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

Optimal. Leaf size=250 \[ \frac {4}{11} \text {RootSum}\left [\text {$\#$1}^5+\text {$\#$1}^4-2 \text {$\#$1}^3-\text {$\#$1}^2+\text {$\#$1}-1\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-2 \text {$\#$1}^3 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-7 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-2 \text {$\#$1} \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )+7 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{5 \text {$\#$1}^4+4 \text {$\#$1}^3-6 \text {$\#$1}^2-2 \text {$\#$1}+1}\& \right ]+\frac {2}{55} \left (7 \sqrt {5}-5\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right )-\frac {2}{55} \left (5+7 \sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}-1\right ) \]

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Rubi [F]  time = 0.46, 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-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*(5 + 7*Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(5 - 7*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sq
rt[1 + Sqrt[1 + x]]])/55 + (4*Log[2 + Sqrt[1 + x] - Sqrt[1 + Sqrt[1 + x]] + 2*(1 + Sqrt[1 + x])^(3/2) - (1 + S
qrt[1 + x])^2 - (1 + Sqrt[1 + x])^(5/2)])/55 + (136*Defer[Subst][Defer[Int][(-1 + x - x^2 - 2*x^3 + x^4 + x^5)
^(-1), x], x, Sqrt[1 + Sqrt[1 + x]]])/55 - (32*Defer[Subst][Defer[Int][x/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x
], x, Sqrt[1 + Sqrt[1 + x]]])/55 - (116*Defer[Subst][Defer[Int][x^2/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x], x,
Sqrt[1 + Sqrt[1 + x]]])/55 - (56*Defer[Subst][Defer[Int][x^3/(-1 + x - x^2 - 2*x^3 + x^4 + x^5), x], x, Sqrt[
1 + Sqrt[1 + x]]])/55

Rubi steps

\begin {align*} \int \frac {1}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{-x \sqrt {1+x}+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {-1+x^2}{1-x^2+4 x^3-4 x^5+x^7} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (\frac {4-x}{11 \left (-1-x+x^2\right )}+\frac {7-2 x-7 x^2-2 x^3+x^4}{11 \left (-1+x-x^2-2 x^3+x^4+x^5\right )}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{11} \operatorname {Subst}\left (\int \frac {4-x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {4}{11} \operatorname {Subst}\left (\int \frac {7-2 x-7 x^2-2 x^3+x^4}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{55} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )+\frac {4}{55} \operatorname {Subst}\left (\int \frac {34-8 x-29 x^2-14 x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {1}{55} \left (2 \left (5-7 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {1}{55} \left (2 \left (5+7 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-\frac {2}{55} \left (5+7 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{55} \left (5-7 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{55} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )+\frac {4}{55} \operatorname {Subst}\left (\int \left (\frac {34}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {8 x}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {29 x^2}{-1+x-x^2-2 x^3+x^4+x^5}-\frac {14 x^3}{-1+x-x^2-2 x^3+x^4+x^5}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-\frac {2}{55} \left (5+7 \sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{55} \left (5-7 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{55} \log \left (2+\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}+2 \left (1+\sqrt {1+x}\right )^{3/2}-\left (1+\sqrt {1+x}\right )^2-\left (1+\sqrt {1+x}\right )^{5/2}\right )-\frac {32}{55} \operatorname {Subst}\left (\int \frac {x}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {56}{55} \operatorname {Subst}\left (\int \frac {x^3}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-\frac {116}{55} \operatorname {Subst}\left (\int \frac {x^2}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {136}{55} \operatorname {Subst}\left (\int \frac {1}{-1+x-x^2-2 x^3+x^4+x^5} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}

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Mathematica [B]  time = 3.56, size = 1883, normalized size = 7.53

