3.31.3 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\)
Optimal. Leaf size=402 \[ \frac {\sqrt {x^2+1} \left (\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (-2 x^2+5 x-1\right )+\left (2 x^2-x+1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (-2 x^3+5 x^2-2 x-3\right )+\left (2 x^3-x^2+2 x-1\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{16 \left (2 x^4+3 x^2+1\right )+16 \sqrt {x^2+1} \left (2 x^3+2 x\right )}-\frac {1}{64} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-4 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-24 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+3 \text {$\#$1}^3-\text {$\#$1}}\& \right ] \]
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Rubi [F] time = 2.01, 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 {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
-1/4*Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((I - x)^2*Sqrt[x + Sqrt[1 + x^2]]), x] + (I/4)*Defer[Int][S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((I - x)*Sqrt[x + Sqrt[1 + x^2]]), x] - Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 +
x^2]]]/((I + x)^2*Sqrt[x + Sqrt[1 + x^2]]), x]/4 + (I/4)*Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((I + x
)*Sqrt[x + Sqrt[1 + x^2]]), x]
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (i-x)^2 \sqrt {x+\sqrt {1+x^2}}}-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (i+x)^2 \sqrt {x+\sqrt {1+x^2}}}-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (-1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (-1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{2} \int \left (-\frac {i \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (i-x) \sqrt {x+\sqrt {1+x^2}}}-\frac {i \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (i+x) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=\frac {1}{4} i \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i-x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{4} i \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i+x) \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{4} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx-\frac {1}{4} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(i+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [F] time = 3.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]), x]
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IntegrateAlgebraic [A] time = 0.63, size = 518, normalized size = 1.29 \begin {gather*} \frac {\left (-1+2 x-x^2+2 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-3-2 x+5 x^2-2 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (1-x+2 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-1+5 x-2 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{16 \sqrt {1+x^2} \left (2 x+2 x^3\right )+16 \left (1+3 x^2+2 x^4\right )}+\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{64} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {36 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+136 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x^2)^2*Sqrt[x + Sqrt[1 + x^2]]),x]
[Out]
((-1 + 2*x - x^2 + 2*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-3 - 2*x + 5*x^2 - 2*x^3)*Sqrt[x + Sqrt[1 + x^2
]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((1 - x + 2*x^2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-1
+ 5*x - 2*x^2)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(16*Sqrt[1 + x^2]*(2*x + 2*x^3) + 1
6*(1 + 3*x^2 + 2*x^4)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]
] - #1] + 5*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/2 - RootSum[2
- 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (36*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 136*Log[Sqrt[1 + Sqrt
[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 4*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + Log[Sqrt[1 + Sqrt[x + S
qrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/64
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fricas [B] time = 1.61, size = 3327, normalized size = 8.28
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="fricas")
[Out]
1/64*(sqrt(2)*(x^2 + 1)*sqrt(sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 33
37/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37
/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 204
8*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) +
3337/2097152) - 93816) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 512*sqrt(-49125/524288*I*sqrt(2) +
3337/2097152) - 37)*log(1/4*(10485760*(50643935151*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2)
+ 3337/2097152) + 74) + 35783410727342*sqrt(2))*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/
2097152) - 37/2048)^2 + 375216256868333649920*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) +
3337/2097152) - 37/2048)^2 + (531040149448949760*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(
2) + 3337/2097152) - 37/2048)^2 - 9369128002935*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 33
37/2097152) + 74) - 33052532426664649*sqrt(2))*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097
152) + 74) - sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2
- 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I*s
qrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*s
qrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 93816)*
(5*(22131399660987*I*sqrt(2) + 103718779189248*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 39531061928516)*(
-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 78186752439242270*I*sqrt(2) + 36642
2125847982080*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 224700673505967572) - 33052532426664649*sqrt(2)*(4
37*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 74) - 3979900971776820892*sqrt(2))*sqrt(sqrt
(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 157286
