3.507 \(\int (a^{k x}-a^{l x}) \, dx\)

Optimal. Leaf size=28 \[ \frac{a^{k x}}{k \log (a)}-\frac{a^{l x}}{l \log (a)} \]

[Out]

a^(k*x)/(k*Log[a]) - a^(l*x)/(l*Log[a])

________________________________________________________________________________________

Rubi [A]  time = 0.0083801, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2194} \[ \frac{a^{k x}}{k \log (a)}-\frac{a^{l x}}{l \log (a)} \]

Antiderivative was successfully verified.

[In]

Int[a^(k*x) - a^(l*x),x]

[Out]

a^(k*x)/(k*Log[a]) - a^(l*x)/(l*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 (a^{k x}-a^{l x}\right ) \, dx &=\int a^{k x} \, dx-\int a^{l x} \, dx\\ &=\frac{a^{k x}}{k \log (a)}-\frac{a^{l x}}{l \log (a)}\\ \end{align*}

Mathematica [A]  time = 0.0061269, size = 28, normalized size = 1. \[ \frac{a^{k x}}{k \log (a)}-\frac{a^{l x}}{l \log (a)} \]

Antiderivative was successfully verified.

[In]

Integrate[a^(k*x) - a^(l*x),x]

[Out]

a^(k*x)/(k*Log[a]) - a^(l*x)/(l*Log[a])

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 29, normalized size = 1. \begin{align*}{\frac{{a}^{kx}}{k\ln \left ( a \right ) }}-{\frac{{a}^{lx}}{l\ln \left ( a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^(k*x)-a^(l*x),x)

[Out]

a^(k*x)/k/ln(a)-a^(l*x)/l/ln(a)

________________________________________________________________________________________

Maxima [A]  time = 0.934184, size = 38, normalized size = 1.36 \begin{align*} \frac{a^{k x}}{k \log \left (a\right )} - \frac{a^{l x}}{l \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^(k*x)-a^(l*x),x, algorithm="maxima")

[Out]

a^(k*x)/(k*log(a)) - a^(l*x)/(l*log(a))

________________________________________________________________________________________

Fricas [A]  time = 1.87292, size = 53, normalized size = 1.89 \begin{align*} -\frac{a^{l x} k - a^{k x} l}{k l \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^(k*x)-a^(l*x),x, algorithm="fricas")

[Out]

-(a^(l*x)*k - a^(k*x)*l)/(k*l*log(a))

________________________________________________________________________________________

Sympy [A]  time = 0.287002, size = 29, normalized size = 1.04 \begin{align*} \begin{cases} \frac{a^{k x}}{k \log{\left (a \right )}} & \text{for}\: k \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} - \begin{cases} \frac{a^{l x}}{l \log{\left (a \right )}} & \text{for}\: l \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(a**(k*x)-a**(l*x),x)

[Out]

Piecewise((a**(k*x)/(k*log(a)), Ne(k*log(a), 0)), (x, True)) - Piecewise((a**(l*x)/(l*log(a)), Ne(l*log(a), 0)
), (x, True))

________________________________________________________________________________________

Giac [A]  time = 1.12463, size = 38, normalized size = 1.36 \begin{align*} \frac{a^{k x}}{k \log \left (a\right )} - \frac{a^{l x}}{l \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^(k*x)-a^(l*x),x, algorithm="giac")

[Out]

a^(k*x)/(k*log(a)) - a^(l*x)/(l*log(a))