[next] [prev] [prev-tail] [tail] [up]

3.28.27 \(\int \frac {1}{x^2-\sqrt {1+x} \sqrt {1+\sqrt {1+x}}} \, dx\) [2727]

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

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [F]
time = 0.34, 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 \text {Subst}\left (\int \frac {x}{-x \sqrt {1+x}+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {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 \text {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} \text {Subst}\left (\int \frac {4-x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {4}{11} \text {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} \text {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 ) \text {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 ) \text {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} \text {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} \text {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} \text {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} \text {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} \text {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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 247, normalized size = 0.99 \begin {gather*} \frac {2}{55} \left (\left (-5+7 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )-\left (5+7 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )+10 \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 ]\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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]]] - (5 + 7*Sqrt[5])*Log[-1 + Sqrt[5] + 2*Sqrt[1
+ Sqrt[1 + x]]] + 10*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) & ]))/55

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.46, size = 2204, normalized size = 8.82

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]

-1/11*sum((-5*_R^4+12*_R^3-9*_R^2+_R-2)/(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))-1/11*sum((3*_R^8-3*_R^6-10*_R^4-2)/(5*_R^9-20*_R^7+24*_R^5-10*_R^3+3*_R)*ln((1+(1+
x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^10-5*_Z^8+8*_Z^6-5*_Z^4+3*_Z^2-1))+1/11*sum((19*_R^4-39*_R^3+10*_R^2+5*_R+12)
/(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))-1/11*sum((5
*_R^4+12*_R^3+9*_R^2+_R+2)/(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))+1/11*sum((-4*_R^8+26*_R^6-38*_R^4+11*_R^2-12)/(5*_R^9-20*_R^7+24*_R^5-10*_R^3+3*_R)*ln((1+(1+x)
^(1/2))^(1/2)-_R),_R=RootOf(_Z^10-5*_Z^8+8*_Z^6-5*_Z^4+3*_Z^2-1))+1/11*sum((-19*_R^4-39*_R^3-10*_R^2+5*_R-12)/
(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))-1/11*sum((9*
_R^4-15*_R^3+3*_R^2-4*_R+8)/(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))+1/11*sum((-3*_R^4+6*_R^3-2*_R^2+3*_R-1)/(5*_R^4+4*_R^3+6*_R^2+2*_R+1)*ln(x-_R),_R=RootOf(_Z^5+
_Z^4+2*_Z^3+_Z^2+_Z+1))+1/11*sum((-4*_R^8+15*_R^6-16*_R^4+11*_R^2-1)/(5*_R^9-20*_R^7+24*_R^5-10*_R^3+3*_R)*ln(
(1+(1+x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^10-5*_Z^8+8*_Z^6-5*_Z^4+3*_Z^2-1))+1/11*sum((-3*_R^8+14*_R^6-23*_R^4+11
*_R^2-9)/(5*_R^9-20*_R^7+24*_R^5-10*_R^3+3*_R)*ln((1+(1+x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^10-5*_Z^8+8*_Z^6-5*_Z
^4+3*_Z^2-1))-7/55*5^(1/2)*arctanh(1/5*(-1+2*x)*5^(1/2))+1/11*sum((6*_R^4-4*_R^3-13*_R^2-_R+9)/(5*_R^4-6*_R^2-
2*_R+2)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^5-2*_Z^3-_Z^2+2*_Z+1))-14/55*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1
/2)+1)*5^(1/2))+1/11*sum((-3*_R^4-5*_R^3-_R^2-5*_R+1)/(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))+1/11*sum((-7*_R^4+_R^3+6*_R^2+3*_R-5)/(5*_R^4-6*_R^2-2*_R+2)*ln((1+x
)^(1/2)-_R),_R=RootOf(_Z^5-2*_Z^3-_Z^2+2*_Z+1))-1/11*sum((-9*_R^4-4*_R^3-17*_R^2-2*_R-14)/(5*_R^4+4*_R^3+6*_R^
2+2*_R+1)*ln(x-_R),_R=RootOf(_Z^5+_Z^4+2*_Z^3+_Z^2+_Z+1))-1/11*sum((-9*_R^4-15*_R^3-3*_R^2-4*_R-8)/(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))+1/11*ln((1+x)^(1/2)+(1+
(1+x)^(1/2))^(1/2))-1/11*ln((1+x)^(1/2)-(1+(1+x)^(1/2))^(1/2))-1/11*sum((5*_R^4+_R^3+7*_R^2-5*_R+9)/(5*_R^4+4*
_R^3+6*_R^2+2*_R+1)*ln(x-_R),_R=RootOf(_Z^5+_Z^4+2*_Z^3+_Z^2+_Z+1))+2/11*sum((12*_R^8-45*_R^6+48*_R^4-22*_R^2+
14)/(5*_R^9-20*_R^7+24*_R^5-10*_R^3+3*_R)*ln((1+(1+x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^10-5*_Z^8+8*_Z^6-5*_Z^4+3*
_Z^2-1))+2/11*sum((13*_R^4+29*_R^3+8*_R^2-4*_R+14)/(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))+1/11*sum((-6*_R^4-21*_R^3-13*_R^2+_R-9)/(5*_R^4+4*_R^3-6*_R^2-2*_R+1)*l
n((1+(1+x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^5+_Z^4-2*_Z^3-_Z^2+_Z-1))+1/11*sum((7*_R^4+_R^3-6*_R^2+3*_R+5)/(5*_R^
4-6*_R^2+2*_R+2)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^5-2*_Z^3+_Z^2+2*_Z-1))+2/11*sum((-13*_R^4+29*_R^3-8*_R^2-4*_R
-14)/(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))-1/22*ln
(x-(1+x)^(1/2))+1/22*ln(x+(1+x)^(1/2))+1/11*sum((3*_R^4-5*_R^3+_R^2-5*_R-1)/(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))-1/11*sum((10*_R^8-32*_R^6+29*_R^4-11*_R^2+8)/(
5*_R^9-20*_R^7+24*_R^5-10*_R^3+3*_R)*ln((1+(1+x)^(1/2))^(1/2)-_R),_R=RootOf(_Z^10-5*_Z^8+8*_Z^6-5*_Z^4+3*_Z^2-
1))-14/55*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1/2)-1)*5^(1/2))+1/11*sum((6*_R^4-21*_R^3+13*_R^2+_R+9)/(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))+1/11*sum((-6*_R^
4-4*_R^3+13*_R^2-_R-9)/(5*_R^4-6*_R^2+2*_R+2)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^5-2*_Z^3+_Z^2+2*_Z-1))-7/55*5^(1
/2)*arctanh(1/5*(-1+2*(1+x)^(1/2))*5^(1/2))-7/55*5^(1/2)*arctanh(1/5*(2*(1+x)^(1/2)+1)*5^(1/2))-1/22*ln(x^2-x-
1)

________________________________________________________________________________________

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(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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]