3.30.100 \(\int \frac {\sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x^2)^2} \, dx\) [3000]

Optimal. Leaf size=399 \[ \frac {\left (-1-2 x-x^2-2 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (1+18 x+x^2+26 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-1-x-2 x^2\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (5+x+26 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{16 \sqrt {1+x^2} \left (2 x+2 x^3\right )+16 \left (1+3 x^2+2 x^4\right )}+\frac {1}{64} \text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {8 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-6 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.19, 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 {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1+x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/4*Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(I - x)^2, x] + (I/4)*Defer[Int][(
Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(I - x), x] - Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(I + x)^2, x]/4 + (I/4)*Defer[Int][(Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]])/(I + x), x]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 427, normalized size = 1.07 \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 (1+18 x+x^2+26 x^3\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-1-x-2 x^2+\left (5+x+26 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 {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-\text {$\#$1}+\text {$\#$1}^3}\&\right ]-\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {24 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-26 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4+19 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+3 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(-1 - 2*x - x^2 - 2*x^3 + (1 + 18*x + x^2 + 26*x^3)*Sqrt[x + Sqrt[1 + x^
2]] + Sqrt[1 + x^2]*(-1 - x - 2*x^2 + (5 + x + 26*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] + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2)/(-#1 + #1^3) & ] - RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^
6 + #1^8 & , (24*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 26*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*
#1^2 - 16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4 + 19*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1
^6)/(-#1 + 3*#1^3 - 3*#1^5 + #1^7) & ])/64

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.86, size = 3329, normalized size = 8.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(sqrt(2)*(x^2 + 1)*sqrt(sqrt(2)*sqrt(-1572864*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) - 1052
71/2097152) + 37/2048)^2 - 1572864*(-117/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) +
37/2048)^2 - 1/16*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) - 74)*(-117*I*sqrt(2) +
2048*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 222) + 4329/2*I*sqrt(2) - 37888*sqrt(2645/524288*I*sqrt(2)
- 105271/2097152) - 59480) + 512*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 512*sqrt(-2645/524288*I*sqrt(
2) - 105271/2097152) + 37)*log(1/4*(2097152*(267626275*sqrt(2)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(
2) - 105271/2097152) - 74) - 461246050066*sqrt(2))*(-117/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 10
5271/2097152) + 37/2048)^2 - 967303076388012032*sqrt(2)*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) -
105271/2097152) + 37/2048)^2 + 5*(112250595573760*sqrt(2)*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2
) - 105271/2097152) + 37/2048)^2 + 1980434435*sqrt(2)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 1052
71/2097152) - 74) + 859779825444*sqrt(2))*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152)
- 74) - sqrt(-1572864*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 15
72864*(-117/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1/16*(117*I*sqrt
(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) - 74)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(
2) - 105271/2097152) + 222) + 4329/2*I*sqrt(2) - 37888*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 59480)*(
(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) - 74)*(-31312274175*I*sqrt(2) + 5480986112
00*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 481050394416) + 53965787857722*I*sqrt(2) - 944631910535168*s
qrt(2645/524288*I*sqrt(2) - 105271/2097152) - 136787816132412) + 4298899127220*sqrt(2)*(-117*I*sqrt(2) + 2048*
sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 74) - 23674754806209592*sqrt(2))*sqrt(sqrt(2)*sqrt(-1572864*(11
7/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1572864*(-117/4096*I*sqrt(2
) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1/16*(117*I*sqrt(2) + 2048*sqrt(-2645/524
288*I*sqrt(2) - 105271/2097152) - 74)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 22
2) + 4329/2*I*sqrt(2) - 37888*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 59480) + 512*sqrt(2645/524288*I*s
qrt(2) - 105271/2097152) + 512*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37) + 100991068027313397*sqrt(s
qrt(x + sqrt(x^2 + 1)) + 1)) - sqrt(2)*(x^2 + 1)*sqrt(sqrt(2)*sqrt(-1572864*(117/4096*I*sqrt(2) - 1/2*sqrt(264
5/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1572864*(-117/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqr
t(2) - 105271/2097152) + 37/2048)^2 - 1/16*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152)
- 74)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 222) + 4329/2*I*sqrt(2) - 37888*s
qrt(2645/524288*I*sqrt(2) - 105271/2097152) - 59480) + 512*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 512*
sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37)*log(-1/4*(2097152*(267626275*sqrt(2)*(-117*I*sqrt(2) + 204
8*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 74) - 461246050066*sqrt(2))*(-117/4096*I*sqrt(2) - 1/2*sqrt(-
2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 967303076388012032*sqrt(2)*(117/4096*I*sqrt(2) - 1/2*sq
rt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 + 5*(112250595573760*sqrt(2)*(117/4096*I*sqrt(2) - 1/2
*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 + 1980434435*sqrt(2)*(-117*I*sqrt(2) + 2048*sqrt(26
45/524288*I*sqrt(2) - 105271/2097152) - 74) + 859779825444*sqrt(2))*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*
sqrt(2) - 105271/2097152) - 74) - sqrt(-1572864*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) - 105271/
2097152) + 37/2048)^2 - 1572864*(-117/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37/
2048)^2 - 1/16*(117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) - 74)*(-117*I*sqrt(2) + 204
8*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 222) + 4329/2*I*sqrt(2) - 37888*sqrt(2645/524288*I*sqrt(2) -
105271/2097152) - 59480)*((117*I*sqrt(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) - 74)*(-31312274
175*I*sqrt(2) + 548098611200*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 481050394416) + 53965787857722*I*s
qrt(2) - 944631910535168*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 136787816132412) + 4298899127220*sqrt(
2)*(-117*I*sqrt(2) + 2048*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) - 74) - 23674754806209592*sqrt(2))*sqrt
(sqrt(2)*sqrt(-1572864*(117/4096*I*sqrt(2) - 1/2*sqrt(2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1
572864*(-117/4096*I*sqrt(2) - 1/2*sqrt(-2645/524288*I*sqrt(2) - 105271/2097152) + 37/2048)^2 - 1/16*(117*I*sqr
t(2) + 2048*sqrt(-2645/524288*I*sqrt(2) - 10527...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((x+(x^2+1)^(1/2))^(1/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(x^2+1)^2,x, algorithm="giac")

[Out]

Timed out

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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}\,\sqrt {x+\sqrt {x^2+1}}}{{\left (x^2+1\right )}^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 + (x^2 + 1)^(1/2))^(1/2))/(x^2 + 1)^2,x)

[Out]

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

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