[next] [prev] [prev-tail] [tail] [up]

3.31.3 \(\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx\) [3003]

Optimal. Leaf size=402 \[ \frac {\left (-1+2 x-x^2+2 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-3-2 x+5 x^2-2 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (1-x+2 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-1+5 x-2 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{16 \sqrt {1+x^2} \left (2 x+2 x^3\right )+16 \left (1+3 x^2+2 x^4\right )}-\frac {1}{64} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-24 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [F]
time = 1.48, 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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.54, size = 444, normalized size = 1.10 \begin {gather*} \frac {1}{64} \left (\frac {4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (-1+2 x-x^2+2 x^3-\left (3+2 x-5 x^2+2 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (1-x+2 x^2+\left (-1+5 x-2 x^2\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{\left (1+x^2\right ) \left (1+2 x^2+2 x \sqrt {1+x^2}\right )}+32 \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {36 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+136 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-4 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-1 + 2*x - x^2 + 2*x^3 - (3 + 2*x - 5*x^2 + 2*x^3)*Sqrt[x + Sqrt[1 + x^
2]] + Sqrt[1 + x^2]*(1 - x + 2*x^2 + (-1 + 5*x - 2*x^2)*Sqrt[x + Sqrt[1 + x^2]])))/((1 + x^2)*(1 + 2*x^2 + 2*x
*Sqrt[1 + x^2])) + 32*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) & ] - 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 + Sqrt[1
 + x^2]]] - #1]*#1^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/64

________________________________________________________________________________________

Maple [F]
time = 0.01, 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*} \text {Failed to integrate} \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)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.31, size = 3327, normalized size = 8.28 \begin {gather*} \text {Too large to display} \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="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) + 204...

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [F(-1)] Timed out
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

________________________________________________________________________________________

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)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]