\(\int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx\) [7]

Optimal result

Integrand size = 21, antiderivative size = 291 \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=-8 \text {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+x}}}{\sqrt {2}}\right ) \log (1+x)+2 \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+\sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )-\sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right )-\sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )+\sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right ) \] Output:

-8*arctanh((1+(1+x)^(1/2))^(1/2))-arctanh(1/2*(1+(1+x)^(1/2))^(1/2)*2^(1/2 
))*ln(1+x)*2^(1/2)+2*arctanh(1/2*2^(1/2))*ln(1-(1+(1+x)^(1/2))^(1/2))*2^(1 
/2)-2*arctanh(1/2*2^(1/2))*ln(1+(1+(1+x)^(1/2))^(1/2))*2^(1/2)+polylog(2,- 
2^(1/2)*(1-(1+(1+x)^(1/2))^(1/2))/(2-2^(1/2)))*2^(1/2)-polylog(2,2^(1/2)*( 
1-(1+(1+x)^(1/2))^(1/2))/(2+2^(1/2)))*2^(1/2)-polylog(2,-2^(1/2)*(1+(1+(1+ 
x)^(1/2))^(1/2))/(2-2^(1/2)))*2^(1/2)+polylog(2,2^(1/2)*(1+(1+(1+x)^(1/2)) 
^(1/2))/(2+2^(1/2)))*2^(1/2)-2*ln(1+x)/(1+(1+x)^(1/2))^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.19 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.07 \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=-8 \text {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}+\frac {\log (1+x) \left (\log \left (\sqrt {2}-\sqrt {1+\sqrt {1+x}}\right )-\log \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )\right )}{\sqrt {2}}+\sqrt {2} \left (-\log \left (-1+\sqrt {2}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )+\log \left (1+\sqrt {2}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )+\log \left (-1+\sqrt {2}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )-\log \left (1+\sqrt {2}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )-\operatorname {PolyLog}\left (2,-\left (\left (-1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )+\operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )+\operatorname {PolyLog}\left (2,\left (-1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )-\operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )\right ) \] Input:

Integrate[Log[1 + x]/(x*Sqrt[1 + Sqrt[1 + x]]),x]
 

Output:

-8*ArcTanh[Sqrt[1 + Sqrt[1 + x]]] - (2*Log[1 + x])/Sqrt[1 + Sqrt[1 + x]] + 
 (Log[1 + x]*(Log[Sqrt[2] - Sqrt[1 + Sqrt[1 + x]]] - Log[Sqrt[2] + Sqrt[1 
+ Sqrt[1 + x]]]))/Sqrt[2] + Sqrt[2]*(-(Log[-1 + Sqrt[2]]*Log[1 - Sqrt[1 + 
Sqrt[1 + x]]]) + Log[1 + Sqrt[2]]*Log[1 - Sqrt[1 + Sqrt[1 + x]]] + Log[-1 
+ Sqrt[2]]*Log[1 + Sqrt[1 + Sqrt[1 + x]]] - Log[1 + Sqrt[2]]*Log[1 + Sqrt[ 
1 + Sqrt[1 + x]]] - PolyLog[2, -((-1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x] 
]))] + PolyLog[2, (1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x]])] + PolyLog[2, 
 (-1 + Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]])] - PolyLog[2, -((1 + Sqrt[2])* 
(1 + Sqrt[1 + Sqrt[1 + x]]))])
 

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.

\(\displaystyle \int \frac {\log (x+1)}{x \sqrt {\sqrt {x+1}+1}} \, dx\)

\(\Big \downarrow \) 2867

\(\displaystyle \int \frac {\log (x+1)}{x \sqrt {\sqrt {x+1}+1}}dx\)

Input:

Int[Log[1 + x]/(x*Sqrt[1 + Sqrt[1 + x]]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2867

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(AFx_), x_Sy 
mbol] :> Unintegrable[AFx*(a + b*Log[c*(d + e*x)^n])^p, x] /; FreeQ[{a, b, 
c, d, e, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.22 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.59

method	result	size
derivativedivides	\(-\frac {2 \ln \left (1+x \right )}{\sqrt {1+\sqrt {1+x}}}-8 \,\operatorname {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\operatorname {dilog}\left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\operatorname {dilog}\left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )\)	\(172\)
default	\(-\frac {2 \ln \left (1+x \right )}{\sqrt {1+\sqrt {1+x}}}-8 \,\operatorname {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\operatorname {dilog}\left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\operatorname {dilog}\left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )\)	\(172\)

Input:

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

Output:

-2*ln(1+x)/(1+(1+x)^(1/2))^(1/2)-8*arctanh((1+(1+x)^(1/2))^(1/2))+8*Sum(1/ 
8*(1/2*ln((1+(1+x)^(1/2))^(1/2)-_alpha)*ln(1+x)-dilog((1+(1+(1+x)^(1/2))^( 
1/2))/(1+_alpha))-ln((1+(1+x)^(1/2))^(1/2)-_alpha)*ln((1+(1+(1+x)^(1/2))^( 
1/2))/(1+_alpha))-dilog(((1+(1+x)^(1/2))^(1/2)-1)/(-1+_alpha))-ln((1+(1+x) 
^(1/2))^(1/2)-_alpha)*ln(((1+(1+x)^(1/2))^(1/2)-1)/(-1+_alpha)))*_alpha,_a 
lpha=RootOf(_Z^2-2))
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.26 \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\frac {1}{2} \, {\left (\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}\right ) - \frac {4}{\sqrt {\sqrt {x + 1} + 1}}\right )} \log \left (x + 1\right ) + \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} - \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} + \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 4 \, \log \left (\sqrt {\sqrt {x + 1} + 1} + 1\right ) + 4 \, \log \left (\sqrt {\sqrt {x + 1} + 1} - 1\right ) \] Input:

