\(\int b^x \, dx\) [248]

Optimal result

Integrand size = 3, antiderivative size = 8 \[ \int b^x \, dx=\frac {b^x}{\log (b)} \]

[Out]

b^x/ln(b)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2225} \[ \int b^x \, dx=\frac {b^x}{\log (b)} \]

[In]

Int[b^x,x]

[Out]

b^x/Log[b]

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*} \text {integral}& = \frac {b^x}{\log (b)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int b^x \, dx=\frac {b^x}{\log (b)} \]

[In]

Integrate[b^x,x]

[Out]

b^x/Log[b]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

b^x/ln(b)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int b^x \, dx=\frac {b^{x}}{\log \left (b\right )} \]

[In]

integrate(b^x,x, algorithm="fricas")

[Out]

b^x/log(b)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int b^x \, dx=\begin {cases} \frac {b^{x}}{\log {\left (b \right )}} & \text {for}\: \log {\left (b \right )} \neq 0 \\x & \text {otherwise} \end {cases} \]

[In]

integrate(b**x,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int b^x \, dx=\frac {b^{x}}{\log \left (b\right )} \]

[In]

integrate(b^x,x, algorithm="maxima")

[Out]

b^x/log(b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int b^x \, dx=\frac {b^{x}}{\log \left (b\right )} \]

[In]

integrate(b^x,x, algorithm="giac")

[Out]

b^x/log(b)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int b^x \, dx=\frac {b^x}{\ln \left (b\right )} \]

[In]

int(b^x,x)

[Out]

b^x/log(b)