\(\int \frac {4^x}{a-2^x b} \, dx\) [475]

Optimal result

Integrand size = 14, antiderivative size = 32 \[ \int \frac {4^x}{a-2^x b} \, dx=-\frac {2^x}{b \log (2)}-\frac {a \log \left (a-2^x b\right )}{b^2 \log (2)} \]

[Out]

-2^x/b/ln(2)-a*ln(a-2^x*b)/b^2/ln(2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2280, 45} \[ \int \frac {4^x}{a-2^x b} \, dx=-\frac {a \log \left (a-b 2^x\right )}{b^2 \log (2)}-\frac {2^x}{b \log (2)} \]

[In]

Int[4^x/(a - 2^x*b),x]

[Out]

-(2^x/(b*Log[2])) - (a*Log[a - 2^x*b])/(b^2*Log[2])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2280

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[g*h*(Log[G]/(d*e*Log[F]))]}, Dist[Denominator[m]*(G^(f*h - c*g*(h/d))/(d*e*Log[F])), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*((c + d*x)/Denominator[m]))], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{a-b x} \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{b}-\frac {a}{b (-a+b x)}\right ) \, dx,x,2^x\right )}{\log (2)} \\ & = -\frac {2^x}{b \log (2)}-\frac {a \log \left (a-2^x b\right )}{b^2 \log (2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {4^x}{a-2^x b} \, dx=-\frac {2^x b+a \log \left (a-2^x b\right )}{b^2 \log (2)} \]

[In]

Integrate[4^x/(a - 2^x*b),x]

[Out]

-((2^x*b + a*Log[a - 2^x*b])/(b^2*Log[2]))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {2^{x}}{b \ln \left (2\right )}-\frac {a \ln \left (2^{x}-\frac {a}{b}\right )}{\ln \left (2\right ) b^{2}}\)	\(35\)
norman	\(-\frac {{\mathrm e}^{x \ln \left (2\right )}}{\ln \left (2\right ) b}-\frac {a \ln \left (a -{\mathrm e}^{x \ln \left (2\right )} b \right )}{\ln \left (2\right ) b^{2}}\)	\(37\)

[In]

int(4^x/(a-2^x*b),x,method=_RETURNVERBOSE)

[Out]

-2^x/b/ln(2)-a/ln(2)/b^2*ln(2^x-a/b)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {4^x}{a-2^x b} \, dx=-\frac {2^{x} b + a \log \left (2^{x} b - a\right )}{b^{2} \log \left (2\right )} \]

[In]

integrate(4^x/(a-2^x*b),x, algorithm="fricas")

[Out]

-(2^x*b + a*log(2^x*b - a))/(b^2*log(2))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {4^x}{a-2^x b} \, dx=- \frac {a \log {\left (- \frac {a}{b} + e^{\frac {x \log {\left (4 \right )}}{2}} \right )}}{b^{2} \log {\left (2 \right )}} + \begin {cases} - \frac {e^{\frac {x \log {\left (4 \right )}}{2}}}{b \log {\left (2 \right )}} & \text {for}\: b \log {\left (2 \right )} \neq 0 \\- \frac {x}{b} & \text {otherwise} \end {cases} \]

[In]

integrate(4**x/(a-2**x*b),x)

[Out]

-a*log(-a/b + exp(x*log(4)/2))/(b**2*log(2)) + Piecewise((-exp(x*log(4)/2)/(b*log(2)), Ne(b*log(2), 0)), (-x/b
, True))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {4^x}{a-2^x b} \, dx=-\frac {2^{x}}{b \log \left (2\right )} - \frac {a \log \left (2^{x} b - a\right )}{b^{2} \log \left (2\right )} \]

[In]

integrate(4^x/(a-2^x*b),x, algorithm="maxima")

[Out]

-2^x/(b*log(2)) - a*log(2^x*b - a)/(b^2*log(2))

Giac [F]

\[ \int \frac {4^x}{a-2^x b} \, dx=\int { -\frac {4^{x}}{2^{x} b - a} \,d x } \]

[In]

integrate(4^x/(a-2^x*b),x, algorithm="giac")

[Out]

integrate(-4^x/(2^x*b - a), x)

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {4^x}{a-2^x b} \, dx=-\frac {2^x\,b+a\,\ln \left (2^x\,b-a\right )}{b^2\,\ln \left (2\right )} \]

[In]

int(4^x/(a - 2^x*b),x)

[Out]

-(2^x*b + a*log(2^x*b - a))/(b^2*log(2))