Optimal. Leaf size=30 \[ \frac {2^x}{b \log (2)}-\frac {a \log \left (a+2^x b\right )}{b^2 \log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2280, 45}
\begin {gather*} \frac {2^x}{b \log (2)}-\frac {a \log \left (a+b 2^x\right )}{b^2 \log (2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2280
Rubi steps
\begin {align*} \int \frac {4^x}{a+2^x b} \, dx &=\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]
time = 0.09, size = 27, normalized size = 0.90 \begin {gather*} \frac {\frac {2^x}{b}-\frac {a \log \left (a+2^x b\right )}{b^2}}{\log (2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 33, normalized size = 1.10
method | result | size |
risch | \(\frac {2^{x}}{b \ln \left (2\right )}-\frac {a \ln \left (2^{x}+\frac {a}{b}\right )}{\ln \left (2\right ) b^{2}}\) | \(33\) |
norman | \(\frac {{\mathrm e}^{x \ln \left (2\right )}}{\ln \left (2\right ) b}-\frac {a \ln \left (a +{\mathrm e}^{x \ln \left (2\right )} b \right )}{\ln \left (2\right ) b^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 30, normalized size = 1.00 \begin {gather*} \frac {2^{x}}{b \log \left (2\right )} - \frac {a \log \left (2^{x} b + a\right )}{b^{2} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 25, normalized size = 0.83 \begin {gather*} \frac {2^{x} b - a \log \left (2^{x} b + a\right )}{b^{2} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.14, size = 41, normalized size = 1.37 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.60, size = 26, normalized size = 0.87 \begin {gather*} -\frac {a\,\ln \left (a+2^x\,b\right )-2^x\,b}{b^2\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________