3.1.8 \(\int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx\) [8]

Optimal. Leaf size=308 \[ -16 \sqrt {1+\sqrt {1+x}}+16 \tanh ^{-1}\left (\sqrt {1+\sqrt {1+x}}\right )+4 \sqrt {1+\sqrt {1+x}} \log (1+x)-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+x}}}{\sqrt {2}}\right ) \log (1+x)+4 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-4 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+2 \sqrt {2} \text {Li}_2\left (-\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )-2 \sqrt {2} \text {Li}_2\left (\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right )-2 \sqrt {2} \text {Li}_2\left (-\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )+2 \sqrt {2} \text {Li}_2\left (\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right ) \]

[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 = 0.03, 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 {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx \end {gather*}

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] Leaf count is larger than twice the leaf count of optimal. \(654\) vs. \(2(308)=616\).
time = 0.44, size = 654, normalized size = 2.12 \begin {gather*} -16 \sqrt {1+\sqrt {1+x}}+4 \sqrt {1+\sqrt {1+x}} \log (1+x)+\sqrt {2} \log (1+x) \log \left (\sqrt {2}-\sqrt {1+\sqrt {1+x}}\right )-8 \log \left (-1+\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \log \left (\sqrt {2}-\sqrt {1+\sqrt {1+x}}\right ) \log \left (-1+\sqrt {1+\sqrt {1+x}}\right )+8 \log \left (1+\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \log \left (\sqrt {2}-\sqrt {1+\sqrt {1+x}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )-\sqrt {2} \log (1+x) \log \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )+2 \sqrt {2} \log \left (-1+\sqrt {1+\sqrt {1+x}}\right ) \log \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )+2 \sqrt {2} \log \left (1+\sqrt {1+\sqrt {1+x}}\right ) \log \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \log \left (-1+\sqrt {1+\sqrt {1+x}}\right ) \log \left (\left (-1+\sqrt {2}\right ) \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )\right )-2 \sqrt {2} \log \left (1+\sqrt {1+\sqrt {1+x}}\right ) \log \left (2+\sqrt {2}+\sqrt {1+\sqrt {1+x}}+\sqrt {2} \sqrt {1+\sqrt {1+x}}\right )+2 \sqrt {2} \log \left (-1+\sqrt {1+\sqrt {1+x}}\right ) \log \left (1-\left (1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )+2 \sqrt {2} \log \left (1+\sqrt {1+\sqrt {1+x}}\right ) \log \left (1-\left (-1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )-2 \sqrt {2} \text {Li}_2\left (-\left (\left (-1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )+2 \sqrt {2} \text {Li}_2\left (\left (1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )+2 \sqrt {2} \text {Li}_2\left (\left (-1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )-2 \sqrt {2} \text {Li}_2\left (-\left (\left (1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )\right ) \end {gather*}

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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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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.01, size = 199, normalized size = 0.65

method	result	size
derivativedivides	\(4 \ln \left (1+x \right ) \sqrt {1+\sqrt {1+x}}-16 \sqrt {1+\sqrt {1+x}}-8 \ln \left (\sqrt {1+\sqrt {1+x}}-1\right )+8 \ln \left (1+\sqrt {1+\sqrt {1+x}}\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\dilog \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{4}\right )\)	\(199\)
default	\(4 \ln \left (1+x \right ) \sqrt {1+\sqrt {1+x}}-16 \sqrt {1+\sqrt {1+x}}-8 \ln \left (\sqrt {1+\sqrt {1+x}}-1\right )+8 \ln \left (1+\sqrt {1+\sqrt {1+x}}\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\dilog \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{4}\right )\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*ln(1+x)*(1+(1+x)^(1/2))^(1/2)-16*(1+(1+x)^(1/2))^(1/2)-8*ln((1+(1+x)^(1/2))^(1/2)-1)+8*ln(1+(1+(1+x)^(1/2))^
(1/2))+8*Sum(1/4*(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+_al
pha))-ln((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)
)

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Maxima [A]
time = 0.38, size = 378, normalized size = 1.23 \begin {gather*} {\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 {gather*}

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

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] 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)*(1+(1+x)^(1/2))^(1/2)/x,x)

[Out]

Could not integrate

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

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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