3.22.43 \(\int \frac {1+x^2}{(-1+x^2) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\)

Optimal. Leaf size=156 \[ \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x+1}+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}}\& \right ]+\frac {-4 (-30 x-151)+\sqrt {x+1} \left (8-24 \sqrt {\sqrt {x+1}+1}\right )+64 \sqrt {\sqrt {x+1}+1}}{105 \sqrt {\sqrt {\sqrt {x+1}+1}+1}}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right ) \]

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Rubi [F]  time = 0.96, 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 {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

4/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] + (16*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 - (24*(1 + Sqrt[1 + Sqrt[1 + x]])
^(5/2))/5 + (8*(1 + Sqrt[1 + Sqrt[1 + x]])^(7/2))/7 + 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[2
]] + 16*Defer[Subst][Defer[Int][x^2/(-2 + 4*x^4 - 4*x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] - 24*D
efer[Subst][Defer[Int][x^4/(-2 + 4*x^4 - 4*x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] + 8*Defer[Subst
][Defer[Int][x^6/(-2 + 4*x^4 - 4*x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]]

Rubi steps

\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {2-2 x^2+x^4}{x \left (-2+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x+4 x^5-4 x^7+x^9}{\sqrt {1+x} \left (1+x^2-3 x^4+x^6\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=8 \operatorname {Subst}\left (\int \frac {-1+x^2+4 \left (-1+x^2\right )^5-4 \left (-1+x^2\right )^7+\left (-1+x^2\right )^9}{1+\left (-1+x^2\right )^2-3 \left (-1+x^2\right )^4+\left (-1+x^2\right )^6} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (-\frac {1}{2 x^2}+2 x^2-3 x^4+x^6-\frac {1}{2 \left (-2+x^2\right )}+\frac {x^2 \left (2-3 x^2+x^4\right )}{-2+4 x^4-4 x^6+x^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-4 \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+8 \operatorname {Subst}\left (\int \frac {x^2 \left (2-3 x^2+x^4\right )}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right )+8 \operatorname {Subst}\left (\int \left (\frac {2 x^2}{-2+4 x^4-4 x^6+x^8}-\frac {3 x^4}{-2+4 x^4-4 x^6+x^8}+\frac {x^6}{-2+4 x^4-4 x^6+x^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right )+8 \operatorname {Subst}\left (\int \frac {x^6}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+16 \operatorname {Subst}\left (\int \frac {x^2}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-24 \operatorname {Subst}\left (\int \frac {x^4}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 302, normalized size = 1.94 \begin {gather*} 8 \left (\frac {1}{8} \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4+4 \text {$\#$1}^2-1\&,\frac {2 \text {$\#$1}^4 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )-3 \text {$\#$1}^2 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )+\log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )}{2 \text {$\#$1}^7-2 \text {$\#$1}^3+\text {$\#$1}}\&\right ]+\frac {1}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {3}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {2}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}+\frac {1}{2 \sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\frac {\log \left (\sqrt {2}-\frac {2}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}\right )}{4 \sqrt {2}}+\frac {\log \left (\frac {2}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}+\sqrt {2}\right )}{4 \sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

8*(1/(2*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]) + (2*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 - (3*(1 + Sqrt[1 + Sqrt[1 +
x]])^(5/2))/5 + (1 + Sqrt[1 + Sqrt[1 + x]])^(7/2)/7 - Log[Sqrt[2] - 2/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]]/(4*Sqr
t[2]) + Log[Sqrt[2] + 2/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]]/(4*Sqrt[2]) + RootSum[-1 + 4*#1^2 - 4*#1^4 + 2*#1^8 &
, (Log[1/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1] - 3*Log[1/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]*#1^2 + 2*Log[1
/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]*#1^4)/(#1 - 2*#1^3 + 2*#1^7) & ]/8)

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IntegrateAlgebraic [A]  time = 0.00, size = 153, normalized size = 0.98 \begin {gather*} \frac {-4 \sqrt {1+\sqrt {1+x}} \left (-16+6 \sqrt {1+x}\right )-4 \left (-121-2 \sqrt {1+x}-30 (1+x)\right )}{105 \sqrt {1+\sqrt {1+\sqrt {1+x}}}}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right )+\text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[1 + Sqrt[1 + x]]*(-16 + 6*Sqrt[1 + x]) - 4*(-121 - 2*Sqrt[1 + x] - 30*(1 + x)))/(105*Sqrt[1 + Sqrt[1
+ Sqrt[1 + x]]]) + 2*Sqrt[2]*ArcTanh[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]/Sqrt[2]] + RootSum[-2 + 4*#1^4 - 4*#1^6 +
#1^8 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/#1 & ]

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fricas [B]  time = 1.30, size = 2034, normalized size = 13.04

