Optimal. Leaf size=21 \[ (a+b \log (3)) \log (x)-b \text {Li}_2\left (-\frac {e x}{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2439, 2438}
\begin {gather*} \log (x) (a+b \log (3))-b \text {PolyLog}\left (2,-\frac {e x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2438
Rule 2439
Rubi steps
\begin {align*} \int \frac {a+b \log (3+e x)}{x} \, dx &=(a+b \log (3)) \log (x)+b \int \frac {\log \left (1+\frac {e x}{3}\right )}{x} \, dx\\ &=(a+b \log (3)) \log (x)-b \text {Li}_2\left (-\frac {e x}{3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 22, normalized size = 1.05 \begin {gather*} a \log (x)+b \log (3) \log (x)-b \text {Li}_2\left (-\frac {e x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs.
\(2(19)=38\).
time = 0.18, size = 46, normalized size = 2.19
method | result | size |
risch | \(\ln \left (x \right ) a -\ln \left (\frac {e x}{3}+1\right ) \ln \left (-\frac {e x}{3}\right ) b +\ln \left (e x +3\right ) \ln \left (-\frac {e x}{3}\right ) b -\dilog \left (\frac {e x}{3}+1\right ) b\) | \(44\) |
derivativedivides | \(a \ln \left (e x \right )+\ln \left (e x +3\right ) \ln \left (-\frac {e x}{3}\right ) b -\ln \left (\frac {e x}{3}+1\right ) \ln \left (-\frac {e x}{3}\right ) b -\dilog \left (\frac {e x}{3}+1\right ) b\) | \(46\) |
default | \(a \ln \left (e x \right )+\ln \left (e x +3\right ) \ln \left (-\frac {e x}{3}\right ) b -\ln \left (\frac {e x}{3}+1\right ) \ln \left (-\frac {e x}{3}\right ) b -\dilog \left (\frac {e x}{3}+1\right ) b\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 30, normalized size = 1.43 \begin {gather*} {\left (\log \left (x e + 3\right ) \log \left (-\frac {1}{3} \, x e\right ) + {\rm Li}_2\left (\frac {1}{3} \, x e + 1\right )\right )} b + a \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [A]
time = 2.00, size = 94, normalized size = 4.48 \begin {gather*} a \log {\left (x \right )} + b \left (\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (3 \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (3 \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (3 \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (3 \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {otherwise} \end {cases}\right ) \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 = 0.08, size = 25, normalized size = 1.19 \begin {gather*} b\,{\mathrm {Li}}_{\mathrm {2}}\left (-\frac {e\,x}{3}\right )+a\,\ln \left (x\right )+b\,\ln \left (e\,x+3\right )\,\ln \left (-\frac {e\,x}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________