Integrand size = 7, antiderivative size = 14 \[ \int a^x b^x \, dx=\frac {a^x b^x}{\log (a)+\log (b)} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2325, 2225} \[ \int a^x b^x \, dx=\frac {a^x b^x}{\log (a)+\log (b)} \]
[In]
[Out]
Rule 2225
Rule 2325
Rubi steps \begin{align*} \text {integral}& = \int e^{x (\log (a)+\log (b))} \, dx \\ & = \frac {a^x b^x}{\log (a)+\log (b)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int a^x b^x \, dx=\frac {a^x b^x}{\log (a)+\log (b)} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
gosper | \(\frac {a^{x} b^{x}}{\ln \left (a \right )+\ln \left (b \right )}\) | \(15\) |
risch | \(\frac {a^{x} b^{x}}{\ln \left (a \right )+\ln \left (b \right )}\) | \(15\) |
parallelrisch | \(\frac {a^{x} b^{x}}{\ln \left (a \right )+\ln \left (b \right )}\) | \(15\) |
norman | \(\frac {{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\ln \left (a \right )+\ln \left (b \right )}\) | \(19\) |
meijerg | \(-\frac {1-{\mathrm e}^{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}}{\ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}\) | \(36\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int a^x b^x \, dx=\frac {a^{x} b^{x}}{\log \left (a\right ) + \log \left (b\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.21 \[ \int a^x b^x \, dx=\begin {cases} \frac {a^{x} b^{x}}{\log {\left (a \right )} + \log {\left (b \right )}} & \text {for}\: a \neq \frac {1}{b} \\\frac {b^{x} \left (\frac {1}{b}\right )^{x}}{\log {\left (\frac {1}{b} \right )} + \log {\left (b \right )}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int a^x b^x \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 16.93 \[ \int a^x b^x \, dx=2 \, {\left (\frac {2 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\left (a\right ) - \frac {1}{2} \, \pi x \mathrm {sgn}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi \mathrm {sgn}\left (a\right ) - \pi \mathrm {sgn}\left (b\right )\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}^{2}} + \frac {{\left (2 \, \pi - \pi \mathrm {sgn}\left (a\right ) - \pi \mathrm {sgn}\left (b\right )\right )} \sin \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\left (a\right ) - \frac {1}{2} \, \pi x \mathrm {sgn}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi \mathrm {sgn}\left (a\right ) - \pi \mathrm {sgn}\left (b\right )\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}\right )} + i \, {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\left (a\right ) + \frac {1}{2} i \, \pi x \mathrm {sgn}\left (b\right ) - i \, \pi x\right )}}{-2 i \, \pi + i \, \pi \mathrm {sgn}\left (a\right ) + i \, \pi \mathrm {sgn}\left (b\right ) + 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\left (a\right ) - \frac {1}{2} i \, \pi x \mathrm {sgn}\left (b\right ) + i \, \pi x\right )}}{2 i \, \pi - i \, \pi \mathrm {sgn}\left (a\right ) - i \, \pi \mathrm {sgn}\left (b\right ) + 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}\right )} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int a^x b^x \, dx=\frac {a^x\,b^x}{\ln \left (a\right )+\ln \left (b\right )} \]
[In]
[Out]