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(Sqrt[10*(3 + Sqrt[5])]*(-1 + 3*Sqrt[5])*ArcTanh[Sqrt[2/(3 - Sqrt[5])]*Sqrt[1 + Sqrt[1 + x]]]) - 2*Sqrt[10/(
3 + Sqrt[5])]*(1 + 3*Sqrt[5])*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*Sqrt[1 + Sqrt[1 + x]]] - 14*Sqrt[5]*ArcTanh[(-1 +
2*Sqrt[1 + x])/Sqrt[5]] - 5*Log[-x + Sqrt[1 + x]] + 10*RootSum[-1 + 2*#1 + #1^2 - 2*#1^3 + #1^5 & , (-4*Log[Sq
rt[1 + x] - #1] + 2*Log[Sqrt[1 + x] - #1]*#1 + 7*Log[Sqrt[1 + x] - #1]*#1^2 - 3*Log[Sqrt[1 + x] - #1]*#1^3 + L
og[Sqrt[1 + x] - #1]*#1^4)/(2 + 2*#1 - 6*#1^2 + 5*#1^4) & ] + 10*(3*RootSum[1 + #1 + #1^2 - 2*#1^3 - #1^4 + #1
^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] + 3*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 + x]] -
#1]*#1^2 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 + 2*#1 - 6*#1^2
- 4*#1^3 + 5*#1^4) & ] - RootSum[1 + #1 + #1^2 - 2*#1^3 - #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] + L
og[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#
1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 + 2*#1 - 6*#1^2 - 4*#1^3 + 5*#1^4) & ] - 2*RootSum[1 + #1 + #1^
2 - 2*#1^3 - #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] + 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[S
qrt[1 + Sqrt[1 + x]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)
/(1 + 2*#1 - 6*#1^2 - 4*#1^3 + 5*#1^4) & ] - RootSum[1 + #1 + #1^2 - 2*#1^3 - #1^4 + #1^5 & , (Log[Sqrt[1 + Sq
rt[1 + x]] - #1] - Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 +
Sqrt[1 + x]] - #1]*#1^4)/(1 + 2*#1 - 6*#1^2 - 4*#1^3 + 5*#1^4) & ] + 6*RootSum[-1 - #1 + 2*#1^2 + #1^3 - #1^4
+ #1^5 & , (-(Log[1/Sqrt[1 + Sqrt[1 + x]] - #1]*#1) - Log[1/Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + Log[1/Sqrt[1 +
Sqrt[1 + x]] - #1]*#1^3)/(-1 + 4*#1 + 3*#1^2 - 4*#1^3 + 5*#1^4) & ] + 2*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4
+ #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 +
x]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1
^2 + 4*#1^3 + 5*#1^4) & ] + RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1]
- Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + Log[Sqrt[1 + Sqrt[1 + x]] - #
1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ] + RootSum[-1 + #1 -
#1^2 - 2*#1^3 + #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] - Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + Log[
Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4)
& ] - 3*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (Log[Sqrt[1 + Sqrt[1 + x]] - #1] - 3*Log[Sqrt[1 + S
qrt[1 + x]] - #1]*#1 - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^3 + Log[S
qrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ] + 6*RootSum[1 - #1 - 2*#1^2 + #1^3
+ #1^4 + #1^5 & , (-(Log[1/Sqrt[1 + Sqrt[1 + x]] - #1]*#1) + Log[1/Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 + Log[1/Sq
rt[1 + Sqrt[1 + x]] - #1]*#1^3)/(-1 - 4*#1 + 3*#1^2 + 4*#1^3 + 5*#1^4) & ]))/55

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IntegrateAlgebraic [A]  time = 0.00, size = 250, normalized size = 1.00 \begin {gather*} \frac {2}{55} \left (-5+7 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{55} \left (5+7 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )+\frac {4}{11} \text {RootSum}\left [-1+\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^5\&,\frac {7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )-2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}-7 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^4}{1-2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+5 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-5 + 7*Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/55 - (2*(5 + 7*Sqrt[5])*Log[-1 + Sqrt[5] + 2*S
qrt[1 + Sqrt[1 + x]]])/55 + (4*RootSum[-1 + #1 - #1^2 - 2*#1^3 + #1^4 + #1^5 & , (7*Log[Sqrt[1 + Sqrt[1 + x]]
- #1] - 2*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1 - 7*Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^2 - 2*Log[Sqrt[1 + Sqrt[1
+ x]] - #1]*#1^3 + Log[Sqrt[1 + Sqrt[1 + x]] - #1]*#1^4)/(1 - 2*#1 - 6*#1^2 + 4*#1^3 + 5*#1^4) & ])/11

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.50, size = 126, normalized size = 0.50

method	result	size
derivativedivides	\(-\frac {2 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )}{11}-\frac {28 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{55}+\frac {4 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{5}+\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-\textit {\_Z}^{2}+\textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{3}-7 \textit {\_R}^{2}-2 \textit {\_R} +7\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-6 \textit {\_R}^{2}-2 \textit {\_R} +1}\right )}{11}\)	\(126\)
default	\(\text {Expression too large to display}\)	\(2204\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*ln((1+x)^(1/2)-(1+(1+x)^(1/2))^(1/2))-28/55*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1/2)-1)*5^(1/2))+4/1
1*sum((_R^4-2*_R^3-7*_R^2-2*_R+7)/(5*_R^4+4*_R^3-6*_R^2-2*_R+1)*ln((1+(1+x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^5+_Z
^4-2*_Z^3-_Z^2+_Z-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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