Optimal. Leaf size=26 \[ -\frac {a^x b^x}{x}+\text {Ei}(x (\log (a)+\log (b))) (\log (a)+\log (b)) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2325, 2208,
2209} \begin {gather*} (\log (a)+\log (b)) \text {Ei}(x (\log (a)+\log (b)))-\frac {a^x b^x}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2208
Rule 2209
Rule 2325
Rubi steps
\begin {align*} \int \frac {a^x b^x}{x^2} \, dx &=\int \frac {e^{x (\log (a)+\log (b))}}{x^2} \, dx\\ &=-\frac {a^x b^x}{x}-(-\log (a)-\log (b)) \int \frac {e^{x (\log (a)+\log (b))}}{x} \, dx\\ &=-\frac {a^x b^x}{x}+\text {Ei}(x (\log (a)+\log (b))) (\log (a)+\log (b))\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^x b^x}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.03, size = 160, normalized size = 6.15
method | result | size |
meijerg | \(-\ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right ) \left (-\frac {2+2 x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}{2 x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}+\frac {{\mathrm e}^{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}}{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}+\ln \left (-x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right )+\expIntegral \left (1, -x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right )+1-\ln \left (x \right )-i \pi -\ln \left (\ln \left (b \right )\right )-\ln \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )+\frac {1}{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}\right )\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 16, normalized size = 0.62 \begin {gather*} {\left (\log \left (a\right ) + \log \left (b\right )\right )} \Gamma \left (-1, -x {\left (\log \left (a\right ) + \log \left (b\right )\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 34, normalized size = 1.31 \begin {gather*} -\frac {a^{x} b^{x} - {\left (x \log \left (a\right ) + x \log \left (b\right )\right )} {\rm Ei}\left (x \log \left (a\right ) + x \log \left (b\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{x} b^{x}}{x^{2}}\, dx \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 = 3.50, size = 28, normalized size = 1.08 \begin {gather*} -\mathrm {expint}\left (-x\,\left (\ln \left (a\right )+\ln \left (b\right )\right )\right )\,\left (\ln \left (a\right )+\ln \left (b\right )\right )-\frac {a^x\,b^x}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________