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

Optimal. Leaf size=159 \[ -\text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^5-2 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )-2 \text {$\#$1}^2 \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )+2 \text {$\#$1} \log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )+\log \left (\sqrt {\sqrt {x^2+1}+x}-\text {$\#$1}\right )}{2 \text {$\#$1}^5+5 \text {$\#$1}^2-2 \text {$\#$1}}\& \right ]-\log \left (\sqrt {x^2+1}+x\right )+x \]

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Rubi [F]  time = 4.00, 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 {x^2-\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x + Defer[Int][(-1 - 2*x^5 + x^8)^(-1), x] - Defer[Int][x^2/(-1 - 2*x^5 + x^8), x] - Defer[Int][x^4/(-1 - 2*x^
5 + x^8), x] + Defer[Int][x^5/(-1 - 2*x^5 + x^8), x] + Defer[Int][(x^3*Sqrt[1 + x^2])/(-1 - 2*x^5 + x^8), x] +
Defer[Int][(x^4*Sqrt[1 + x^2])/(-1 - 2*x^5 + x^8), x] - Defer[Int][(x^6*Sqrt[1 + x^2])/(-1 - 2*x^5 + x^8), x]
+ Defer[Int][Sqrt[x + Sqrt[1 + x^2]]/(-1 - 2*x^5 + x^8), x] + Defer[Int][(x^2*Sqrt[x + Sqrt[1 + x^2]])/(-1 -
2*x^5 + x^8), x] + Defer[Int][(x^3*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^8), x] - Defer[Int][(x^6*Sqrt[x +
Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^8), x] - Defer[Int][(x*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^
8), x] - Defer[Int][(x^2*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^8), x] + Defer[Int][(x^4*Sqrt[
1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(-1 - 2*x^5 + x^8), x]

Rubi steps

\begin {align*} \int \frac {x^2-\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {x^2}{x^2+\sqrt {x+\sqrt {1+x^2}}}-\frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=\int \frac {x^2}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx-\int \frac {\sqrt {1+x^2}}{x^2+\sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=\int \left (1+\frac {x^4 \sqrt {1+x^2}}{-1-2 x^5+x^8}+\frac {1+x^5}{-1-2 x^5+x^8}+\frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {x^6 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {x^2 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx-\int \left (\frac {x^2 \left (1+x^2\right )}{-1-2 x^5+x^8}+\frac {x^3 \sqrt {1+x^2} \left (-1+x^3\right )}{-1-2 x^5+x^8}+\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}-\frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx\\ &=x+\int \frac {x^4 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx-\int \frac {x^2 \left (1+x^2\right )}{-1-2 x^5+x^8} \, dx-\int \frac {x^3 \sqrt {1+x^2} \left (-1+x^3\right )}{-1-2 x^5+x^8} \, dx+\int \frac {1+x^5}{-1-2 x^5+x^8} \, dx+\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^6 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^2 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {\left (1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx\\ &=x+\int \frac {x^4 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx+\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^6 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^2 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \left (\frac {x^2}{-1-2 x^5+x^8}+\frac {x^4}{-1-2 x^5+x^8}\right ) \, dx+\int \left (\frac {1}{-1-2 x^5+x^8}+\frac {x^5}{-1-2 x^5+x^8}\right ) \, dx-\int \left (-\frac {x^3 \sqrt {1+x^2}}{-1-2 x^5+x^8}+\frac {x^6 \sqrt {1+x^2}}{-1-2 x^5+x^8}\right ) \, dx+\int \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}+\frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8}\right ) \, dx\\ &=x+\int \frac {1}{-1-2 x^5+x^8} \, dx-\int \frac {x^2}{-1-2 x^5+x^8} \, dx-\int \frac {x^4}{-1-2 x^5+x^8} \, dx+\int \frac {x^5}{-1-2 x^5+x^8} \, dx+\int \frac {x^3 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1-2 x^5+x^8} \, dx+\int \frac {\sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x^3 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^6 \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx-\int \frac {x^2 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{-1-2 x^5+x^8} \, dx\\ \end {align*}

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Mathematica [B]  time = 15.82, size = 14154, normalized size = 89.02 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.94, size = 159, normalized size = 1.00 \begin {gather*} x-\log \left (x+\sqrt {1+x^2}\right )-\text {RootSum}\left [1-2 \text {$\#$1}^4+4 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+2 \log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}+5 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((x^2-(x^2+1)^(1/2))/(x^2+(x+(x^2+1)^(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 {x^{2} - \sqrt {x^{2} + 1}}{x^{2} + \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-\sqrt {x^{2}+1}}{x^{2}+\sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x^{2} + \sqrt {x^{2} + 1}}{2 \, x} + \int \frac {x^{6} - x^{3} + x^{2} - {\left (x^{4} + x^{2} - x\right )} \sqrt {x^{2} + 1} + 1}{2 \, {\left (x^{6} + 2 \, \sqrt {x + \sqrt {x^{2} + 1}} x^{4} + x^{3} + \sqrt {x^{2} + 1} x^{2}\right )}}\,{d x} - \int \frac {1}{2 \, \sqrt {x^{2} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2 + sqrt(x^2 + 1))/x + integrate(1/2*(x^6 - x^3 + x^2 - (x^4 + x^2 - x)*sqrt(x^2 + 1) + 1)/(x^6 + 2*sqr
t(x + sqrt(x^2 + 1))*x^4 + x^3 + sqrt(x^2 + 1)*x^2), x) - integrate(1/2/sqrt(x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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