3.30 \(\int \frac{\log \left (x+\sqrt{1+x}\right )}{1+x^2} \, dx\)

Optimal. Leaf size=981 \[ \text{result too large to display} \]

[Out]

(I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 + I
] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + (I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log
[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] -
(I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sq
rt[1 - I] - Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] +
2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] - Sqrt[5])] + (I/2)*Log[Sqrt[1 + I] + Sqrt[1 +
x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] - Sqrt[5])] + (I/2)*Lo
g[Sqrt[1 + I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I
] - Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1
+ x])/(1 - 2*Sqrt[1 - I] + Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log
[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*Log[Sqrt[1
+ I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqrt
[5])] + (I/2)*Log[Sqrt[1 + I] - Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(
1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/2)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]))
/(1 + 2*Sqrt[1 - I] - Sqrt[5])] - (I/2)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]
))/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*PolyLog[2, (2*(Sqrt[1 + I] - Sqrt[1 +
x]))/(1 + 2*Sqrt[1 + I] - Sqrt[5])] + (I/2)*PolyLog[2, (2*(Sqrt[1 + I] - Sqrt[1
+ x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/2)*PolyLog[2, (-2*(Sqrt[1 - I] + Sqrt
[1 + x]))/(1 - 2*Sqrt[1 - I] - Sqrt[5])] - (I/2)*PolyLog[2, (-2*(Sqrt[1 - I] + S
qrt[1 + x]))/(1 - 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*PolyLog[2, (-2*(Sqrt[1 + I]
+ Sqrt[1 + x]))/(1 - 2*Sqrt[1 + I] - Sqrt[5])] + (I/2)*PolyLog[2, (-2*(Sqrt[1 +
I] + Sqrt[1 + x]))/(1 - 2*Sqrt[1 + I] + Sqrt[5])]

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Rubi [A]  time = 2.28, antiderivative size = 981, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{x+1}\right ) \log \left (x+\sqrt{x+1}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{x+1}\right ) \log \left (x+\sqrt{x+1}\right )+\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1-i}\right ) \log \left (x+\sqrt{x+1}\right )-\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1+i}\right ) \log \left (x+\sqrt{x+1}\right )-\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1-i}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1-i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1+i}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1+i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1-i}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1-i}+\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1+i}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1+i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{1-2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{1-2 \sqrt{1+i}+\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Log[x + Sqrt[1 + x]]/(1 + x^2),x]

[Out]

(I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 + I
] - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + (I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log
[x + Sqrt[1 + x]] - (I/2)*Log[Sqrt[1 + I] + Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] -
(I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sq
rt[1 - I] - Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] +
2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] - Sqrt[5])] + (I/2)*Log[Sqrt[1 + I] + Sqrt[1 +
x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] - Sqrt[5])] + (I/2)*Lo
g[Sqrt[1 + I] - Sqrt[1 + x]]*Log[(1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I
] - Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1
+ x])/(1 - 2*Sqrt[1 - I] + Sqrt[5])] - (I/2)*Log[Sqrt[1 - I] - Sqrt[1 + x]]*Log
[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*Log[Sqrt[1
+ I] + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqrt
[5])] + (I/2)*Log[Sqrt[1 + I] - Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(
1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/2)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]))
/(1 + 2*Sqrt[1 - I] - Sqrt[5])] - (I/2)*PolyLog[2, (2*(Sqrt[1 - I] - Sqrt[1 + x]
))/(1 + 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*PolyLog[2, (2*(Sqrt[1 + I] - Sqrt[1 +
x]))/(1 + 2*Sqrt[1 + I] - Sqrt[5])] + (I/2)*PolyLog[2, (2*(Sqrt[1 + I] - Sqrt[1
+ x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])] - (I/2)*PolyLog[2, (-2*(Sqrt[1 - I] + Sqrt
[1 + x]))/(1 - 2*Sqrt[1 - I] - Sqrt[5])] - (I/2)*PolyLog[2, (-2*(Sqrt[1 - I] + S
qrt[1 + x]))/(1 - 2*Sqrt[1 - I] + Sqrt[5])] + (I/2)*PolyLog[2, (-2*(Sqrt[1 + I]
+ Sqrt[1 + x]))/(1 - 2*Sqrt[1 + I] - Sqrt[5])] + (I/2)*PolyLog[2, (-2*(Sqrt[1 +
I] + Sqrt[1 + x]))/(1 - 2*Sqrt[1 + I] + Sqrt[5])]

