3.30.71 \(\int \frac {\sqrt {1+x^2} (1+x^4) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx\)
Optimal. Leaf size=375 \[ -\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\& \right ]+\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2+2}\& \right ]+\frac {251}{128} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (-2560 x^3-2048 x^2-345 x-184\right )+\sqrt {x^2+1} \left (\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \left (-2560 x^2-2048 x+935\right )+\left (-35840 x^3+3072 x^2-175104 x-282\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (-35840 x^4+3072 x^3-193024 x^2+1254 x-78032\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{80640 \sqrt {x^2+1} x+40320 \left (2 x^2+1\right )} \]
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Rubi [F] time = 2.07, 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+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]
[Out]
-Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[1 + x^2], x] + Defer[Int][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]
/((1 - x)*Sqrt[1 + x^2]), x] - Defer[Int][(x^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/Sqrt[1 + x^2], x] + Defer[In
t][Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]/((1 + x)*Sqrt[1 + x^2]), x]
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2} \left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{1-x^4} \, dx &=\int \frac {\left (1+x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx\\ &=\int \left (-\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}}-\frac {x^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}}+\frac {2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\left (1-x^2\right ) \sqrt {1+x^2}} \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx-\int \frac {x^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx\\ &=2 \int \left (\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1-x) \sqrt {1+x^2}}+\frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{2 (1+x) \sqrt {1+x^2}}\right ) \, dx-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx-\int \frac {x^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx\\ &=-\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx+\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1-x) \sqrt {1+x^2}} \, dx-\int \frac {x^2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx+\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{(1+x) \sqrt {1+x^2}} \, dx\\ \end {align*}
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Mathematica [B] time = 1.81, size = 968, normalized size = 2.58 \begin {gather*} -\frac {\left (x^2+\sqrt {x^2+1} x+1\right ) \left (71680 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^4+71680 \sqrt {x^2+1} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^3+5120 \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^3-6144 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^3+158130 \log \left (1-\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\right ) x^2-158130 \log \left (\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}+1\right ) x^2-6144 \sqrt {x^2+1} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^2+5120 \sqrt {x^2+1} \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^2+4096 \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^2+386048 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x^2+158130 \sqrt {x^2+1} \log \left (1-\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\right ) x-158130 \sqrt {x^2+1} \log \left (\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}+1\right ) x+350208 \sqrt {x^2+1} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x+4096 \sqrt {x^2+1} \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x+690 \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x-2508 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1} x+79065 \log \left (1-\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\right )-79065 \log \left (\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}+1\right )+80640 \left (2 x^2+2 \sqrt {x^2+1} x+1\right ) \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\&,\frac {\log \left (\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\&\right ]-80640 \left (2 x^2+2 \sqrt {x^2+1} x+1\right ) \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+8 \text {$\#$1}^4-8 \text {$\#$1}^2+2\&,\frac {\log \left (\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}-\text {$\#$1}\right ) \text {$\#$1}}{\text {$\#$1}^4-2 \text {$\#$1}^2+2}\&\right ]+564 \sqrt {x^2+1} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}-1870 \sqrt {x^2+1} \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}+368 \sqrt {x+\sqrt {x^2+1}} \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}+156064 \sqrt {\sqrt {x+\sqrt {x^2+1}}+1}\right )}{80640 \sqrt {x^2+1} \left (x+\sqrt {x^2+1}\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]
[Out]
-1/80640*((1 + x^2 + x*Sqrt[1 + x^2])*(156064*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 2508*x*Sqrt[1 + Sqrt[x + Sqr
t[1 + x^2]]] + 386048*x^2*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 6144*x^3*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 716
