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

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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ x-\frac {a^{m x}}{m \log (a)} \]

Antiderivative was successfully verified.

[In]

Integrate[1 - a^(m*x),x]

[Out]

x - a^(m*x)/(m*Log[a])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1-a^{m x}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1 - a^(m*x),x]

[Out]

Could not integrate

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fricas [A]  time = 0.98, size = 21, normalized size = 1.31 \[ \frac {m x \log \relax (a) - a^{m x}}{m \log \relax (a)} \]

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))

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giac [A]  time = 0.59, size = 16, normalized size = 1.00 \[ x - \frac {a^{m x}}{m \log \relax (a)} \]

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))

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maple [A]  time = 0.03, size = 17, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.68, size = 16, normalized size = 1.00 \[ x - \frac {a^{m x}}{m \log \relax (a)} \]

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))

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mupad [B]  time = 0.30, size = 16, normalized size = 1.00 \[ x-\frac {a^{m\,x}}{m\,\ln \relax (a)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - a^(m*x),x)

[Out]

x - a^(m*x)/(m*log(a))

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sympy [A]  time = 0.10, size = 19, normalized size = 1.19 \[ x + \begin {cases} - \frac {a^{m x}}{m \log {\relax (a )}} & \text {for}\: m \log {\relax (a )} \neq 0 \\- x & \text {otherwise} \end {cases} \]

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))

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