3.2.51 \(\int a^x \, dx\) [151]

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

[Out]

a^x/ln(a)

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {2225} \begin {gather*} \frac {a^x}{\log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a^x,x]

[Out]

a^x/Log[a]

Rule 2225

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*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {a^x}{\log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a^x,x]

[Out]

a^x/Log[a]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.81, size = 16, normalized size = 2.00 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {a^x}{\text {Log}\left [a\right ]},\text {Log}\left [a\right ]\text {!=}0\right \}\right \},x\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[a^x,x]')

[Out]

Piecewise[{{a ^ x / Log[a], Log[a] != 0}}, x]

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Maple [A]
time = 0.01, size = 9, normalized size = 1.12

method	result	size
gosper	\(\frac {a^{x}}{\ln \left (a \right )}\)	\(9\)
derivativedivides	\(\frac {a^{x}}{\ln \left (a \right )}\)	\(9\)
default	\(\frac {a^{x}}{\ln \left (a \right )}\)	\(9\)
risch	\(\frac {a^{x}}{\ln \left (a \right )}\)	\(9\)
norman	\(\frac {{\mathrm e}^{x \ln \left (a \right )}}{\ln \left (a \right )}\)	\(11\)
meijerg	\(-\frac {1-{\mathrm e}^{x \ln \left (a \right )}}{\ln \left (a \right )}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x,x,method=_RETURNVERBOSE)

[Out]

a^x/ln(a)

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Maxima [A]
time = 0.25, size = 8, normalized size = 1.00 \begin {gather*} \frac {a^{x}}{\log \left (a\right )} \end {gather*}

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 = 0.34, size = 8, normalized size = 1.00 \begin {gather*} \frac {a^{x}}{\log \left (a\right )} \end {gather*}

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 = 0.04, size = 8, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {a^{x}}{\log {\left (a \right )}} & \text {for}\: \log {\left (a \right )} \neq 0 \\x & \text {otherwise} \end {cases} \end {gather*}

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 = 0.00, size = 7, normalized size = 0.88 \begin {gather*} \frac {a^{x}}{\ln a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x,x)

[Out]

a^x/log(a)

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Mupad [B]
time = 0.16, size = 8, normalized size = 1.00 \begin {gather*} \frac {a^x}{\ln \left (a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x,x)

[Out]

a^x/log(a)

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