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

Output:

1/2*(sqrt(2)*log(-(sqrt(2) - sqrt(sqrt(x + 1) + 1))/(sqrt(2) + sqrt(sqrt(x 
 + 1) + 1))) - 4/sqrt(sqrt(x + 1) + 1))*log(x + 1) + sqrt(2)*(log(sqrt(2) 
+ sqrt(sqrt(x + 1) + 1))*log(-(sqrt(2) + sqrt(sqrt(x + 1) + 1))/(sqrt(2) + 
 1) + 1) + dilog((sqrt(2) + sqrt(sqrt(x + 1) + 1))/(sqrt(2) + 1))) - sqrt( 
2)*(log(-sqrt(2) + sqrt(sqrt(x + 1) + 1))*log(-(sqrt(2) - sqrt(sqrt(x + 1) 
 + 1))/(sqrt(2) + 1) + 1) + dilog((sqrt(2) - sqrt(sqrt(x + 1) + 1))/(sqrt( 
2) + 1))) + sqrt(2)*(log(sqrt(2) + sqrt(sqrt(x + 1) + 1))*log(-(sqrt(2) + 
sqrt(sqrt(x + 1) + 1))/(sqrt(2) - 1) + 1) + dilog((sqrt(2) + sqrt(sqrt(x + 
 1) + 1))/(sqrt(2) - 1))) - sqrt(2)*(log(-sqrt(2) + sqrt(sqrt(x + 1) + 1)) 
*log(-(sqrt(2) - sqrt(sqrt(x + 1) + 1))/(sqrt(2) - 1) + 1) + dilog((sqrt(2 
) - sqrt(sqrt(x + 1) + 1))/(sqrt(2) - 1))) - 4*log(sqrt(sqrt(x + 1) + 1) + 
 1) + 4*log(sqrt(sqrt(x + 1) + 1) - 1)
 

Giac [F]

\[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\int { \frac {\log \left (x + 1\right )}{x \sqrt {\sqrt {x + 1} + 1}} \,d x } \] Input:

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

Output:

integrate(log(x + 1)/(x*sqrt(sqrt(x + 1) + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {\ln \left (x+1\right )}{x\,\sqrt {\sqrt {x+1}+1}} \,d x \] Input:

int(log(x + 1)/(x*((x + 1)^(1/2) + 1)^(1/2)),x)
 

Output:

int(log(x + 1)/(x*((x + 1)^(1/2) + 1)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx=\frac {4 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x +1}+1}-\sqrt {2}\right )-4 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x +1}+1}+\sqrt {2}\right )-2 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \left (\int \frac {\sqrt {\sqrt {x +1}+1}\, \mathrm {log}\left (x +1\right )}{x^{3}+x^{2}}d x \right )-2 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \left (\int \frac {\sqrt {\sqrt {x +1}+1}\, \mathrm {log}\left (x +1\right )}{x^{2}+x}d x \right )-2 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \mathrm {log}\left (x +1\right )}{x^{3}+x^{2}}d x \right )-4 \sqrt {\sqrt {x +1}+1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x +1}+1}-\sqrt {2}\right )+4 \sqrt {\sqrt {x +1}+1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {x +1}+1}+\sqrt {2}\right )+2 \sqrt {\sqrt {x +1}+1}\, \left (\int \frac {\sqrt {\sqrt {x +1}+1}\, \mathrm {log}\left (x +1\right )}{x^{3}+x^{2}}d x \right )+2 \sqrt {\sqrt {x +1}+1}\, \left (\int \frac {\sqrt {\sqrt {x +1}+1}\, \mathrm {log}\left (x +1\right )}{x^{2}+x}d x \right )+2 \sqrt {\sqrt {x +1}+1}\, \left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}\, \mathrm {log}\left (x +1\right )}{x^{3}+x^{2}}d x \right )-4 \sqrt {x +1}\, \mathrm {log}\left (x +1\right )}{\sqrt {\sqrt {x +1}+1}\, \left (\sqrt {x +1}-1\right )} \] Input:

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

Output:

(2*(2*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*sqrt(2)*log(sqrt(sqrt(x + 1) + 1) 
- sqrt(2)) - 2*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*sqrt(2)*log(sqrt(sqrt(x + 
 1) + 1) + sqrt(2)) - sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*int((sqrt(sqrt(x + 
 1) + 1)*log(x + 1))/(x**3 + x**2),x) - sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)* 
int((sqrt(sqrt(x + 1) + 1)*log(x + 1))/(x**2 + x),x) - sqrt(x + 1)*sqrt(sq 
rt(x + 1) + 1)*int((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1)*log(x + 1))/(x**3 + 
x**2),x) - 2*sqrt(sqrt(x + 1) + 1)*sqrt(2)*log(sqrt(sqrt(x + 1) + 1) - sqr 
t(2)) + 2*sqrt(sqrt(x + 1) + 1)*sqrt(2)*log(sqrt(sqrt(x + 1) + 1) + sqrt(2 
)) + sqrt(sqrt(x + 1) + 1)*int((sqrt(sqrt(x + 1) + 1)*log(x + 1))/(x**3 + 
x**2),x) + sqrt(sqrt(x + 1) + 1)*int((sqrt(sqrt(x + 1) + 1)*log(x + 1))/(x 
**2 + x),x) + sqrt(sqrt(x + 1) + 1)*int((sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) 
*log(x + 1))/(x**3 + x**2),x) - 2*sqrt(x + 1)*log(x + 1)))/(sqrt(sqrt(x + 
1) + 1)*(sqrt(x + 1) - 1))