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

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

[Out]

16*arctanh((1+(1+x)^(1/2))^(1/2))-2*arctanh(1/2*(1+(1+x)^(1/2))^(1/2)*2^(1/2))*ln(1+x)*2^(1/2)+4*arctanh(1/2*2
^(1/2))*ln(1-(1+(1+x)^(1/2))^(1/2))*2^(1/2)-4*arctanh(1/2*2^(1/2))*ln(1+(1+(1+x)^(1/2))^(1/2))*2^(1/2)+2*polyl
og(2,-2^(1/2)*(1-(1+(1+x)^(1/2))^(1/2))/(2-2^(1/2)))*2^(1/2)-2*polylog(2,2^(1/2)*(1-(1+(1+x)^(1/2))^(1/2))/(2+
2^(1/2)))*2^(1/2)-2*polylog(2,-2^(1/2)*(1+(1+(1+x)^(1/2))^(1/2))/(2-2^(1/2)))*2^(1/2)+2*polylog(2,2^(1/2)*(1+(
1+(1+x)^(1/2))^(1/2))/(2+2^(1/2)))*2^(1/2)-16*(1+(1+x)^(1/2))^(1/2)+4*ln(1+x)*(1+(1+x)^(1/2))^(1/2)

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Rubi [F]  time = 0.04, 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 = {} \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx \]

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*}

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Mathematica [B]  time = 0.60, size = 654, normalized size = 2.12 \[ -2 \sqrt {2} \operatorname {PolyLog}\left (2,-\left (\left (\sqrt {2}-1\right ) \left (\sqrt {\sqrt {x+1}+1}-1\right )\right )\right )+2 \sqrt {2} \operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) \left (\sqrt {\sqrt {x+1}+1}-1\right )\right )+2 \sqrt {2} \operatorname {PolyLog}\left (2,\left (\sqrt {2}-1\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )\right )-2 \sqrt {2} \operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )\right )\right )-16 \sqrt {\sqrt {x+1}+1}+\sqrt {2} \log \left (\sqrt {2}-\sqrt {\sqrt {x+1}+1}\right ) \log (x+1)-\sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}+\sqrt {2}\right ) \log (x+1)+4 \sqrt {\sqrt {x+1}+1} \log (x+1)-2 \sqrt {2} \log \left (\sqrt {2}-\sqrt {\sqrt {x+1}+1}\right ) \log \left (\sqrt {\sqrt {x+1}+1}-1\right )-8 \log \left (\sqrt {\sqrt {x+1}+1}-1\right )-2 \sqrt {2} \log \left (\sqrt {2}-\sqrt {\sqrt {x+1}+1}\right ) \log \left (\sqrt {\sqrt {x+1}+1}+1\right )+8 \log \left (\sqrt {\sqrt {x+1}+1}+1\right )+2 \sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}-1\right ) \log \left (\sqrt {\sqrt {x+1}+1}+\sqrt {2}\right )+2 \sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}+1\right ) \log \left (\sqrt {\sqrt {x+1}+1}+\sqrt {2}\right )-2 \sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}-1\right ) \log \left (\left (\sqrt {2}-1\right ) \left (\sqrt {\sqrt {x+1}+1}+\sqrt {2}\right )\right )-2 \sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}+1\right ) \log \left (\sqrt {2} \sqrt {\sqrt {x+1}+1}+\sqrt {\sqrt {x+1}+1}+\sqrt {2}+2\right )+2 \sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}-1\right ) \log \left (1-\left (1+\sqrt {2}\right ) \left (\sqrt {\sqrt {x+1}+1}-1\right )\right )+2 \sqrt {2} \log \left (\sqrt {\sqrt {x+1}+1}+1\right ) \log \left (1-\left (\sqrt {2}-1\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )\right ) \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\sqrt {x + 1} + 1} \log \left (x + 1\right )}{x}\,{d x} \]

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)

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maple [C]  time = 0.02, size = 198, normalized size = 0.64 \[ 4 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (-2 \ln \left (\frac {1+\sqrt {1+\sqrt {x +1}}}{\underline {\hspace {1.25 ex}}\alpha +1}\right ) \ln \left (-\underline {\hspace {1.25 ex}}\alpha +\sqrt {1+\sqrt {x +1}}\right )-2 \ln \left (\frac {\sqrt {1+\sqrt {x +1}}-1}{\underline {\hspace {1.25 ex}}\alpha -1}\right ) \ln \left (-\underline {\hspace {1.25 ex}}\alpha +\sqrt {1+\sqrt {x +1}}\right )+\ln \left (x +1\right ) \ln \left (-\underline {\hspace {1.25 ex}}\alpha +\sqrt {1+\sqrt {x +1}}\right )-2 \dilog \left (\frac {1+\sqrt {1+\sqrt {x +1}}}{\underline {\hspace {1.25 ex}}\alpha +1}\right )-2 \dilog \left (\frac {\sqrt {1+\sqrt {x +1}}-1}{\underline {\hspace {1.25 ex}}\alpha -1}\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{4}\right )+4 \sqrt {1+\sqrt {x +1}}\, \ln \left (x +1\right )+8 \ln \left (1+\sqrt {1+\sqrt {x +1}}\right )-8 \ln \left (\sqrt {1+\sqrt {x +1}}-1\right )-16 \sqrt {1+\sqrt {x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*ln(x+1)*(1+(x+1)^(1/2))^(1/2)-16*(1+(x+1)^(1/2))^(1/2)-8*ln((1+(x+1)^(1/2))^(1/2)-1)+8*ln(1+(1+(x+1)^(1/2))^
(1/2))+4*Sum(1/4*(ln(x+1)*ln(-_alpha+(1+(x+1)^(1/2))^(1/2))-2*dilog((1+(1+(x+1)^(1/2))^(1/2))/(_alpha+1))-2*ln
((1+(1+(x+1)^(1/2))^(1/2))/(_alpha+1))*ln(-_alpha+(1+(x+1)^(1/2))^(1/2))-2*dilog(((1+(x+1)^(1/2))^(1/2)-1)/(_a
lpha-1))-2*ln(((1+(x+1)^(1/2))^(1/2)-1)/(_alpha-1))*ln(-_alpha+(1+(x+1)^(1/2))^(1/2)))*_alpha,_alpha=RootOf(_Z
^2-2))

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maxima [A]  time = 1.28, size = 378, normalized size = 1.23 \[ {\left (\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}\right ) + 4 \, \sqrt {\sqrt {x + 1} + 1}\right )} \log \left (x + 1\right ) + 2 \, \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} - 2 \, \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} + 2 \, \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 2 \, \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 16 \, \sqrt {\sqrt {x + 1} + 1} + 8 \, \log \left (\sqrt {\sqrt {x + 1} + 1} + 1\right ) - 8 \, \log \left (\sqrt {\sqrt {x + 1} + 1} - 1\right ) \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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