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

Optimal. Leaf size=308 \[ 2 \sqrt{2} \text{PolyLog}\left (2,-\frac{\sqrt{2} \left (1-\sqrt{\sqrt{x+1}+1}\right )}{2-\sqrt{2}}\right )-2 \sqrt{2} \text{PolyLog}\left (2,\frac{\sqrt{2} \left (1-\sqrt{\sqrt{x+1}+1}\right )}{2+\sqrt{2}}\right )-2 \sqrt{2} \text{PolyLog}\left (2,-\frac{\sqrt{2} \left (\sqrt{\sqrt{x+1}+1}+1\right )}{2-\sqrt{2}}\right )+2 \sqrt{2} \text{PolyLog}\left (2,\frac{\sqrt{2} \left (\sqrt{\sqrt{x+1}+1}+1\right )}{2+\sqrt{2}}\right )-16 \sqrt{\sqrt{x+1}+1}+4 \sqrt{\sqrt{x+1}+1} \log (x+1)+16 \tanh ^{-1}\left (\sqrt{\sqrt{x+1}+1}\right )-2 \sqrt{2} \log (x+1) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{x+1}+1}}{\sqrt{2}}\right )+4 \sqrt{2} \tanh ^{-1}\left (\frac{1}{\sqrt{2}}\right ) \log \left (1-\sqrt{\sqrt{x+1}+1}\right )-4 \sqrt{2} \tanh ^{-1}\left (\frac{1}{\sqrt{2}}\right ) \log \left (\sqrt{\sqrt{x+1}+1}+1\right ) \]

[Out]

-16*Sqrt[1 + Sqrt[1 + x]] + 16*ArcTanh[Sqrt[1 + Sqrt[1 + x]]] + 4*Sqrt[1 + Sqrt[1 + x]]*Log[1 + x] - 2*Sqrt[2]
*ArcTanh[Sqrt[1 + Sqrt[1 + x]]/Sqrt[2]]*Log[1 + x] + 4*Sqrt[2]*ArcTanh[1/Sqrt[2]]*Log[1 - Sqrt[1 + Sqrt[1 + x]
]] - 4*Sqrt[2]*ArcTanh[1/Sqrt[2]]*Log[1 + Sqrt[1 + Sqrt[1 + x]]] + 2*Sqrt[2]*PolyLog[2, -((Sqrt[2]*(1 - Sqrt[1
+ Sqrt[1 + x]]))/(2 - Sqrt[2]))] - 2*Sqrt[2]*PolyLog[2, (Sqrt[2]*(1 - Sqrt[1 + Sqrt[1 + x]]))/(2 + Sqrt[2])]
- 2*Sqrt[2]*PolyLog[2, -((Sqrt[2]*(1 + Sqrt[1 + Sqrt[1 + x]]))/(2 - Sqrt[2]))] + 2*Sqrt[2]*PolyLog[2, (Sqrt[2]
*(1 + Sqrt[1 + Sqrt[1 + x]]))/(2 + Sqrt[2])]

