Optimal. Leaf size=32 \[ -\frac {2^x}{b \log (2)}-\frac {a \log \left (a-2^x b\right )}{b^2 \log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2280, 45}
\begin {gather*} -\frac {a \log \left (a-b 2^x\right )}{b^2 \log (2)}-\frac {2^x}{b \log (2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2280
Rubi steps
\begin {align*} \int \frac {2^{2 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 = 0.01, size = 26, normalized size = 0.81 \begin {gather*} -\frac {2^x b+a \log \left (a-2^x b\right )}{b^2 \log (2)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 35, normalized size = 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 | \(-\frac {{\mathrm e}^{x \ln \left (2\right )}}{\ln \left (2\right ) b}-\frac {a \ln \left (a -{\mathrm e}^{x \ln \left (2\right )} b \right )}{\ln \left (2\right ) b^{2}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 33, normalized size = 1.03 \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 = 0.37, size = 27, normalized size = 0.84 \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 = 0.07, size = 34, normalized size = 1.06 \begin {gather*} - \frac {a \log {\left (2^{x} - \frac {a}{b} \right )}}{b^{2} \log {\left (2 \right )}} + \begin {cases} - \frac {2^{x}}{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 [A]
time = 1.94, size = 34, normalized size = 1.06 \begin {gather*} -\frac {2^{x}}{b \log \left (2\right )} - \frac {a \log \left ({\left | 2^{x} b - a \right |}\right )}{b^{2} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.58, size = 27, normalized size = 0.84 \begin {gather*} -\frac {2^x\,b+a\,\ln \left (2^x\,b-a\right )}{b^2\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________