\(\int (1-a^{m x}) \, dx\) [517]

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int \left (1-a^{m x}\right ) \, dx=x-\frac {a^{m x}}{m \log (a)} \]

[Out]

x-a^(m*x)/m/ln(a)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2225} \[ \int \left (1-a^{m x}\right ) \, dx=x-\frac {a^{m x}}{m \log (a)} \]

[In]

Int[1 - a^(m*x),x]

[Out]

x - a^(m*x)/(m*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*} \text {integral}& = x-\int a^{m x} \, dx \\ & = x-\frac {a^{m x}}{m \log (a)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1-a^{m x}\right ) \, dx=x-\frac {a^{m x}}{m \log (a)} \]

[In]

Integrate[1 - a^(m*x),x]

[Out]

x - a^(m*x)/(m*Log[a])

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
default	\(x -\frac {a^{m x}}{m \ln \left (a \right )}\)	\(17\)
risch	\(x -\frac {a^{m x}}{m \ln \left (a \right )}\)	\(17\)
parallelrisch	\(x -\frac {a^{m x}}{m \ln \left (a \right )}\)	\(17\)
parts	\(x -\frac {a^{m x}}{m \ln \left (a \right )}\)	\(17\)
norman	\(x -\frac {{\mathrm e}^{m x \ln \left (a \right )}}{m \ln \left (a \right )}\)	\(18\)
derivativedivides	\(\frac {-a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\)	\(23\)

[In]

int(1-a^(m*x),x,method=_RETURNVERBOSE)

[Out]

x-a^(m*x)/m/ln(a)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \left (1-a^{m x}\right ) \, dx=\frac {m x \log \left (a\right ) - a^{m x}}{m \log \left (a\right )} \]

[In]

integrate(1-a^(m*x),x, algorithm="fricas")

[Out]

(m*x*log(a) - a^(m*x))/(m*log(a))

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \left (1-a^{m x}\right ) \, dx=x + \begin {cases} - \frac {a^{m x}}{m \log {\left (a \right )}} & \text {for}\: m \log {\left (a \right )} \neq 0 \\- x & \text {otherwise} \end {cases} \]

[In]

integrate(1-a**(m*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1-a^{m x}\right ) \, dx=x - \frac {a^{m x}}{m \log \left (a\right )} \]

[In]

integrate(1-a^(m*x),x, algorithm="maxima")

[Out]

x - a^(m*x)/(m*log(a))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1-a^{m x}\right ) \, dx=x - \frac {a^{m x}}{m \log \left (a\right )} \]

[In]

integrate(1-a^(m*x),x, algorithm="giac")

[Out]

x - a^(m*x)/(m*log(a))

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (1-a^{m x}\right ) \, dx=x-\frac {a^{m\,x}}{m\,\ln \left (a\right )} \]

[In]

int(1 - a^(m*x),x)

[Out]

x - a^(m*x)/(m*log(a))