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x+(1+x)**(1/2))/(x**2+1),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 1.05394, size = 868, normalized size = 0.88 \[ \frac{1}{2} i \left (2 i \tan ^{-1}(x) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+\log \left (\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-\log \left (\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{-1+2 \sqrt{1-i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{-1+2 \sqrt{1+i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+2 i \tan ^{-1}(x) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )+\log \left (\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-\log \left (\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )+\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{-1+2 \sqrt{1-i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{-1+2 \sqrt{1+i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-2 i \tan ^{-1}(x) \log \left (x+\sqrt{x+1}\right )+\text{PolyLog}\left (2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+2 \sqrt{1-i}+\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+2 \sqrt{1+i}+\sqrt{5}}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1-i}-\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1+i}-\sqrt{5}}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1-i}+\sqrt{5}}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1-i}+\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1+i}+\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1+i}+\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[x + Sqrt[1 + x]]/(1 + x^2),x]

[Out]

(I/2)*((2*I)*ArcTan[x]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I]
- Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] - Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]
] - Log[(2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] - Sqrt[5])]*Log[1/2 -
Sqrt[5]/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 -
I] + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] + Sqrt[
1 + x]))/(-1 + 2*Sqrt[1 + I] + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + (2
*I)*ArcTan[x]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] - Sqrt[1
+ x]))/(1 + 2*Sqrt[1 - I] + Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - Log[(
2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])]*Log[(1 + Sqrt[5])/
2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 - I] - Sqr
t[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] + Sqrt[1 + x]))/
(-1 + 2*Sqrt[1 + I] - Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - (2*I)*ArcTa
n[x]*Log[x + Sqrt[1 + x]] + PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + 2*Sq
rt[1 - I] + Sqrt[5])] - PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + 2*Sqrt[1
+ I] + Sqrt[5])] + PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I]
- Sqrt[5])] - PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] - Sqrt
[5])] + PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 - I] + Sqrt[5])]
+ PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5])] - Poly
Log[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqrt[5])] - PolyLog[2,
(1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] + Sqrt[5])])

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Maple [C]  time = 0.114, size = 730, normalized size = 0.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x+(1+x)^(1/2))/(x^2+1),x)

[Out]

1/2*I*ln(x+(1+x)^(1/2))*ln((1+x)^(1/2)-(1-I)^(1/2))+1/2*I*ln(x+(1+x)^(1/2))*ln((
1-I)^(1/2)+(1+x)^(1/2))-1/2*I*ln((1-I)^(1/2)+(1+x)^(1/2))*ln((1-5^(1/2)+2*(1+x)^
(1/2))/(1-2*(1-I)^(1/2)-5^(1/2)))+1/2*I*dilog((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+
I)^(1/2)+5^(1/2)))+1/2*I*dilog((1+5^(1/2)+2*(1+x)^(1/2))/(1-2*(1+I)^(1/2)+5^(1/2
)))+1/2*I*ln((1+I)^(1/2)+(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1-2*(1+I)^(1
/2)+5^(1/2)))-1/2*I*dilog((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)-5^(1/2)))-1
/2*I*ln(x+(1+x)^(1/2))*ln((1+x)^(1/2)-(1+I)^(1/2))+1/2*I*ln((1+I)^(1/2)+(1+x)^(1
/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))+1/2*I*ln((1+x)^(1/2
)-(1+I)^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)-5^(1/2)))+1/2*I*dil
og((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)-5^(1/2)))-1/2*I*ln((1+x)^(1/2)-(1-
I)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)+5^(1/2)))-1/2*I*ln((1-I)
^(1/2)+(1+x)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1-2*(1-I)^(1/2)+5^(1/2)))-1/2*
I*dilog((1-5^(1/2)+2*(1+x)^(1/2))/(1-2*(1-I)^(1/2)-5^(1/2)))+1/2*I*ln((1+x)^(1/2
)-(1+I)^(1/2))*ln((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1+I)^(1/2)+5^(1/2)))-1/2*I*dil
og((1+5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)+5^(1/2)))-1/2*I*ln(x+(1+x)^(1/2))*
ln((1+I)^(1/2)+(1+x)^(1/2))-1/2*I*dilog((1+5^(1/2)+2*(1+x)^(1/2))/(1-2*(1-I)^(1/
2)+5^(1/2)))+1/2*I*dilog((1-5^(1/2)+2*(1+x)^(1/2))/(1-2*(1+I)^(1/2)-5^(1/2)))-1/
2*I*ln((1+x)^(1/2)-(1-I)^(1/2))*ln((1-5^(1/2)+2*(1+x)^(1/2))/(1+2*(1-I)^(1/2)-5^
(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x + sqrt(x + 1))/(x^2 + 1),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\log \left (x + \sqrt{x + 1}\right )}{x^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x + sqrt(x + 1))/(x^2 + 1),x, algorithm="fricas")

[Out]

integral(log(x + sqrt(x + 1))/(x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log{\left (x + \sqrt{x + 1} \right )}}{x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x+(1+x)**(1/2))/(x**2+1),x)

[Out]

Integral(log(x + sqrt(x + 1))/(x**2 + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\log \left (x + \sqrt{x + 1}\right )}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x + sqrt(x + 1))/(x^2 + 1),x, algorithm="giac")

[Out]

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