3.32.6 \(\int \frac {(1+x^2)^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x^2)^2} \, dx\) [3106]

Optimal. Leaf size=603 \[ \frac {\left (-803792+690 x-6024144 x^2+10470 x^3-1112160 x^4+1128 x^5+5861376 x^6-12288 x^7+143360 x^8\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{40320 \left (-1+x^2\right ) \sqrt {1+x^2} \left (4 x+8 x^3\right )+40320 \left (-1+x^2\right ) \left (1+8 x^2+8 x^4\right )}-\frac {2299}{128} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ]-\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}+13 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2+4 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 1.59, 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 {\left (1+x^2\right )^{5/2} \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[((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

Defer[Int][((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x)^2, x]/4 + Defer[Int][((1 + x^2)^(5/2)*S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x), x]/4 + Defer[Int][((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]
])/(1 + x)^2, x]/4 + Defer[Int][((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 + x), x]/4

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right )^2} \, dx &=\int \left (\frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1-x)^2}+\frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{4 (1+x)^2}+\frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 \left (1-x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{2} \int \left (\frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x)}+\frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x)}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x)^2} \, dx+\frac {1}{4} \int \frac {\left (1+x^2\right )^{5/2} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1+x} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.10, size = 852, normalized size = 1.41 \begin {gather*} \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}} \left (2 \left (-401896+345 x-3012072 x^2+5235 x^3-556080 x^4+564 x^5+2930688 x^6-6144 x^7+71680 x^8\right )+\left (-184+1525 x-6328 x^2-8025 x^3-1680 x^4-3740 x^5+8192 x^6+10240 x^7\right ) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-282-3314848 x+5298 x^2-3989088 x^3+7272 x^4+5789696 x^5-12288 x^6+143360 x^7+\left (935-2416 x-2315 x^2-5776 x^3-8860 x^4+8192 x^5+10240 x^6\right ) \sqrt {x+\sqrt {1+x^2}}\right )\right )}{40320 \left (-1+x^2\right ) \left (1+8 x^2+8 x^4+4 \sqrt {1+x^2} \left (x+2 x^3\right )\right )}-\frac {2299}{128} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )+4 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-\frac {1}{4} \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {14 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]-4 \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]+\frac {1}{4} \text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-16 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+4 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^2)^(5/2)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^2)^2,x]

[Out]

(Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]*(2*(-401896 + 345*x - 3012072*x^2 + 5235*x^3 - 556080*x^4 + 564*x^5 + 29306
88*x^6 - 6144*x^7 + 71680*x^8) + (-184 + 1525*x - 6328*x^2 - 8025*x^3 - 1680*x^4 - 3740*x^5 + 8192*x^6 + 10240
*x^7)*Sqrt[x + Sqrt[1 + x^2]] + Sqrt[1 + x^2]*(-282 - 3314848*x + 5298*x^2 - 3989088*x^3 + 7272*x^4 + 5789696*
x^5 - 12288*x^6 + 143360*x^7 + (935 - 2416*x - 2315*x^2 - 5776*x^3 - 8860*x^4 + 8192*x^5 + 10240*x^6)*Sqrt[x +
Sqrt[1 + x^2]])))/(40320*(-1 + x^2)*(1 + 8*x^2 + 8*x^4 + 4*Sqrt[1 + x^2]*(x + 2*x^3))) - (2299*ArcTanh[Sqrt[1
+ Sqrt[x + Sqrt[1 + x^2]]]])/128 + 4*RootSum[-2 + 4*#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 + Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*
#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ] - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , (14*Log[Sqrt[1 + Sqrt[x + Sqrt[1
+ x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]
- #1]*#1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ]/4 - 4*RootSum[2 - 8*#1^2 + 8*#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 + Log[Sqrt[1 + Sqrt[x + Sqrt
[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & ,
(-16*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1] - 2*Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^2 + 3*Log
[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1^4)/(-2*#1 + 4*#1^3 - 3*#1^5 + #1^7) & ]/4

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right )^{\frac {5}{2}} \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\left (-x^{2}+1\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)^(5/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x)

[Out]

int((x^2+1)^(5/2)*(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^2+1)^(5/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="maxima")

[Out]

integrate((x^2 + 1)^(5/2)*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 = 2.42, size = 6743, normalized size = 11.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(5/2)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^2+1)^2,x, algorithm="fricas")

[Out]

