3.517 \(\int (1-a^{m x}) \, dx\)

Optimal. Leaf size=16 \[ x-\frac{a^{m x}}{m \log (a)} \]

[Out]

x - a^(m*x)/(m*Log[a])

________________________________________________________________________________________

Rubi [A]  time = 0.0050012, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2194} \[ x-\frac{a^{m x}}{m \log (a)} \]

Antiderivative was successfully verified.

[In]

Int[1 - a^(m*x),x]

[Out]

x - a^(m*x)/(m*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 \left (1-a^{m x}\right ) \, dx &=x-\int a^{m x} \, dx\\ &=x-\frac{a^{m x}}{m \log (a)}\\ \end{align*}

Mathematica [A]  time = 0.0042407, size = 16, normalized size = 1. \[ x-\frac{a^{m x}}{m \log (a)} \]

Antiderivative was successfully verified.

[In]

Integrate[1 - a^(m*x),x]

[Out]

x - a^(m*x)/(m*Log[a])

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 17, normalized size = 1.1 \begin{align*} x-{\frac{{a}^{mx}}{m\ln \left ( a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1-a^(m*x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.926793, size = 22, normalized size = 1.38 \begin{align*} x - \frac{a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - a^(m*x)/(m*log(a))

________________________________________________________________________________________

Fricas [A]  time = 1.83819, size = 47, normalized size = 2.94 \begin{align*} \frac{m x \log \left (a\right ) - a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(m*x*log(a) - a^(m*x))/(m*log(a))

________________________________________________________________________________________

Sympy [A]  time = 0.090836, size = 19, normalized size = 1.19 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.16075, size = 22, normalized size = 1.38 \begin{align*} x - \frac{a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - a^(m*x)/(m*log(a))