4*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) +
2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) +
3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 93816) + 512*sqr
t(49125/524288*I*sqrt(2) + 3337/2097152) + 512*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37) + 2632189421
55746172797*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt
(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(
49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) +
3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sq
rt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 93816) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/20
97152) + 512*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37)*log(-1/4*(10485760*(50643935151*sqrt(2)*(437*I
*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 35783410727342*sqrt(2))*(437/4096*I*sqrt(2
) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 + 375216256868333649920*sqrt(2)*(-437/4096*I
*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 + (531040149448949760*sqrt(2)*(-437/40
96*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 9369128002935*sqrt(2)*(437*I*sqr
t(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 74) - 33052532426664649*sqrt(2))*(-437*I*sqrt(2) + 2
048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) - sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/5
24288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2)
+ 3337/2097152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(
-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(4912
5/524288*I*sqrt(2) + 3337/2097152) - 93816)*(5*(22131399660987*I*sqrt(2) + 103718779189248*sqrt(49125/524288*I
*sqrt(2) + 3337/2097152) + 39531061928516)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152)
+ 74) + 78186752439242270*I*sqrt(2) + 366422125847982080*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 224700
673505967572) - 33052532426664649*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) +
74) - 3979900971776820892*sqrt(2))*sqrt(sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*s
qrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/20
97152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sq
rt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*
I*sqrt(2) + 3337/2097152) - 93816) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 512*sqrt(-49125/524288*
I*sqrt(2) + 3337/2097152) - 37) + 263218942155746172797*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + sqrt(2)*(x^2 + 1)
*sqrt(-sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)
^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I
*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I
*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 93816
) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 512*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37)*l
og(1/4*(10485760*(50643935151*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 74)
+ 35783410727342*sqrt(2))*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2
+ 375216256868333649920*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/20
48)^2 + (531040149448949760*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 3
7/2048)^2 - 9369128002935*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 74) - 33
052532426664649*sqrt(2))*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + sqrt(-157
2864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-437/4096*
I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(491
25/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152)
+ 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 93816)*(5*(22131399660987*I*s
qrt(2) + 103718779189248*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 39531061928516)*(-437*I*sqrt(2) + 2048*
sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 78186752439242270*I*sqrt(2) + 366422125847982080*sqrt(491
25/524288*I*sqrt(2) + 3337/2097152) + 224700673505967572) - 33052532426664649*sqrt(2)*(437*I*sqrt(2) + 2048*sq
rt(49125/524288*I*sqrt(2) + 3337/2097152) + 74) - 3979900971776820892*sqrt(2))*sqrt(-sqrt(2)*sqrt(-1572864*(43
7/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2
) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/52428
8*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) +
16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 93816) + 512*sqrt(49125/524288*I*sqrt
(2) + 3337/2097152) + 512*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37) + 263218942155746172797*sqrt(sqrt
(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(-sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-4912
5/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(
2) + 3337/2097152) - 37/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222
)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(4
9125/524288*I*sqrt(2) + 3337/2097152) - 93816) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 512*sqrt(-4
9125/524288*I*sqrt(2) + 3337/2097152) - 37)*log(-1/4*(10485760*(50643935151*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt
(49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 35783410727342*sqrt(2))*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/
524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 + 375216256868333649920*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*sqrt(
49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 + (531040149448949760*sqrt(2)*(-437/4096*I*sqrt(2) - 1/2*s
qrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 9369128002935*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(491
25/524288*I*sqrt(2) + 3337/2097152) + 74) - 33052532426664649*sqrt(2))*(-437*I*sqrt(2) + 2048*sqrt(-49125/5242
88*I*sqrt(2) + 3337/2097152) + 74) + sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 33