-1/80640*(20160*sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142)*log(1/4*sqrt
(1/2)*(547633*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^3 + (1095266*sqrt(1/2)*sqrt(14933*sqrt
(2) + 18583) + 547633*sqrt(2) + 820211864)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142)^2 + 31105
5544*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^2 - (547633*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 1
8583) + sqrt(2) - 142)^2 + 622111088*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + 311055544*sqrt(2) + 210463757752)
*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142) + 51091968368*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583)
+ 25545984184*sqrt(2) - 23818438149128)*sqrt(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142) + 78292
579460375*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 20160*sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(14933*sqrt(2) +
18583) - sqrt(2) + 142)*log(-1/4*sqrt(1/2)*(547633*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^
3 + (1095266*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + 547633*sqrt(2) + 820211864)*(2*sqrt(1/2)*sqrt(14933*sqrt(
2) + 18583) - sqrt(2) + 142)^2 + 311055544*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^2 - (5476
33*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^2 + 622111088*sqrt(1/2)*sqrt(14933*sqrt(2) + 1858
3) + 311055544*sqrt(2) + 210463757752)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142) + 51091968368
*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + 25545984184*sqrt(2) - 23818438149128)*sqrt(2*sqrt(1/2)*sqrt(14933*sqr
t(2) + 18583) - sqrt(2) + 142) + 78292579460375*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 20160*sqrt(1/2)*(x^2 - 1)
*sqrt(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) + 144)*log(1/4*sqrt(1/2)*((438047238*sqrt(1/2)*sqrt(14
593*sqrt(2) - 18193) - 219023619*sqrt(2) + 310702217138)*(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) +
144)^2 + 219023619*(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - sqrt(2) - 144)^3 - (219023619*(2*sqrt(1/2)*sqrt(
14593*sqrt(2) - 18193) - sqrt(2) - 144)^2 + 252315209088*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - 126157604544*
sqrt(2) + 80733215480968)*(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) + 144) + 126157604544*(2*sqrt(1/2
)*sqrt(14593*sqrt(2) - 18193) - sqrt(2) - 144)^2 + 86375906577792*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - 4318
7953288896*sqrt(2) + 15300143559468424)*sqrt(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) + 144) + 292828
84968104501*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 20160*sqrt(1/2)*(x^2 - 1)*sqrt(2*sqrt(1/2)*sqrt(14593*sqrt(2)
- 18193) + sqrt(2) + 144)*log(-1/4*sqrt(1/2)*((438047238*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - 219023619*sq
rt(2) + 310702217138)*(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) + 144)^2 + 219023619*(2*sqrt(1/2)*sqr
t(14593*sqrt(2) - 18193) - sqrt(2) - 144)^3 - (219023619*(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - sqrt(2) -
144)^2 + 252315209088*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - 126157604544*sqrt(2) + 80733215480968)*(2*sqrt(1
/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) + 144) + 126157604544*(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - sqr
t(2) - 144)^2 + 86375906577792*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) - 43187953288896*sqrt(2) + 15300143559468
424)*sqrt(2*sqrt(1/2)*sqrt(14593*sqrt(2) - 18193) + sqrt(2) + 144) + 29282884968104501*sqrt(sqrt(x + sqrt(x^2
+ 1)) + 1)) - 20160*(x^2 - 1)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142
)^2 + 1/16*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) + 426)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583)
- sqrt(2) + 142) - 3/32*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142)^2 - 71*sqrt(1/2)*sqrt(14933*
sqrt(2) + 18583) - 71/2*sqrt(2) + 9292) + 1/2*sqrt(2) + 71)*log(1/8*((1095266*sqrt(1/2)*sqrt(14933*sqrt(2) + 1
8583) + 547633*sqrt(2) + 820211864)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142)^2 + 897975750*(2
*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^2 - (547633*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) +
sqrt(2) - 142)^2 + 622111088*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + 311055544*sqrt(2) + 210463757752)*(2*sqr
t(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142) + 4*sqrt(-3/32*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) +
sqrt(2) - 142)^2 + 1/16*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) + 426)*(2*sqrt(1/2)*sqrt(14933*sqrt
(2) + 18583) - sqrt(2) + 142) - 3/32*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142)^2 - 71*sqrt(1/2
)*sqrt(14933*sqrt(2) + 18583) - 71/2*sqrt(2) + 9292)*((547633*sqrt(2)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583)
+ sqrt(2) - 142) + 897975750*sqrt(2))*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sqrt(2) + 142) - 897975750*s
qrt(2)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142) - 255416581000*sqrt(2)) + 509267290000*sqrt(1
/2)*sqrt(14933*sqrt(2) + 18583) + 254633645000*sqrt(2) - 57075233236000)*sqrt(sqrt(2)*sqrt(-3/32*(2*sqrt(1/2)*
sqrt(14933*sqrt(2) + 18583) + sqrt(2) - 142)^2 + 1/16*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) + sqrt(2) + 426
)*(2*sqrt(1/2)*sqrt(14933*sqrt(2) + 18583) - sq...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)**(5/2)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**2+1)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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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^2+1)^(5/2)*(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}\,{\left (x^2+1\right )}^{5/2}}{{\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^2 + 1)^(5/2))/(x^2 - 1)^2,x)

[Out]

int((((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(5/2))/(x^2 - 1)^2, x)

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