80*x^4*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 564*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 350208*x*Sqrt
[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 6144*x^2*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 71680
*x^3*Sqrt[1 + x^2]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 368*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 +
x^2]]] + 690*x*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 4096*x^2*Sqrt[x + Sqrt[1 + x^2]]*Sq
rt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 5120*x^3*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - 1870*Sq
rt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 4096*x*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 +
x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + 5120*x^2*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sq
rt[1 + x^2]]] + 79065*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + 158130*x^2*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1
+ x^2]]]] + 158130*x*Sqrt[1 + x^2]*Log[1 - Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 79065*Log[1 + Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]]] - 158130*x^2*Log[1 + Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] - 158130*x*Sqrt[1 + x^2]*Log[1 + S
qrt[1 + Sqrt[x + Sqrt[1 + x^2]]]] + 80640*(1 + 2*x^2 + 2*x*Sqrt[1 + x^2])*RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8
& , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]/(-2*#1 + #1^3) & ] - 80640*(1 + 2*x^2 + 2*x*Sqrt[1 + x^2])*Roo
tSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]*#1)/(2 - 2*#1^2 + #1
^4) & ]))/(Sqrt[1 + x^2]*(x + Sqrt[1 + x^2])^3)
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IntegrateAlgebraic [A] time = 0.61, size = 375, normalized size = 1.00 \begin {gather*} \frac {\left (-78032+1254 x-193024 x^2+3072 x^3-35840 x^4\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (-184-345 x-2048 x^2-2560 x^3\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\sqrt {1+x^2} \left (\left (-282-175104 x+3072 x^2-35840 x^3\right ) \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}+\left (935-2048 x-2560 x^2\right ) \sqrt {x+\sqrt {1+x^2}} \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )}{80640 x \sqrt {1+x^2}+40320 \left (1+2 x^2\right )}+\frac {251}{128} \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right )-\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [2-8 \text {$\#$1}^2+8 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{2-2 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(Sqrt[1 + x^2]*(1 + x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]])/(1 - x^4),x]
[Out]
((-78032 + 1254*x - 193024*x^2 + 3072*x^3 - 35840*x^4)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (-184 - 345*x - 204
8*x^2 - 2560*x^3)*Sqrt[x + Sqrt[1 + x^2]]*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + Sqrt[1 + x^2]*((-282 - 175104*x
+ 3072*x^2 - 35840*x^3)*Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] + (935 - 2048*x - 2560*x^2)*Sqrt[x + Sqrt[1 + x^2]]*
Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]]))/(80640*x*Sqrt[1 + x^2] + 40320*(1 + 2*x^2)) + (251*ArcTanh[Sqrt[1 + Sqrt[x
+ Sqrt[1 + x^2]]]])/128 - RootSum[-2 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] - #1]
/(-2*#1 + #1^3) & ] + RootSum[2 - 8*#1^2 + 8*#1^4 - 4*#1^6 + #1^8 & , (Log[Sqrt[1 + Sqrt[x + Sqrt[1 + x^2]]] -
#1]*#1)/(2 - 2*#1^2 + #1^4) & ]
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fricas [B] time = 0.72, size = 2212, normalized size = 5.90
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="fricas")
[Out]
-1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2)
+ 1)*(sqrt(2) - 2)*arctan(1/8*sqrt(2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(2))*sqrt(-2*sqrt(sqrt(2) + 2)*(s
qrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) +
4*sqrt(sqrt(2) + 2) + 4)*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2)
+ 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(3/4) - 1/4*((2*sqrt(2) - 3)*sqrt(s
qrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2
*sqrt(2))*(4*sqrt(2) + 8)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(
2) - 1) + sqrt(sqrt(2) + 1)*(sqrt(2) - 1)) - 1/2*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(
2) + 8)^(1/4)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 2)*arctan(1/8*sqrt(-2*(sqrt(sqrt(2) + 2)*(sqrt(2)
- 1) + sqrt(2))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4)*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt
(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1))*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqr
t(2) + 8)^(3/4) - 1/4*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1) + (2*sqrt(2) - 3)*sqrt(sqrt(2) + 1)
)*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(3/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)
- sqrt(sqrt(2) + 2)*sqrt(sqrt(2) + 1)*(sqrt(2) - 1) - sqrt(sqrt(2) + 1)*(sqrt(2) - 1)) + 1/2*4^(1/4)*2^(7/8)*
sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2) - 1)*arctan(1/8*4^(3/4)*2^(3/8)*sqrt(2*4^(1/4)*2^(
