3.570 \(\int a^x b^{-x} \, dx\)

Optimal. Leaf size=18 \[ \frac{a^x b^{-x}}{\log (a)-\log (b)} \]

[Out]

a^x/(b^x*(Log[a] - Log[b]))

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Rubi [A]  time = 0.0211684, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2287, 2194} \[ \frac{a^x b^{-x}}{\log (a)-\log (b)} \]

Antiderivative was successfully verified.

[In]

Int[a^x/b^x,x]

[Out]

a^x/(b^x*(Log[a] - Log[b]))

Rule 2287

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

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 a^x b^{-x} \, dx &=\int e^{x (\log (a)-\log (b))} \, dx\\ &=\frac{a^x b^{-x}}{\log (a)-\log (b)}\\ \end{align*}

Mathematica [A]  time = 0.0096114, size = 18, normalized size = 1. \[ \frac{a^x b^{-x}}{\log (a)-\log (b)} \]

Antiderivative was successfully verified.

[In]

Integrate[a^x/b^x,x]

[Out]

a^x/(b^x*(Log[a] - Log[b]))

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Maple [A]  time = 0.003, size = 19, normalized size = 1.1 \begin{align*}{\frac{{a}^{x}}{{b}^{x} \left ( \ln \left ( a \right ) -\ln \left ( b \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x/(b^x),x)

[Out]

a^x/(b^x)/(ln(a)-ln(b))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x/(b^x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28415, size = 39, normalized size = 2.17 \begin{align*} \frac{a^{x}}{b^{x}{\left (\log \left (a\right ) - \log \left (b\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x/(b^x),x, algorithm="fricas")

[Out]

a^x/(b^x*(log(a) - log(b)))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x/(b**x),x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.49597, size = 312, normalized size = 17.33 \begin{align*} 2 \,{\left (\frac{2 \,{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi x \mathrm{sgn}\left (a\right ) + \frac{1}{2} \, \pi x \mathrm{sgn}\left (b\right )\right )}{{\left (\pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )}^{2} + 4 \,{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}^{2}} - \frac{{\left (\pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )} \sin \left (-\frac{1}{2} \, \pi x \mathrm{sgn}\left (a\right ) + \frac{1}{2} \, \pi x \mathrm{sgn}\left (b\right )\right )}{{\left (\pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )}^{2} + 4 \,{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}\right )} - \frac{{\left (\frac{i e^{\left (\frac{1}{2} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} - \pi{\left (\mathrm{sgn}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i \mathrm{sgn}\left (a\right ) - \pi i \mathrm{sgn}\left (b\right ) + 2 \, \log \left ({\left | a \right |}\right ) - 2 \, \log \left ({\left | b \right |}\right )} + \frac{i e^{\left (-\frac{1}{2} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} - \pi{\left (\mathrm{sgn}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i \mathrm{sgn}\left (a\right ) - \pi i \mathrm{sgn}\left (b\right ) - 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )}\right )} e^{\left (x{\left (\log \left ({\left | a \right |}\right ) - \log \left ({\left | b \right |}\right )\right )}\right )}}{i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x/(b^x),x, algorithm="giac")

[Out]

2*(2*(log(abs(a)) - log(abs(b)))*cos(-1/2*pi*x*sgn(a) + 1/2*pi*x*sgn(b))/((pi*sgn(a) - pi*sgn(b))^2 + 4*(log(a
bs(a)) - log(abs(b)))^2) - (pi*sgn(a) - pi*sgn(b))*sin(-1/2*pi*x*sgn(a) + 1/2*pi*x*sgn(b))/((pi*sgn(a) - pi*sg
n(b))^2 + 4*(log(abs(a)) - log(abs(b)))^2))*e^(x*(log(abs(a)) - log(abs(b)))) - (i*e^(1/2*(pi*(sgn(a) - 1) - p
i*(sgn(b) - 1))*i*x)/(pi*i*sgn(a) - pi*i*sgn(b) + 2*log(abs(a)) - 2*log(abs(b))) + i*e^(-1/2*(pi*(sgn(a) - 1)
- pi*(sgn(b) - 1))*i*x)/(pi*i*sgn(a) - pi*i*sgn(b) - 2*log(abs(a)) + 2*log(abs(b))))*e^(x*(log(abs(a)) - log(a
bs(b))))/i