3.31.41 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^{5/2}} \, dx\)
Optimal. Leaf size=452 \[ \frac {1}{256} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\& ,\frac {2 \text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-7 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+8 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+40 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+3 \text {$\#$1}^3-\text {$\#$1}}\& \right ]+\frac {\sqrt {x^2+1} \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (-20 x^4+30 x^3-17 x^2-166 x-1\right )+\left (24 x^4+8 x^3+22 x^2+4 x+2\right ) \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (-20 x^5+30 x^4-27 x^3-151 x^2-7 x-117\right )+\left (24 x^5+8 x^4+34 x^3+8 x^2+10 x\right ) \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{192 \left (4 x^6+9 x^4+6 x^2+1\right )+192 \sqrt {x^2+1} \left (4 x^5+7 x^3+3 x\right )} \]
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Rubi [F] time = 0.14, 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 )^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]
[Out]
Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2), x]
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx &=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^{5/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.52, size = 435, normalized size = 0.96 \begin {gather*} \frac {3 \left (x^2+\sqrt {x^2+1} x+1\right )^3 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+6 \text {$\#$1}^4-4 \text {$\#$1}^2+2\&,\frac {2 \text {$\#$1}^6 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )-7 \text {$\#$1}^4 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+8 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )+40 \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^7-3 \text {$\#$1}^5+3 \text {$\#$1}^3-\text {$\#$1}}\&\right ]-4 \left (\sqrt {\sqrt {x^2+1}+x}+1\right )^{3/2} \left (-6 \sqrt {x^2+1}+11 \sqrt {\sqrt {x^2+1}+x}-6 x-15\right ) \left (x^2+\sqrt {x^2+1} x+1\right )^2-16 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (x \left (\sqrt {\sqrt {x^2+1}+x}-1\right )-\sqrt {x^2+1}+\sqrt {x^2+1} \sqrt {\sqrt {x^2+1}+x}+\sqrt {\sqrt {x^2+1}+x}+49\right ) \left (x^2+\sqrt {x^2+1} x+1\right )+256 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{768 \left (x^2+1\right )^{3/2} \left (\sqrt {x^2+1}+x\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]
[Out]
(256*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 4*(1 + x^2 + x*Sqrt[1 + x^2])^2*(1 + Sqrt[x + Sqrt[1 + x^2]])^(3/2)*(
-15 - 6*x - 6*Sqrt[1 + x^2] + 11*Sqrt[x + Sqrt[1 + x^2]]) - 16*(1 + x^2 + x*Sqrt[1 + x^2])*Sqrt[1 + Sqrt[x + S
qrt[1 + x^2]]]*(49 - Sqrt[1 + x^2] + Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]] + x*(-1 +
Sqrt[x + Sqrt[1 + x^2]])) + 3*(1 + x^2 + x*Sqrt[1 + x^2])^3*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (
40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 8*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 7*Log[Sq
rt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3
- 3*#1^5 + #1^7) & ])/(768*(1 + x^2)^(3/2)*(x + Sqrt[1 + x^2])^3)
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IntegrateAlgebraic [A] time = 1.09, size = 819, normalized size = 1.81 \begin {gather*} \frac {\left (-117-7 x-151 x^2-27 x^3+30 x^4-20 x^5\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (10 x+8 x^2+34 x^3+8 x^4+24 x^5\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-1-166 x-17 x^2+30 x^3-20 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (2+4 x+22 x^2+8 x^3+24 x^4\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{192 \sqrt {1+x^2} \left (3 x+7 x^3+4 x^5\right )+192 \left (1+6 x^2+9 x^4+4 x^6\right )}+\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {10 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {23}{256} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {40 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+40 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-25 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {3}{32} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {90 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+70 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-45 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+11 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/(1 + x^2)^(5/2),x]
[Out]
((-117 - 7*x - 151*x^2 - 27*x^3 + 30*x^4 - 20*x^5)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (10*x + 8*x^2 + 34*x^3
+ 8*x^4 + 24*x^5)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-1 - 166*x - 17*
x^2 + 30*x^3 - 20*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (2 + 4*x + 22*x^2 + 8*x^3 + 24*x^4)*Sqrt[x + Sqrt[1
+ x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(192*Sqrt[1 + x^2]*(3*x + 7*x^3 + 4*x^5) + 192*(1 + 6*x^2 + 9*x^4
+ 4*x^6)) + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , (10*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] +
6*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 + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ]/2 + (23*RootSum[2 - 4*#1^2 +
6*#1^4 - 4*#1^6 + #1^8 & , (40*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 40*Log[Sqrt[1 + Sqrt[x + Sqrt[1