1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*
sqrt(x + sqrt(x^2 + 1)) + 4*2^(1/4) + 4)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2)
- 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4) - 1/4*4^(3/4)*2^(3/8)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2
) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(x + sqrt(x^
2 + 1)) + 1) - 2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) - (sqrt(2) + 1)*sqrt(sqrt(2) - 1)) + 1/2*4^(1/4)*2^(7/8
)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(2) - 1)*arctan(1/8*4^(3/4)*2^(3/8)*sqrt(-2*4^(1/4)*
2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) +
4*sqrt(x + sqrt(x^2 + 1)) + 4*2^(1/4) + 4)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt
(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4) - 1/4*4^(3/4)*2^(3/8)*(2^(1/4)*(sqrt(2) + 1)*sqrt(sqr
t(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1))*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*sqrt(sqrt(x + sqrt
(x^2 + 1)) + 1) + 2^(1/4)*(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + (sqrt(2) + 1)*sqrt(sqrt(2) - 1)) - 1/8*4^(1/4)*2^(
1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*log(2*4^(1/4)*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(
2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sqrt(x + sqrt(x^2 + 1)) + 4
*2^(1/4) + 4) + 1/8*4^(1/4)*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(2^(3/4) + 2)*log(-2*4^(1/4
)*2^(1/8)*sqrt(-2*2^(1/4)*(sqrt(2) + 1) + 2*sqrt(2) + 4)*(sqrt(2) + 2^(1/4))*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)
+ 4*sqrt(x + sqrt(x^2 + 1)) + 4*2^(1/4) + 4) - 1/8*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(sqrt
(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(4*sqrt(2) + 8)^(1/4)*log(2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(2))*sqrt(
-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 4*sq
rt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4) + 1/8*sqrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(s
qrt(sqrt(2) + 2)*(sqrt(2) - 2) - 2)*(4*sqrt(2) + 8)^(1/4)*log(-2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + sqrt(2))*s
qrt(-2*sqrt(sqrt(2) + 2)*(sqrt(2) - 1) + 2*sqrt(2))*(4*sqrt(2) + 8)^(1/4)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) +
4*sqrt(x + sqrt(x^2 + 1)) + 4*sqrt(sqrt(2) + 2) + 4) + 1/40320*(1120*x^2 - 2*sqrt(x^2 + 1)*(9520*x + 141) + (1
680*x^2 - 5*sqrt(x^2 + 1)*(336*x - 187) - 2215*x - 184)*sqrt(x + sqrt(x^2 + 1)) + 1818*x - 78032)*sqrt(sqrt(x
+ sqrt(x^2 + 1)) + 1) - 2*sqrt(2*sqrt(sqrt(2) + 1) - 2)*arctan(1/2*sqrt(sqrt(x + sqrt(x^2 + 1)) + sqrt(sqrt(2)
+ 1))*sqrt(2*sqrt(sqrt(2) + 1) - 2)*(sqrt(sqrt(2) + 1) + 1) - 1/2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)*sqrt(2*sq
rt(sqrt(2) + 1) - 2)*(sqrt(sqrt(2) + 1) + 1)) + 1/2*sqrt(2*sqrt(sqrt(2) + 1) + 2)*log(sqrt(2)*sqrt(2*sqrt(sqrt
(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(sqrt(2) + 1) + 2)*log(-sqrt(2)*sqrt(2*s
qrt(sqrt(2) + 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(2*sqrt(sqrt(2) - 1) + 2)*log(sqrt(2)*s
qrt(2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(2*sqrt(sqrt(2) - 1) + 2)*log(-s
qrt(2)*sqrt(2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) - 1/2*sqrt(-2*sqrt(sqrt(2) - 1) +
2)*log(sqrt(2)*sqrt(-2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 1/2*sqrt(-2*sqrt(sqrt(2
) - 1) + 2)*log(-sqrt(2)*sqrt(-2*sqrt(sqrt(2) - 1) + 2) + 2*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)) + 251/256*log(s
qrt(sqrt(x + sqrt(x^2 + 1)) + 1) + 1) - 251/256*log(sqrt(sqrt(x + sqrt(x^2 + 1)) + 1) - 1)
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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((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="giac")
[Out]
Timed out
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+1}\, \left (x^{4}+1\right ) \sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{-x^{4}+1}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)
[Out]
int((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{4} + 1\right )} \sqrt {x^{2} + 1} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{4} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2+1)^(1/2)*(x^4+1)*(1+(x+(x^2+1)^(1/2))^(1/2))^(1/2)/(-x^4+1),x, algorithm="maxima")
[Out]
-integrate((x^4 + 1)*sqrt(x^2 + 1)*sqrt(sqrt(x + sqrt(x^2 + 1)) + 1)/(x^4 - 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}\,\sqrt {x^2+1}\,\left (x^4+1\right )}{x^4-1} \,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)^(1/2)*(x^4 + 1))/(x^4 - 1),x)
[Out]
int(-(((x + (x^2 + 1)^(1/2))^(1/2) + 1)^(1/2)*(x^2 + 1)^(1/2)*(x^4 + 1))/(x^4 - 1), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx - \int \frac {x^{4} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{x^{2} \sqrt {x^{2} + 1} - \sqrt {x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2+1)**(1/2)*(x**4+1)*(1+(x+(x**2+1)**(1/2))**(1/2))**(1/2)/(-x**4+1),x)
[Out]
-Integral(sqrt(sqrt(x + sqrt(x**2 + 1)) + 1)/(x**2*sqrt(x**2 + 1) - sqrt(x**2 + 1)), x) - Integral(x**4*sqrt(s
qrt(x + sqrt(x**2 + 1)) + 1)/(x**2*sqrt(x**2 + 1) - sqrt(x**2 + 1)), x)
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