\(\int (a^{k x}-a^{l x}) \, dx\) [507]

Optimal result

Integrand size = 13, antiderivative size = 28 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log (a)}-\frac {a^{l x}}{l \log (a)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2225} \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log (a)}-\frac {a^{l x}}{l \log (a)} \]

[In]

Int[a^(k*x) - a^(l*x),x]

[Out]

a^(k*x)/(k*Log[a]) - a^(l*x)/(l*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}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log (a)}-\frac {a^{l x}}{l \log (a)} \]

[In]

Integrate[a^(k*x) - a^(l*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {a^{k x} l -a^{l x} k}{\ln \left (a \right ) k l}\)	\(28\)
default	\(\frac {a^{k x}}{k \ln \left (a \right )}-\frac {a^{l x}}{l \ln \left (a \right )}\)	\(29\)
risch	\(\frac {a^{k x}}{k \ln \left (a \right )}-\frac {a^{l x}}{l \ln \left (a \right )}\)	\(29\)
parts	\(\frac {a^{k x}}{k \ln \left (a \right )}-\frac {a^{l x}}{l \ln \left (a \right )}\)	\(29\)
norman	\(\frac {{\mathrm e}^{k x \ln \left (a \right )}}{k \ln \left (a \right )}-\frac {{\mathrm e}^{l x \ln \left (a \right )}}{l \ln \left (a \right )}\)	\(31\)
meijerg	\(-\frac {1-{\mathrm e}^{k x \ln \left (a \right )}}{k \ln \left (a \right )}+\frac {1-{\mathrm e}^{l x \ln \left (a \right )}}{l \ln \left (a \right )}\)	\(39\)

[In]

int(a^(k*x)-a^(l*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=-\frac {a^{l x} k - a^{k x} l}{k l \log \left (a\right )} \]

[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] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\begin {cases} \frac {a^{k x}}{k \log {\left (a \right )}} & \text {for}\: k \log {\left (a \right )} \neq 0 \\x & \text {otherwise} \end {cases} - \begin {cases} \frac {a^{l x}}{l \log {\left (a \right )}} & \text {for}\: l \log {\left (a \right )} \neq 0 \\x & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log \left (a\right )} - \frac {a^{l x}}{l \log \left (a\right )} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log \left (a\right )} - \frac {a^{l x}}{l \log \left (a\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \left (a^{k x}-a^{l x}\right ) \, dx=\frac {a^{k\,x}\,l-a^{l\,x}\,k}{k\,l\,\ln \left (a\right )} \]

[In]

int(a^(k*x) - a^(l*x),x)

[Out]

(a^(k*x)*l - a^(l*x)*k)/(k*l*log(a))