37/2097152) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37
/2048)^2 - 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 204
8*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) +
3337/2097152) - 93816)*(5*(22131399660987*I*sqrt(2) + 103718779189248*sqrt(49125/524288*I*sqrt(2) + 3337/2097
152) + 39531061928516)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 78186752439
242270*I*sqrt(2) + 366422125847982080*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 224700673505967572) - 3305
2532426664649*sqrt(2)*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 74) - 39799009717768
20892*sqrt(2))*sqrt(-sqrt(2)*sqrt(-1572864*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/20971
52) - 37/2048)^2 - 1572864*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2
- 1/16*(437*I*sqrt(2) + 2048*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 222)*(-437*I*sqrt(2) + 2048*sqrt(-
49125/524288*I*sqrt(2) + 3337/2097152) + 74) + 16169/2*I*sqrt(2) + 37888*sqrt(49125/524288*I*sqrt(2) + 3337/20
97152) - 93816) + 512*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 512*sqrt(-49125/524288*I*sqrt(2) + 3337/20
97152) - 37) + 263218942155746172797*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 64*(x^2 + 1)*sqrt(437/4096*I*sqrt(2)
- 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)*log(32*(10485760*(22131399660987*I*sqrt(2) + 10
3718779189248*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 39531061928516)*(437/4096*I*sqrt(2) - 1/2*sqrt(-49
125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 2175140452142898216960*(-437/4096*I*sqrt(2) - 1/2*sqrt(491
25/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^3 - 157187884236889128960*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125
/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 + (531040149448949760*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524
288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 4094308937282595*I*sqrt(2) - 19187974150010880*sqrt(49125/524288*
I*sqrt(2) + 3337/2097152) - 33745847898881839)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097
152) + 74) + 10835865245014150005*I*sqrt(2) + 50782270072743659520*sqrt(49125/524288*I*sqrt(2) + 3337/2097152)
+ 25256976385450515878)*sqrt(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)
+ 263218942155746172797*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 64*(x^2 + 1)*sqrt(437/4096*I*sqrt(2) - 1/2*sqrt(
-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)*log(-32*(10485760*(22131399660987*I*sqrt(2) + 1037187791892
48*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 39531061928516)*(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*
I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 2175140452142898216960*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I
*sqrt(2) + 3337/2097152) - 37/2048)^3 - 157187884236889128960*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*s
qrt(2) + 3337/2097152) - 37/2048)^2 + (531040149448949760*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(
2) + 3337/2097152) - 37/2048)^2 - 4094308937282595*I*sqrt(2) - 19187974150010880*sqrt(49125/524288*I*sqrt(2) +
3337/2097152) - 33745847898881839)*(-437*I*sqrt(2) + 2048*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) + 74)
+ 10835865245014150005*I*sqrt(2) + 50782270072743659520*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) + 25256976
385450515878)*sqrt(437/4096*I*sqrt(2) - 1/2*sqrt(-49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048) + 26321894
2155746172797*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 64*(x^2 + 1)*sqrt(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/5242
88*I*sqrt(2) + 3337/2097152) - 37/2048)*log(64*(1087570226071449108480*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/5
24288*I*sqrt(2) + 3337/2097152) - 37/2048)^3 + 266202070552611389440*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524
288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 12639910957733300809*I*sqrt(2) - 59236928241276430336*sqrt(49125/
524288*I*sqrt(2) + 3337/2097152) - 10389861018925133634)*sqrt(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sq
rt(2) + 3337/2097152) - 37/2048) + 263218942155746172797*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 64*(x^2 + 1)*sqr
t(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)*log(-64*(108757022607144910
8480*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^3 + 2662020705526113894
40*(-437/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048)^2 - 12639910957733300809*
I*sqrt(2) - 59236928241276430336*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 10389861018925133634)*sqrt(-437
/4096*I*sqrt(2) - 1/2*sqrt(49125/524288*I*sqrt(2) + 3337/2097152) - 37/2048) + 263218942155746172797*sqrt(sqrt
(x + sqrt(x^2 + 1)) + 1)) - 4*(x^2 + (11*x^2 - sqrt(x^2 + 1)*(11*x - 1) + 3)*sqrt(x + sqrt(x^2 + 1)) - sqrt(x^
2 + 1)*(x + 1) + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1))/(x^2 + 1)
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="giac")
[Out]
Timed out
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x)
[Out]
int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{{\left (x^{2} + 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2/(x+(x^2+1)^(1/2))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/((x^2 + 1)^2*sqrt(x + sqrt(x^2 + 1))), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{{\left (x^2+1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/((x^2 + 1)^2*(x + (x^2 + 1)^(1/2))^(1/2)),x)
[Out]
int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/((x^2 + 1)^2*(x + (x^2 + 1)^(1/2))^(1/2)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(x**2+1)**2/(x+(x**2+1)**(1/2))**(1/2),x)
[Out]
Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(sqrt(x + sqrt(x**2 + 1))*(x**2 + 1)**2), x)
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