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

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

[Out]

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

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Rubi [F]  time = 0.0486268, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log (1+x)}{x \sqrt{1+\sqrt{1+x}}} \, dx \]

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 = 0.919844, size = 430, normalized size = 1.48 \[ -\sqrt{2} \left (2 \text{PolyLog}\left (2,\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right )-\text{PolyLog}\left (2,-\left (\sqrt{2}-2\right ) \left (\frac{1}{\sqrt{\sqrt{x+1}+1}}+1\right )\right )+\text{PolyLog}\left (2,\left (1+\sqrt{2}\right ) \left (\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}-1\right )\right )+\log \left (1+\sqrt{2}\right ) \log \left (\frac{1}{\sqrt{\sqrt{x+1}+1}}+1\right )+\log \left (-\left (2+\sqrt{2}\right ) \left (\frac{1}{\sqrt{\sqrt{x+1}+1}}-1\right )\right ) \log \left (1-\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right )\right )+\sqrt{2} \left (2 \text{PolyLog}\left (2,-\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right )-\text{PolyLog}\left (2,\left (\sqrt{2}-2\right ) \left (\frac{1}{\sqrt{\sqrt{x+1}+1}}-1\right )\right )+\text{PolyLog}\left (2,-\left (1+\sqrt{2}\right ) \left (\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}+1\right )\right )+\log \left (1+\sqrt{2}\right ) \log \left (1-\frac{1}{\sqrt{\sqrt{x+1}+1}}\right )+\log \left (\left (2+\sqrt{2}\right ) \left (\frac{1}{\sqrt{\sqrt{x+1}+1}}+1\right )\right ) \log \left (\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}+1\right )\right )-\frac{2 \log (x+1)}{\sqrt{\sqrt{x+1}+1}}+\frac{\log (x+1) \log \left (1-\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right )}{\sqrt{2}}-\frac{\log (x+1) \log \left (\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}+1\right )}{\sqrt{2}}-8 \tanh ^{-1}\left (\frac{1}{\sqrt{\sqrt{x+1}+1}}\right ) \]

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]]))])

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Maple [C]  time = 0.096, size = 171, normalized size = 0.6 \begin{align*} -2\,{\frac{\ln \left ( 1+x \right ) }{\sqrt{1+\sqrt{1+x}}}}-8\,{\it Artanh} \left ( \sqrt{1+\sqrt{1+x}} \right ) +4\,\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{2}-2 \right ) }1/8\, \left ( \ln \left ( \sqrt{1+\sqrt{1+x}}-{\it \_alpha} \right ) \ln \left ( 1+x \right ) -2\,{\it dilog} \left ({\frac{1+\sqrt{1+\sqrt{1+x}}}{1+{\it \_alpha}}} \right ) -2\,\ln \left ( \sqrt{1+\sqrt{1+x}}-{\it \_alpha} \right ) \ln \left ({\frac{1+\sqrt{1+\sqrt{1+x}}}{1+{\it \_alpha}}} \right ) -2\,{\it dilog} \left ({\frac{-1+\sqrt{1+\sqrt{1+x}}}{-1+{\it \_alpha}}} \right ) -2\,\ln \left ( \sqrt{1+\sqrt{1+x}}-{\it \_alpha} \right ) \ln \left ({\frac{-1+\sqrt{1+\sqrt{1+x}}}{-1+{\it \_alpha}}} \right ) \right ){\it \_alpha} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(1+x)/(1+(1+x)^(1/2))^(1/2)-8*arctanh((1+(1+x)^(1/2))^(1/2))+4*Sum(1/8*(ln((1+(1+x)^(1/2))^(1/2)-_alpha)*
ln(1+x)-2*dilog((1+(1+(1+x)^(1/2))^(1/2))/(1+_alpha))-2*ln((1+(1+x)^(1/2))^(1/2)-_alpha)*ln((1+(1+(1+x)^(1/2))
^(1/2))/(1+_alpha))-2*dilog((-1+(1+(1+x)^(1/2))^(1/2))/(-1+_alpha))-2*ln((1+(1+x)^(1/2))^(1/2)-_alpha)*ln((-1+
(1+(1+x)^(1/2))^(1/2))/(-1+_alpha)))*_alpha,_alpha=RootOf(_Z^2-2))

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Maxima [A]  time = 1.6204, size = 494, normalized size = 1.7 \begin{align*} \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{align*}

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)

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

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: UnboundLocalError

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

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x + 1\right )}{x \sqrt{\sqrt{x + 1} + 1}}\,{d x} \end{align*}

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="giac")

[Out]

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