\(\int \log (\sqrt {3}+\tan (x)) \, dx\) [275]

Optimal result

Integrand size = 9, antiderivative size = 108 \[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=-\frac {1}{2} i \left (\left (\log \left (\frac {i-\tan (x)}{i+\sqrt {3}}\right )-\log \left (\frac {i+\tan (x)}{i-\sqrt {3}}\right )\right ) \log \left (\sqrt {3}+\tan (x)\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {3}+\tan (x)}{-i+\sqrt {3}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {3}+\tan (x)}{i+\sqrt {3}}\right )\right ) \] Output:

-1/2*I*((ln((I-tan(x))/(3^(1/2)+I))-ln((I+tan(x))/(I-3^(1/2))))*ln(3^(1/2) 
+tan(x))-polylog(2,(3^(1/2)+tan(x))/(-I+3^(1/2)))+polylog(2,(3^(1/2)+tan(x 
))/(3^(1/2)+I)))
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.22 \[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=-\frac {1}{2} i \log \left (\frac {i-\tan (x)}{i+\sqrt {3}}\right ) \log \left (\sqrt {3}+\tan (x)\right )+\frac {1}{2} i \log \left (\frac {i+\tan (x)}{i-\sqrt {3}}\right ) \log \left (\sqrt {3}+\tan (x)\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {\sqrt {3}+\tan (x)}{i-\sqrt {3}}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {\sqrt {3}+\tan (x)}{i+\sqrt {3}}\right ) \] Input:

Integrate[Log[Sqrt[3] + Tan[x]],x]
 

Output:

(-1/2*I)*Log[(I - Tan[x])/(I + Sqrt[3])]*Log[Sqrt[3] + Tan[x]] + (I/2)*Log 
[(I + Tan[x])/(I - Sqrt[3])]*Log[Sqrt[3] + Tan[x]] + (I/2)*PolyLog[2, -((S 
qrt[3] + Tan[x])/(I - Sqrt[3]))] - (I/2)*PolyLog[2, (Sqrt[3] + Tan[x])/(I 
+ Sqrt[3])]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \log \left (\tan (x)+\sqrt {3}\right ) \, dx\)

\(\Big \downarrow \) 3028

\(\displaystyle x \log \left (\tan (x)+\sqrt {3}\right )-\int \frac {x \sec ^2(x)}{\tan (x)+\sqrt {3}}dx\)

\(\Big \downarrow \) 7299

\(\displaystyle x \log \left (\tan (x)+\sqrt {3}\right )-\int \frac {x \sec ^2(x)}{\tan (x)+\sqrt {3}}dx\)

Input:

Int[Log[Sqrt[3] + Tan[x]],x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\frac {i \ln \left (\sqrt {3}+\tan \left (x \right )\right ) \ln \left (\frac {i-\tan \left (x \right )}{\sqrt {3}+i}\right )}{2}+\frac {i \ln \left (\sqrt {3}+\tan \left (x \right )\right ) \ln \left (\frac {-i-\tan \left (x \right )}{-i+\sqrt {3}}\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {i-\tan \left (x \right )}{\sqrt {3}+i}\right )}{2}+\frac {i \operatorname {dilog}\left (\frac {-i-\tan \left (x \right )}{-i+\sqrt {3}}\right )}{2}\)	\(96\)
default	\(-\frac {i \ln \left (\sqrt {3}+\tan \left (x \right )\right ) \ln \left (\frac {i-\tan \left (x \right )}{\sqrt {3}+i}\right )}{2}+\frac {i \ln \left (\sqrt {3}+\tan \left (x \right )\right ) \ln \left (\frac {-i-\tan \left (x \right )}{-i+\sqrt {3}}\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {i-\tan \left (x \right )}{\sqrt {3}+i}\right )}{2}+\frac {i \operatorname {dilog}\left (\frac {-i-\tan \left (x \right )}{-i+\sqrt {3}}\right )}{2}\)	\(96\)
risch	\(\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )}{{\mathrm e}^{2 i x}+1}\right )}^{2} x}{2}+x \ln \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}+1\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}+1}\right ) \operatorname {csgn}\left (i \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )\right ) \operatorname {csgn}\left (\frac {i \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )}{{\mathrm e}^{2 i x}+1}\right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )}{{\mathrm e}^{2 i x}+1}\right )}^{3} x}{2}-i \operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )-\ln \left (\frac {-2 \,{\mathrm e}^{i x}-i \sqrt {3}+1}{1-i \sqrt {3}}\right ) x -i \operatorname {dilog}\left (\frac {2 \,{\mathrm e}^{i x}}{1-i \sqrt {3}}\right )+i \operatorname {dilog}\left (\frac {-i \sqrt {3}+2 \,{\mathrm e}^{i x}+1}{1-i \sqrt {3}}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\sqrt {3}\, {\mathrm e}^{2 i x}-i {\mathrm e}^{2 i x}+\sqrt {3}+i\right )}{{\mathrm e}^{2 i x}+1}\right )}^{2} x}{2}-i \ln \left (\frac {-2 \,{\mathrm e}^{i x}-i \sqrt {3}+1}{1-i \sqrt {3}}\right ) \ln \left (\frac {2 \,{\mathrm e}^{i x}}{1-i \sqrt {3}}\right )-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+i {\mathrm e}^{i x}\right )-x \ln \left (\frac {-i \sqrt {3}+2 \,{\mathrm e}^{i x}+1}{1-i \sqrt {3}}\right )-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1-i {\mathrm e}^{i x}\right )-i \operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )\)	\(522\)

Input:

int(ln(3^(1/2)+tan(x)),x,method=_RETURNVERBOSE)
 

Output:

-1/2*I*ln(3^(1/2)+tan(x))*ln((I-tan(x))/(3^(1/2)+I))+1/2*I*ln(3^(1/2)+tan( 
x))*ln((-I-tan(x))/(-I+3^(1/2)))-1/2*I*dilog((I-tan(x))/(3^(1/2)+I))+1/2*I 
*dilog((-I-tan(x))/(-I+3^(1/2)))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (75) = 150\).

Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.43 \[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=x \log \left (\sqrt {3} + \tan \left (x\right )\right ) - \frac {1}{2} \, x \log \left (\frac {{\left (i \, \sqrt {3} + 1\right )} \tan \left (x\right )^{2} + 2 \, {\left (\sqrt {3} + i\right )} \tan \left (x\right ) - i \, \sqrt {3} + 3}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}}\right ) - \frac {1}{2} \, x \log \left (\frac {{\left (-i \, \sqrt {3} + 1\right )} \tan \left (x\right )^{2} + 2 \, {\left (\sqrt {3} - i\right )} \tan \left (x\right ) + i \, \sqrt {3} + 3}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}}\right ) + \frac {1}{2} \, x \log \left (-\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac {1}{2} \, x \log \left (-\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac {1}{4} i \, {\rm Li}_2\left (-\frac {{\left (i \, \sqrt {3} + 1\right )} \tan \left (x\right )^{2} + 2 \, {\left (\sqrt {3} + i\right )} \tan \left (x\right ) - i \, \sqrt {3} + 3}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-\frac {{\left (-i \, \sqrt {3} + 1\right )} \tan \left (x\right )^{2} + 2 \, {\left (\sqrt {3} - i\right )} \tan \left (x\right ) + i \, \sqrt {3} + 3}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} + 1\right ) + \frac {1}{4} i \, {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) \] Input:

integrate(log(3^(1/2)+tan(x)),x, algorithm="fricas")
 

