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

Optimal. Leaf size=70 \[ \frac {2 \sqrt {x^2+1} x}{3 \sqrt {\sqrt {x^2+1}+1}}+\frac {4 x}{3 \sqrt {\sqrt {x^2+1}+1}}-4 \tan ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^2+1}+1}}\right ) \]

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Rubi [F]  time = 0.57, 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 {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*x^3)/(3*(1 + Sqrt[1 + x^2])^(3/2)) + (2*x)/Sqrt[1 + Sqrt[1 + x^2]] - I*Defer[Int][Sqrt[1 + Sqrt[1 + x^2]]/(
I - x), x] - I*Defer[Int][Sqrt[1 + Sqrt[1 + x^2]]/(I + x), x]

Rubi steps

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

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Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.13, size = 70, normalized size = 1.00 \begin {gather*} \frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}-4 \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) - 4*ArcTan[x/Sqrt[1 + Sqrt
[1 + x^2]]]

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fricas [A]  time = 1.57, size = 87, normalized size = 1.24 \begin {gather*} \frac {3 \, x \arctan \left (\frac {4 \, {\left (x^{4} - 12 \, x^{2} + {\left (5 \, x^{2} - 3\right )} \sqrt {x^{2} + 1} + 3\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{5} - 46 \, x^{3} + 17 \, x}\right ) + 2 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*x*arctan(4*(x^4 - 12*x^2 + (5*x^2 - 3)*sqrt(x^2 + 1) + 3)*sqrt(sqrt(x^2 + 1) + 1)/(x^5 - 46*x^3 + 17*x)
) + 2*(x^2 + sqrt(x^2 + 1) - 1)*sqrt(sqrt(x^2 + 1) + 1))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right ) \sqrt {1+\sqrt {x^{2}+1}}}{x^{2}+1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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