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

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

[Out]

x - a^(m*x)/(m*Log[a])

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Rubi [A]  time = 0.0117776, 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 \[ x-\frac{a^{m x}}{m \log (a)} \]

Antiderivative was successfully verified.

[In]  Int[1 - a^(m*x),x]

[Out]

x - a^(m*x)/(m*Log[a])

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Rubi in Sympy [A]  time = 0.91116, size = 10, normalized size = 0.62 \[ - \frac{a^{m x}}{m \log{\left (a \right )}} + x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1-a**(m*x),x)

[Out]

-a**(m*x)/(m*log(a)) + x

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Mathematica [A]  time = 0.00714906, 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])

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

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)

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Maxima [A]  time = 1.34517, size = 22, normalized size = 1.38 \[ x - \frac{a^{m x}}{m \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-a^(m*x) + 1,x, algorithm="maxima")

[Out]

x - a^(m*x)/(m*log(a))

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Fricas [A]  time = 0.223231, size = 28, normalized size = 1.75 \[ \frac{m x \log \left (a\right ) - a^{m x}}{m \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-a^(m*x) + 1,x, algorithm="fricas")

[Out]

(m*x*log(a) - a^(m*x))/(m*log(a))

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Sympy [A]  time = 0.084571, size = 19, normalized size = 1.19 \[ x + \begin{cases} - \frac{a^{m x}}{m \log{\left (a \right )}} & \text{for}\: m \log{\left (a \right )} \neq 0 \\- x & \text{otherwise} \end{cases} \]

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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GIAC/XCAS [A]  time = 0.195935, size = 22, normalized size = 1.38 \[ x - \frac{a^{m x}}{m{\rm ln}\left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-a^(m*x) + 1,x, algorithm="giac")

[Out]

x - a^(m*x)/(m*ln(a))