+ x^2]]] - #1]*#1^2 - 25*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 6*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x
^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/256 - (3*RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & ,
(90*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] + 70*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 - 45*Lo
g[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 11*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^6)/(-#1 + 3
*#1^3 - 3*#1^5 + #1^7) & ])/32
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fricas [B] time = 1.81, size = 3420, normalized size = 7.57
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)^(5/2),x, algorithm="fricas")
[Out]
1/768*(3*sqrt(2)*(x^4 + 2*x^2 + 1)*sqrt(sqrt(2)*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435
456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/26843545
6*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2)
+ 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 9
63) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4) + 8192*sqr
t(487277/268435456*I*sqrt(2) + 582919/1073741824) + 8192*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824)
- 963/2)*log(1/4*(2147483648*(29673954056209*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) +
582919/1073741824) + 963) + 24321287697380314*sqrt(2))*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sq
rt(2) + 582919/1073741824) - 963/65536)^2 + 52229567628427796752105472*sqrt(2)*(-131/65536*I*sqrt(2) - 1/2*sqr
t(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + (63724331107212100370432*sqrt(2)*(-131/6553
6*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 57152035512258534*sqrt
(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 17015803710063315259*
sqrt(2))*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) - 4*sqrt(-402653
184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 40265318
4*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*
I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-4
87277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*s
qrt(2) + 582919/1073741824) + 548597/4)*((3887287981363379*I*sqrt(2) + 972356126513856512*sqrt(487277/26843545
6*I*sqrt(2) + 582919/1073741824) + 52897305453509581)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2)
+ 582919/1073741824) + 963) + 3186088688356821134*I*sqrt(2) + 796959955267758129152*sqrt(487277/268435456*I*s
qrt(2) + 582919/1073741824) - 104295807577858357664) + 17015803710063315259*sqrt(2)*(131*I*sqrt(2) + 32768*sqr
t(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 137914241548751312529152*sqrt(2))*sqrt(sqrt(2)*sqrt
(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 -
402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/
16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768
*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/26843
5456*I*sqrt(2) + 582919/1073741824) + 548597/4) + 8192*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) +
8192*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/2) + 2263551091934532801669435*sqrt(sqrt(x +
sqrt(x^2 + 1)) + 1)) - 3*sqrt(2)*(x^4 + 2*x^2 + 1)*sqrt(sqrt(2)*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqr
t(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt
(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/2684
35456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919
/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 5485
97/4) + 8192*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 8192*sqrt(-487277/268435456*I*sqrt(2) + 58
2919/1073741824) - 963/2)*log(-1/4*(2147483648*(29673954056209*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/2684
35456*I*sqrt(2) + 582919/1073741824) + 963) + 24321287697380314*sqrt(2))*(131/65536*I*sqrt(2) - 1/2*sqrt(-4872
77/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + 52229567628427796752105472*sqrt(2)*(-131/65536*I*
sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + (63724331107212100370432*s
qrt(2)*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 57152
035512258534*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 1701
5803710063315259*sqrt(2))*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963)
- 4*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65
536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/6553
6)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqrt(2
) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487
277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4)*((3887287981363379*I*sqrt(2) + 972356126513856512*sqr
t(487277/268435456*I*sqrt(2) + 582919/1073741824) + 52897305453509581)*(-131*I*sqrt(2) + 32768*sqrt(-487277/26
8435456*I*sqrt(2) + 582919/1073741824) + 963) + 3186088688356821134*I*sqrt(2) + 796959955267758129152*sqrt(487
277/268435456*I*sqrt(2) + 582919/1073741824) - 104295807577858357664) + 17015803710063315259*sqrt(2)*(131*I*sq
rt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 137914241548751312529152*sqrt(2))*
sqrt(sqrt(2)*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824)
- 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) -
963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*
