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

Optimal. Leaf size=291 \[ -8 \tanh ^{-1}\left (\sqrt {1+\sqrt {1+x}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+x}}}{\sqrt {2}}\right ) \log (1+x)+2 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+\sqrt {2} \text {Li}_2\left (-\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )-\sqrt {2} \text {Li}_2\left (\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right )-\sqrt {2} \text {Li}_2\left (-\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )+\sqrt {2} \text {Li}_2\left (\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right ) \]

[Out]

-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)

________________________________________________________________________________________

Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx &=\int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.74, size = 430, normalized size = 1.48 \begin {gather*} -8 \tanh ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}+\frac {\log (1+x) \log \left (1-\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )}{\sqrt {2}}-\frac {\log (1+x) \log \left (1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )}{\sqrt {2}}-\sqrt {2} \left (\log \left (1+\sqrt {2}\right ) \log \left (1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )+\log \left (-\left (\left (2+\sqrt {2}\right ) \left (-1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right ) \log \left (1-\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )+2 \text {Li}_2\left (\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )-\text {Li}_2\left (-\left (\left (-2+\sqrt {2}\right ) \left (1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right )+\text {Li}_2\left (\left (1+\sqrt {2}\right ) \left (-1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right )+\sqrt {2} \left (\log \left (1+\sqrt {2}\right ) \log \left (1-\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )+\log \left (\left (2+\sqrt {2}\right ) \left (1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right ) \log \left (1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )-\text {Li}_2\left (\left (-2+\sqrt {2}\right ) \left (-1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right )+\text {Li}_2\left (-\left (\left (1+\sqrt {2}\right ) \left (1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 172, normalized size = 0.59

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))-l
n((1+(1+x)^(1/2))^(1/2)-_alpha)*ln(((1+(1+x)^(1/2))^(1/2)-1)/(-1+_alpha)))*_alpha,_alpha=RootOf(_Z^2-2))-2*ln(
1+x)/(1+(1+x)^(1/2))^(1/2)-8*arctanh((1+(1+x)^(1/2))^(1/2))

________________________________________________________________________________________

Maxima [A]
time = 0.38, size = 366, normalized size = 1.26 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))/(s
qrt(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))/(sq
rt(2) - 1))) - 4*log(sqrt(sqrt(x + 1) + 1) + 1) + 4*log(sqrt(sqrt(x + 1) + 1) - 1)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (x + 1 \right )}}{x \sqrt {\sqrt {x + 1} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F] N/A
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]

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

[Out]

Could not integrate

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (x+1\right )}{x\,\sqrt {\sqrt {x+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________