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(105*(x + 1)*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt
(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8))*log(1/8*((sqrt(2) + sq
rt(2*sqrt(2) + 2))^2*(sqrt(2) - sqrt(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(
2) + 2))^2 - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sq
rt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqr
t(2*sqrt(2) + 2)))*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt
(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)) + 2*sqrt(sqrt(sqrt(x +
1) + 1) + 1)) - 105*(x + 1)*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + s
qrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8))*log(-1/8*((s
qrt(2) + sqrt(2*sqrt(2) + 2))^2*(sqrt(2) - sqrt(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - s
qrt(2*sqrt(2) + 2))^2 - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(s
qrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sq
rt(2) - sqrt(2*sqrt(2) + 2)))*sqrt(2*sqrt(2) + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) +
sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)) + 2*sqrt(sq
rt(sqrt(x + 1) + 1) + 1)) + 105*(x + 1)*sqrt(2*sqrt(2) - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(
sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8))*l
og(1/8*((sqrt(2) + sqrt(2*sqrt(2) + 2))^2*(sqrt(2) - sqrt(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqrt(2) + 2))*(s
qrt(2) - sqrt(2*sqrt(2) + 2))^2 + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2
) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)*(sqrt(2) + sqrt(2*sqrt(2)
+ 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)))*sqrt(2*sqrt(2) - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(
sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)) +
2*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - 105*(x + 1)*sqrt(2*sqrt(2) - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))
^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))
^2 + 8))*log(-1/8*((sqrt(2) + sqrt(2*sqrt(2) + 2))^2*(sqrt(2) - sqrt(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqrt(
2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2))^2 - 1/2*(sqrt(2) + sq
rt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 + 8)*(sqrt(2) + sqr
t(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)))*sqrt(2*sqrt(2) - 2*sqrt(-3/4*(sqrt(2) + sqrt(2*sqrt(2) + 2)
)^2 - 1/2*(sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2)) - 3/4*(sqrt(2) - sqrt(2*sqrt(2) + 2)
)^2 + 8)) + 2*sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - 210*(x + 1)*sqrt(-1/2*sqrt(2) + 1/2*sqrt(2*sqrt(2) + 2))*log(
1/4*((sqrt(2) + sqrt(2*sqrt(2) + 2))^3 + (sqrt(2) + sqrt(2*sqrt(2) + 2))^2*(sqrt(2) - sqrt(2*sqrt(2) + 2)) + (
sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 - 8*sqrt(2) - 8*sqrt(2*sqrt(2) + 2))*sqrt(-1/
2*sqrt(2) + 1/2*sqrt(2*sqrt(2) + 2)) + sqrt(sqrt(sqrt(x + 1) + 1) + 1)) + 210*(x + 1)*sqrt(-1/2*sqrt(2) + 1/2*
sqrt(2*sqrt(2) + 2))*log(-1/4*((sqrt(2) + sqrt(2*sqrt(2) + 2))^3 + (sqrt(2) + sqrt(2*sqrt(2) + 2))^2*(sqrt(2)
- sqrt(2*sqrt(2) + 2)) + (sqrt(2) + sqrt(2*sqrt(2) + 2))*(sqrt(2) - sqrt(2*sqrt(2) + 2))^2 - 8*sqrt(2) - 8*sqr
t(2*sqrt(2) + 2))*sqrt(-1/2*sqrt(2) + 1/2*sqrt(2*sqrt(2) + 2)) + sqrt(sqrt(sqrt(x + 1) + 1) + 1)) + 210*(x + 1
)*sqrt(-1/2*sqrt(2) - 1/2*sqrt(2*sqrt(2) + 2))*log(1/4*((sqrt(2) + sqrt(2*sqrt(2) + 2))^3 - 8*sqrt(2) - 8*sqrt
(2*sqrt(2) + 2) - 16)*sqrt(-1/2*sqrt(2) - 1/2*sqrt(2*sqrt(2) + 2)) + sqrt(sqrt(sqrt(x + 1) + 1) + 1)) - 210*(x
+ 1)*sqrt(-1/2*sqrt(2) - 1/2*sqrt(2*sqrt(2) + 2))*log(-1/4*((sqrt(2) + sqrt(2*sqrt(2) + 2))^3 - 8*sqrt(2) - 8
*sqrt(2*sqrt(2) + 2) - 16)*sqrt(-1/2*sqrt(2) - 1/2*sqrt(2*sqrt(2) + 2)) + sqrt(sqrt(sqrt(x + 1) + 1) + 1)) + 2
10*sqrt(2)*(x + 1)*log((2*(sqrt(2)*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) + sqrt(2)*sqrt(x + 1))*sqrt(sqrt(sqrt(x +
1) + 1) + 1) + x + 4*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) + 4*sqrt(x + 1) + 1)/(x + 1)) - 8*(3*(12*x + 47)*sqrt(
x + 1) - (15*(2*x + 9)*sqrt(x + 1) + 8*x + 8)*sqrt(sqrt(x + 1) + 1) - 8*x - 8)*sqrt(sqrt(sqrt(x + 1) + 1) + 1)
)/(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.17, size = 154, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )+2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )+\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}\)	\(154\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )+2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )+\frac {4}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

8/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)-24/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)+16/3*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)+
sum((_R^6-3*_R^4+2*_R^2)/(_R^7-3*_R^5+2*_R^3)*ln((1+(1+(1+x)^(1/2))^(1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*_Z^6+4*_
Z^4-2))+2*2^(1/2)*arctanh(1/2*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)*2^(1/2))+4/(1+(1+(1+x)^(1/2))^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2-1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+1}{\left (x^2-1\right )\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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