I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*
sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4) + 8192*sqrt(487277/268435456*I*sqrt(2) + 5829
19/1073741824) + 8192*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/2) + 22635510919345328016694
35*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 3*sqrt(2)*(x^4 + 2*x^2 + 1)*sqrt(-sqrt(2)*sqrt(-402653184*(131/65536*I
*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 402653184*(-131/65536*I*
sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 3276
8*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*
I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/
1073741824) + 548597/4) + 8192*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 8192*sqrt(-487277/268435
456*I*sqrt(2) + 582919/1073741824) - 963/2)*log(1/4*(2147483648*(29673954056209*sqrt(2)*(131*I*sqrt(2) + 32768
*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 24321287697380314*sqrt(2))*(131/65536*I*sqrt(2)
- 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + 52229567628427796752105472*sqrt(
2)*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + (63724331
107212100370432*sqrt(2)*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963
/65536)^2 - 57152035512258534*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/10737418
24) + 963) + 17015803710063315259*sqrt(2))*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1
073741824) + 963) + 4*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/107
3741824) - 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/10737
41824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 288
9)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) +
7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4)*((3887287981363379*I*sqrt(2) + 97235
6126513856512*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 52897305453509581)*(-131*I*sqrt(2) + 3276
8*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 3186088688356821134*I*sqrt(2) + 7969599552677
58129152*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 104295807577858357664) + 17015803710063315259*
sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 13791424154875131
2529152*sqrt(2))*sqrt(-sqrt(2)*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 5
82919/1073741824) - 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582
919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/10737418
24) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*
sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4) + 8192*sqrt(487277/26843545
6*I*sqrt(2) + 582919/1073741824) + 8192*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/2) + 22635
51091934532801669435*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 3*sqrt(2)*(x^4 + 2*x^2 + 1)*sqrt(-sqrt(2)*sqrt(-4026
53184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 402653
184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(13
1*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(
-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I
*sqrt(2) + 582919/1073741824) + 548597/4) + 8192*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 8192*s
qrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/2)*log(-1/4*(2147483648*(29673954056209*sqrt(2)*(13
1*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 24321287697380314*sqrt(2))*(
131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + 5222956762842
7796752105472*sqrt(2)*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/6
5536)^2 + (63724331107212100370432*sqrt(2)*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 58291
9/1073741824) - 963/65536)^2 - 57152035512258534*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2
) + 582919/1073741824) + 963) + 17015803710063315259*sqrt(2))*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I
*sqrt(2) + 582919/1073741824) + 963) + 4*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*s
qrt(2) + 582919/1073741824) - 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqr
t(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 58291
9/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 1
26153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4)*((388728798136337
9*I*sqrt(2) + 972356126513856512*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 52897305453509581)*(-1
31*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 3186088688356821134*I*sqrt
(2) + 796959955267758129152*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 104295807577858357664) + 17
015803710063315259*sqrt(2)*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 963)
+ 137914241548751312529152*sqrt(2))*sqrt(-sqrt(2)*sqrt(-402653184*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/2684
35456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 402653184*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435
456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 1/16*(131*I*sqrt(2) + 32768*sqrt(487277/268435456*I*sqrt(2
) + 582919/1073741824) - 2889)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) +
963) + 126153/4*I*sqrt(2) + 7888896*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 548597/4) + 8192*s
qrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 8192*sqrt(-487277/268435456*I*sqrt(2) + 582919/107374182