Output:

x*log(sqrt(3) + tan(x)) - 1/2*x*log(1/2*((I*sqrt(3) + 1)*tan(x)^2 + 2*(sqr 
t(3) + I)*tan(x) - I*sqrt(3) + 3)/(tan(x)^2 + 1)) - 1/2*x*log(1/2*((-I*sqr 
t(3) + 1)*tan(x)^2 + 2*(sqrt(3) - I)*tan(x) + I*sqrt(3) + 3)/(tan(x)^2 + 1 
)) + 1/2*x*log(-2*(I*tan(x) - 1)/(tan(x)^2 + 1)) + 1/2*x*log(-2*(-I*tan(x) 
 - 1)/(tan(x)^2 + 1)) + 1/4*I*dilog(-1/2*((I*sqrt(3) + 1)*tan(x)^2 + 2*(sq 
rt(3) + I)*tan(x) - I*sqrt(3) + 3)/(tan(x)^2 + 1) + 1) - 1/4*I*dilog(-1/2* 
((-I*sqrt(3) + 1)*tan(x)^2 + 2*(sqrt(3) - I)*tan(x) + I*sqrt(3) + 3)/(tan( 
x)^2 + 1) + 1) + 1/4*I*dilog(2*(I*tan(x) - 1)/(tan(x)^2 + 1) + 1) - 1/4*I* 
dilog(2*(-I*tan(x) - 1)/(tan(x)^2 + 1) + 1)
 

Sympy [F]

\[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=\int \log {\left (\tan {\left (x \right )} + \sqrt {3} \right )}\, dx \] Input:

integrate(ln(3**(1/2)+tan(x)),x)
 

Output:

Integral(log(tan(x) + sqrt(3)), x)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=\frac {1}{2} \, \arctan \left (\frac {1}{4} \, \sqrt {3} + \frac {1}{4} \, \tan \left (x\right ), \frac {1}{4} \, \sqrt {3} \tan \left (x\right ) + \frac {3}{4}\right ) \log \left (\tan \left (x\right )^{2} + 1\right ) - \frac {1}{2} \, x \log \left (\frac {1}{4} \, \tan \left (x\right )^{2} + \frac {1}{2} \, \sqrt {3} \tan \left (x\right ) + \frac {3}{4}\right ) + x \log \left (\sqrt {3} + \tan \left (x\right )\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\frac {{\left (\sqrt {3} + i\right )} \tan \left (x\right ) - i \, \sqrt {3} + 1}{2 i \, \sqrt {3} + 2}\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\frac {{\left (\sqrt {3} - i\right )} \tan \left (x\right ) + i \, \sqrt {3} + 1}{2 i \, \sqrt {3} - 2}\right ) \] Input:

integrate(log(3^(1/2)+tan(x)),x, algorithm="maxima")
 

Output:

1/2*arctan2(1/4*sqrt(3) + 1/4*tan(x), 1/4*sqrt(3)*tan(x) + 3/4)*log(tan(x) 
^2 + 1) - 1/2*x*log(1/4*tan(x)^2 + 1/2*sqrt(3)*tan(x) + 3/4) + x*log(sqrt( 
3) + tan(x)) + 1/2*I*dilog(-((sqrt(3) + I)*tan(x) - I*sqrt(3) + 1)/(2*I*sq 
rt(3) + 2)) - 1/2*I*dilog(((sqrt(3) - I)*tan(x) + I*sqrt(3) + 1)/(2*I*sqrt 
(3) - 2))
 

Giac [F]

\[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=\int { \log \left (\sqrt {3} + \tan \left (x\right )\right ) \,d x } \] Input:

integrate(log(3^(1/2)+tan(x)),x, algorithm="giac")
 

Output:

integrate(log(sqrt(3) + tan(x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=\int \ln \left (\mathrm {tan}\left (x\right )+\sqrt {3}\right ) \,d x \] Input:

int(log(tan(x) + 3^(1/2)),x)
 

Output:

int(log(tan(x) + 3^(1/2)), x)
 

Reduce [F]

\[ \int \log \left (\sqrt {3}+\tan (x)\right ) \, dx=\int \mathrm {log}\left (\sqrt {3}+\tan \left (x \right )\right )d x \] Input:

int(log(3^(1/2)+tan(x)),x)
 

Output:

int(log(sqrt(3) + tan(x)),x)