out]

a^x/log(a)

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Sympy [A]  time = 0.082901, size = 8, normalized size = 1. \begin{align*} \begin{cases} \frac{a^{x}}{\log{\left (a \right )}} & \text{for}\: \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x,x)

[Out]

Piecewise((a**x/log(a), Ne(log(a), 0)), (x, True))

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Giac [A]  time = 1.04125, size = 11, normalized size = 1.38 \begin{align*} \frac{a^{x}}{\log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x,x, algorithm="giac")

[Out]

a^x/log(a)