\(\int \frac {a+b \log (3+e x)}{x} \, dx\) [79]

Optimal result

Integrand size = 14, antiderivative size = 21 \[ \int \frac {a+b \log (3+e x)}{x} \, dx=(a+b \log (3)) \log (x)-b \operatorname {PolyLog}\left (2,-\frac {e x}{3}\right ) \]

[Out]

(a+b*ln(3))*ln(x)-b*polylog(2,-1/3*e*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2439, 2438} \[ \int \frac {a+b \log (3+e x)}{x} \, dx=\log (x) (a+b \log (3))-b \operatorname {PolyLog}\left (2,-\frac {e x}{3}\right ) \]

[In]

Int[(a + b*Log[3 + e*x])/x,x]

[Out]

(a + b*Log[3])*Log[x] - b*PolyLog[2, -1/3*(e*x)]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + e*(x/d)]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rubi steps \begin{align*} \text {integral}& = (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] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \log (3+e x)}{x} \, dx=a \log (x)+b \log (3) \log (x)-b \operatorname {PolyLog}\left (2,-\frac {e x}{3}\right ) \]

[In]

Integrate[(a + b*Log[3 + e*x])/x,x]

[Out]

a*Log[x] + b*Log[3]*Log[x] - b*PolyLog[2, -1/3*(e*x)]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(19)=38\).

Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90

method	result	size
parts	\(\ln \left (x \right ) a +b \left (\left (\ln \left (e x +3\right )-\ln \left (\frac {e x}{3}+1\right )\right ) \ln \left (-\frac {e x}{3}\right )-\operatorname {dilog}\left (\frac {e x}{3}+1\right )\right )\)	\(40\)
derivativedivides	\(a \ln \left (e x \right )+b \left (\left (\ln \left (e x +3\right )-\ln \left (\frac {e x}{3}+1\right )\right ) \ln \left (-\frac {e x}{3}\right )-\operatorname {dilog}\left (\frac {e x}{3}+1\right )\right )\)	\(42\)
default	\(a \ln \left (e x \right )+b \left (\left (\ln \left (e x +3\right )-\ln \left (\frac {e x}{3}+1\right )\right ) \ln \left (-\frac {e x}{3}\right )-\operatorname {dilog}\left (\frac {e x}{3}+1\right )\right )\)	\(42\)
risch	\(\ln \left (x \right ) a +\ln \left (e x +3\right ) \ln \left (-\frac {e x}{3}\right ) b -\ln \left (\frac {e x}{3}+1\right ) \ln \left (-\frac {e x}{3}\right ) b -\operatorname {dilog}\left (\frac {e x}{3}+1\right ) b\)	\(44\)

[In]

int((a+b*ln(e*x+3))/x,x,method=_RETURNVERBOSE)

[Out]

ln(x)*a+b*((ln(e*x+3)-ln(1/3*e*x+1))*ln(-1/3*e*x)-dilog(1/3*e*x+1))

Fricas [F]

\[ \int \frac {a+b \log (3+e x)}{x} \, dx=\int { \frac {b \log \left (e x + 3\right ) + a}{x} \,d x } \]

[In]

integrate((a+b*log(e*x+3))/x,x, algorithm="fricas")

[Out]

integral((b*log(e*x + 3) + a)/x, x)

Sympy [A] (verification not implemented)

Time = 1.99 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.48 \[ \int \frac {a+b \log (3+e x)}{x} \, dx=a \log {\left (x \right )} + b \left (\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (3 \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (3 \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (3 \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (3 \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((a+b*ln(e*x+3))/x,x)

[Out]

a*log(x) + b*Piecewise((-polylog(2, e*x*exp_polar(I*pi)/3), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(3)*log(x) - p
olylog(2, e*x*exp_polar(I*pi)/3), Abs(x) < 1), (-log(3)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/3), 1/Abs(x)
< 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(3) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(3) - polyl
og(2, e*x*exp_polar(I*pi)/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \log (3+e x)}{x} \, dx={\left (\log \left (e x + 3\right ) \log \left (-\frac {1}{3} \, e x\right ) + {\rm Li}_2\left (\frac {1}{3} \, e x + 1\right )\right )} b + a \log \left (x\right ) \]

[In]

integrate((a+b*log(e*x+3))/x,x, algorithm="maxima")

[Out]

(log(e*x + 3)*log(-1/3*e*x) + dilog(1/3*e*x + 1))*b + a*log(x)

Giac [F]

\[ \int \frac {a+b \log (3+e x)}{x} \, dx=\int { \frac {b \log \left (e x + 3\right ) + a}{x} \,d x } \]

[In]

integrate((a+b*log(e*x+3))/x,x, algorithm="giac")

[Out]

integrate((b*log(e*x + 3) + a)/x, x)

Mupad [B] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log (3+e x)}{x} \, dx=b\,{\mathrm {Li}}_{\mathrm {2}}\left (-\frac {e\,x}{3}\right )+a\,\ln \left (x\right )+b\,\ln \left (e\,x+3\right )\,\ln \left (-\frac {e\,x}{3}\right ) \]

[In]

int((a + b*log(e*x + 3))/x,x)

[Out]

b*dilog(-(e*x)/3) + a*log(x) + b*log(e*x + 3)*log(-(e*x)/3)