4) - 963/2) + 2263551091934532801669435*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 768*(x^4 + 2*x^2 + 1)*sqrt(131/65
536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)*log(128*(2147483648*(38
87287981363379*I*sqrt(2) + 972356126513856512*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) + 528973054
53509581)*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 41
76237763442252209876631552*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) -
963/65536)^3 - 245466123424981010626904064*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 58291
9/1073741824) - 963/65536)^2 + (63724331107212100370432*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sq
rt(2) + 582919/1073741824) - 963/65536)^2 - 7486916652105867954*I*sqrt(2) - 1872757899665687642112*sqrt(487277
/268435456*I*sqrt(2) + 582919/1073741824) - 38021606488241652983)*(-131*I*sqrt(2) + 32768*sqrt(-487277/2684354
56*I*sqrt(2) + 582919/1073741824) + 963) + 8682296579254920630290*I*sqrt(2) + 2171767132129963658117120*sqrt(4
87277/268435456*I*sqrt(2) + 582919/1073741824) - 73802131019016954849564)*sqrt(131/65536*I*sqrt(2) - 1/2*sqrt(
-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536) + 2263551091934532801669435*sqrt(sqrt(x + sqrt(x
^2 + 1)) + 1)) + 768*(x^4 + 2*x^2 + 1)*sqrt(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 58291
9/1073741824) - 963/65536)*log(-128*(2147483648*(3887287981363379*I*sqrt(2) + 972356126513856512*sqrt(487277/2
68435456*I*sqrt(2) + 582919/1073741824) + 52897305453509581)*(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456
*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 4176237763442252209876631552*(-131/65536*I*sqrt(2) - 1/2*sqrt
(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^3 - 245466123424981010626904064*(-131/65536*I*sq
rt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 + (63724331107212100370432*(-1
31/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^2 - 748691665210586
7954*I*sqrt(2) - 1872757899665687642112*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 380216064882416
52983)*(-131*I*sqrt(2) + 32768*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) + 963) + 8682296579254920
630290*I*sqrt(2) + 2171767132129963658117120*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 7380213101
9016954849564)*sqrt(131/65536*I*sqrt(2) - 1/2*sqrt(-487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/6553
6) + 2263551091934532801669435*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 768*(x^4 + 2*x^2 + 1)*sqrt(-131/65536*I*sq
rt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)*log(128*(41762377634422522098766
31552*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)^3 + 297695
691053408807379009536*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/6
5536)^2 - 6453226293236626331361*I*sqrt(2) - 1614193276158608943710208*sqrt(487277/268435456*I*sqrt(2) + 58291
9/1073741824) - 193724940499278202752967)*sqrt(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 58
2919/1073741824) - 963/65536) + 2263551091934532801669435*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 768*(x^4 + 2*x^
2 + 1)*sqrt(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536)*log(-
128*(4176237763442252209876631552*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2) + 582919/1073741
824) - 963/65536)^3 + 297695691053408807379009536*(-131/65536*I*sqrt(2) - 1/2*sqrt(487277/268435456*I*sqrt(2)
+ 582919/1073741824) - 963/65536)^2 - 6453226293236626331361*I*sqrt(2) - 1614193276158608943710208*sqrt(487277
/268435456*I*sqrt(2) + 582919/1073741824) - 193724940499278202752967)*sqrt(-131/65536*I*sqrt(2) - 1/2*sqrt(487
277/268435456*I*sqrt(2) + 582919/1073741824) - 963/65536) + 2263551091934532801669435*sqrt(sqrt(x + sqrt(x^2 +
1)) + 1)) - 4*(121*x^4 + 4*x^3 + 238*x^2 - (121*x^3 - x^2 + 185*x - 1)*sqrt(x^2 + 1) + 2*(2*x^4 - 2*x^3 + 2*x
^2 - (2*x^3 + x^2 + 2*x + 1)*sqrt(x^2 + 1) - 2*x)*sqrt(x + sqrt(x^2 + 1)) + 4*x + 117)*sqrt(sqrt(x + sqrt(x^2
+ 1)) + 1))/(x^4 + 2*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)^(5/2),x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (x^{2}+1\right )^{\frac {5}{2}}}\, dx\]
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)^(5/2),x)
[Out]
int((1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^(5/2),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 )}^{\frac {5}{2}}}\,{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)^(5/2),x, algorithm="maxima")
[Out]
integrate(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^2 + 1)^(5/2), x)
________________________________________________________________________________________
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 )}^{5/2}} \,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)^(5/2),x)
[Out]
int(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)/(x^2 + 1)^(5/2), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\left (x^{2} + 1\right )^{\frac {5}{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)**(5/2),x)
[Out]
Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2 + 1)**(5/2), x)
________________________________________________________________________________________