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Rubi [F]  time = 0.0432186, 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 = {} \[ \int \frac{\sqrt{1+\sqrt{1+x}} \log (1+x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [B]  time = 0.510365, size = 654, normalized size = 2.12 \[ -2 \sqrt{2} \text{PolyLog}\left (2,-\left (\sqrt{2}-1\right ) \left (\sqrt{\sqrt{x+1}+1}-1\right )\right )+2 \sqrt{2} \text{PolyLog}\left (2,\left (1+\sqrt{2}\right ) \left (\sqrt{\sqrt{x+1}+1}-1\right )\right )+2 \sqrt{2} \text{PolyLog}\left (2,\left (\sqrt{2}-1\right ) \left (\sqrt{\sqrt{x+1}+1}+1\right )\right )-2 \sqrt{2} \text{PolyLog}\left (2,-\left (1+\sqrt{2}\right ) \left (\sqrt{\sqrt{x+1}+1}+1\right )\right )-16 \sqrt{\sqrt{x+1}+1}+\sqrt{2} \log \left (\sqrt{2}-\sqrt{\sqrt{x+1}+1}\right ) \log (x+1)-\sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right ) \log (x+1)+4 \sqrt{\sqrt{x+1}+1} \log (x+1)-2 \sqrt{2} \log \left (\sqrt{2}-\sqrt{\sqrt{x+1}+1}\right ) \log \left (\sqrt{\sqrt{x+1}+1}-1\right )-8 \log \left (\sqrt{\sqrt{x+1}+1}-1\right )-2 \sqrt{2} \log \left (\sqrt{2}-\sqrt{\sqrt{x+1}+1}\right ) \log \left (\sqrt{\sqrt{x+1}+1}+1\right )+8 \log \left (\sqrt{\sqrt{x+1}+1}+1\right )+2 \sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}-1\right ) \log \left (\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right )+2 \sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}+1\right ) \log \left (\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right )-2 \sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}-1\right ) \log \left (\left (\sqrt{2}-1\right ) \left (\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right )\right )-2 \sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}+1\right ) \log \left (\sqrt{2} \sqrt{\sqrt{x+1}+1}+\sqrt{\sqrt{x+1}+1}+\sqrt{2}+2\right )+2 \sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}-1\right ) \log \left (1-\left (1+\sqrt{2}\right ) \left (\sqrt{\sqrt{x+1}+1}-1\right )\right )+2 \sqrt{2} \log \left (\sqrt{\sqrt{x+1}+1}+1\right ) \log \left (1-\left (\sqrt{2}-1\right ) \left (\sqrt{\sqrt{x+1}+1}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-16*Sqrt[1 + Sqrt[1 + x]] + 4*Sqrt[1 + Sqrt[1 + x]]*Log[1 + x] + Sqrt[2]*Log[1 + x]*Log[Sqrt[2] - Sqrt[1 + Sqr
t[1 + x]]] - 8*Log[-1 + Sqrt[1 + Sqrt[1 + x]]] - 2*Sqrt[2]*Log[Sqrt[2] - Sqrt[1 + Sqrt[1 + x]]]*Log[-1 + Sqrt[
1 + Sqrt[1 + x]]] + 8*Log[1 + Sqrt[1 + Sqrt[1 + x]]] - 2*Sqrt[2]*Log[Sqrt[2] - Sqrt[1 + Sqrt[1 + x]]]*Log[1 +
Sqrt[1 + Sqrt[1 + x]]] - Sqrt[2]*Log[1 + x]*Log[Sqrt[2] + Sqrt[1 + Sqrt[1 + x]]] + 2*Sqrt[2]*Log[-1 + Sqrt[1 +
Sqrt[1 + x]]]*Log[Sqrt[2] + Sqrt[1 + Sqrt[1 + x]]] + 2*Sqrt[2]*Log[1 + Sqrt[1 + Sqrt[1 + x]]]*Log[Sqrt[2] + S
qrt[1 + Sqrt[1 + x]]] - 2*Sqrt[2]*Log[-1 + Sqrt[1 + Sqrt[1 + x]]]*Log[(-1 + Sqrt[2])*(Sqrt[2] + Sqrt[1 + Sqrt[
1 + x]])] - 2*Sqrt[2]*Log[1 + Sqrt[1 + Sqrt[1 + x]]]*Log[2 + Sqrt[2] + Sqrt[1 + Sqrt[1 + x]] + Sqrt[2]*Sqrt[1
+ Sqrt[1 + x]]] + 2*Sqrt[2]*Log[-1 + Sqrt[1 + Sqrt[1 + x]]]*Log[1 - (1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x]])
] + 2*Sqrt[2]*Log[1 + Sqrt[1 + Sqrt[1 + x]]]*Log[1 - (-1 + Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]])] - 2*Sqrt[2]*P
olyLog[2, -((-1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x]]))] + 2*Sqrt[2]*PolyLog[2, (1 + Sqrt[2])*(-1 + Sqrt[1 +
Sqrt[1 + x]])] + 2*Sqrt[2]*PolyLog[2, (-1 + Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]])] - 2*Sqrt[2]*PolyLog[2, -((1
+ Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]]))]

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Maple [C]  time = 0.008, size = 198, normalized size = 0.6 \begin{align*} 4\,\ln \left ( 1+x \right ) \sqrt{1+\sqrt{1+x}}-16\,\sqrt{1+\sqrt{1+x}}-8\,\ln \left ( -1+\sqrt{1+\sqrt{1+x}} \right ) +8\,\ln \left ( 1+\sqrt{1+\sqrt{1+x}} \right ) +4\,\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{2}-2 \right ) }1/4\, \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)*(1+(1+x)^(1/2))^(1/2)/x,x)

[Out]

4*ln(1+x)*(1+(1+x)^(1/2))^(1/2)-16*(1+(1+x)^(1/2))^(1/2)-8*ln(-1+(1+(1+x)^(1/2))^(1/2))+8*ln(1+(1+(1+x)^(1/2))
^(1/2))+4*Sum(1/4*(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 = 1.61415, size = 510, normalized size = 1.66 \begin{align*}{\left (\sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{\sqrt{x + 1} + 1}}{\sqrt{2} + \sqrt{\sqrt{x + 1} + 1}}\right ) + 4 \, \sqrt{\sqrt{x + 1} + 1}\right )} \log \left (x + 1\right ) + 2 \, \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 )} - 2 \, \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 )} + 2 \, \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 )} - 2 \, \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 )} - 16 \, \sqrt{\sqrt{x + 1} + 1} + 8 \, \log \left (\sqrt{\sqrt{x + 1} + 1} + 1\right ) - 8 \, \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)*(1+(1+x)^(1/2))^(1/2)/x,x, algorithm="maxima")

[Out]

(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) + 2*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))) - 2*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))) + 2*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))) - 2*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))) - 16*sqrt(sqrt(x + 1) + 1) + 8*log(sqrt(sqrt(x + 1) + 1) + 1) - 8*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)*(1+(1+x)^(1/2))^(1/2)/x,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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