3.15.56 \(\int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx\) [1456]

Optimal. Leaf size=102 \[ x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\& ,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.01, 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 {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x + Log[-1 - 2*x^3 + x^4]/4 + Defer[Int][(-1 - 2*x^3 + x^4)^(-1), x] + (3*Defer[Int][x^2/(-1 - 2*x^3 + x^4), x
])/2 + Defer[Int][(x^2*Sqrt[1 + x^2])/(-1 - 2*x^3 + x^4), x] + Defer[Int][(x^2*Sqrt[x + Sqrt[1 + x^2]])/(-1 -
2*x^3 + x^4), x] - Defer[Int][(x^3*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^3 + x^4), x] - Defer[Int][(x*Sqrt[1 + x^
2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {x}{x+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (1+\frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4}+\frac {1+x^3}{-1-2 x^3+x^4}+\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}-\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4}\right ) \, dx\\ &=x+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {1+x^3}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \frac {4+6 x^2}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {1}{4} \int \left (\frac {4}{-1-2 x^3+x^4}+\frac {6 x^2}{-1-2 x^3+x^4}\right ) \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ &=x+\frac {1}{4} \log \left (-1-2 x^3+x^4\right )+\frac {3}{2} \int \frac {x^2}{-1-2 x^3+x^4} \, dx+\int \frac {1}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2}}{-1-2 x^3+x^4} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^3+x^4} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 102, normalized size = 1.00 \begin {gather*} x-2 \sqrt {x+\sqrt {1+x^2}}+2 \text {RootSum}\left [-1+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 2*Sqrt[x + Sqrt[1 + x^2]] + 2*RootSum[-1 + 2*#1^3 + #1^4 & , (-Log[Sqrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqr
t[x + Sqrt[1 + x^2]] - #1]*#1^3)/(3*#1^2 + 2*#1^3) & ]

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x}{x +\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.23, size = 7672, normalized size = 75.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="fricas")

[Out]

1/86*(sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(12
9) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) + 86*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((
3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqr
t(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)
^(1/3) + 78) + 86)*log(3/14792*(13*sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) +
117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - 1118*sqrt(1/43)*sqrt(1
05*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129)
- 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) + 240*(2/9)
^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) - 344)*(sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129)
- 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) + 86*sq
rt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(
2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1
/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) + 86)^2 + 39/14792*(sqrt(258)*sqrt((2/9)^(2/3)*(3*(
2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(1
29) - 28989)^(1/3)) - 86*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989
)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*
(2557*sqrt(129) - 28989)^(1/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) + 86)^3 + 1/14792*(39*(s
qrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28
989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - 86*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)
^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129)
- 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3)
+ 78) + 86)^2 - 13416*sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/
3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) + 1153776*sqrt(1/43)*sqrt(105*sqrt(25
8)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(
1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) + 240*(2/9)^(2/3)/(25
57*sqrt(129) - 28989)^(1/3) + 78) + 1316488)*(sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129) - 2898
9)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) + 86*sqrt(1/4
3)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*s
qrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) +
240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) + 86) - 39/43*(sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*
(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989
)^(1/3)) - 86*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 1
17*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(
129) - 28989)^(1/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) + 86)^2 + 468/43*sqrt(258)*sqrt((2/
9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)
/(2557*sqrt(129) - 28989)^(1/3)) - 936*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqr
t(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3))
- (2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) + 10228*sq
rt(x + sqrt(x^2 + 1)) + 4) - 1/86*(sqrt(43)*sqrt(-3/172*(sqrt(258)*sqrt((2/9)^(2/3)*(3*(2/9)^(2/3)*(2557*sqrt(
129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) +
86*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1
/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt(129) - 28989)^(1/3)) - (2/9)^(1/3)*(2557*sqrt(129) - 2898
9)^(1/3) + 240*(2/9)^(2/3)/(2557*sqrt(129) - 28989)^(1/3) + 78) + 86)^2 - 3/172*(sqrt(258)*sqrt((2/9)^(2/3)*(3
*(2/9)^(2/3)*(2557*sqrt(129) - 28989)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 28989)^(1/3) - 160)/(2557*sqrt
(129) - 28989)^(1/3)) - 86*sqrt(1/43)*sqrt(105*sqrt(258)*(2/9)^(2/3)/sqrt((3*(2/9)^(2/3)*(2557*sqrt(129) - 289
89)^(2/3) + 117*(2/9)^(1/3)*(2557*sqrt(129) - 2...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{x + \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
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(x/(x+(x+(x^2+1)^(1/2))^(1/2)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{x+\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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