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

Optimal. Leaf size=143 \[ \frac {2 x}{3 \sqrt {\sqrt {x^2+1}+1}}+\sqrt {x^2+1} \left (\frac {2}{3} \sqrt {\sqrt {x^2+1}+1}-\frac {2 x}{3 \sqrt {\sqrt {x^2+1}+1}}\right )-\frac {4}{3} \sqrt {\sqrt {x^2+1}+1}-2 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}-\frac {\sqrt {\sqrt {x^2+1}+1}}{\sqrt {2}}\right ) \]

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Rubi [F]  time = 0.21, 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-\sqrt {1+x^2}}{\sqrt {1+\sqrt {1+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.38, size = 126, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+1} \left (-6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )-4 x^2+4 \sqrt {x^2+1} x+8 \sqrt {x^2+1}-3 \sqrt {2} \sqrt {\sqrt {x^2+1}-1} \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}-1}}{\sqrt {2}}\right )-8 x-2\right )}{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + Sqrt[1 + x^2]]*(-2 - 8*x - 4*x^2 + 8*Sqrt[1 + x^2] + 4*x*Sqrt[1 + x^2] - 3*Sqrt[2]*Sqrt[-1 + Sqrt[1
+ x^2]]*ArcTan[Sqrt[-1 + Sqrt[1 + x^2]]/Sqrt[2]] - 6*Hypergeometric2F1[-1/2, 1, 1/2, 1/2 - Sqrt[1 + x^2]/2]))/
(6*x)

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IntegrateAlgebraic [A]  time = 0.39, size = 143, normalized size = 1.00 \begin {gather*} \frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}-\frac {4}{3} \sqrt {1+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (-\frac {2 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2}{3} \sqrt {1+\sqrt {1+x^2}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}-\frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.45, size = 64, normalized size = 0.45 \begin {gather*} \frac {3 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, {\left (x^{2} - \sqrt {x^{2} + 1} {\left (x + 2\right )} + 2 \, x + 2\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*sqrt(2)*x*arctan(sqrt(2)*sqrt(sqrt(x^2 + 1) + 1)/x) - 2*(x^2 - sqrt(x^2 + 1)*(x + 2) + 2*x + 2)*sqrt(sq
rt(x^2 + 1) + 1))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x-(x^2+1)^(1/2))/(1+(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 {2}{3} \, {\left (\sqrt {x^{2} + 1} + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} - \int \frac {\sqrt {x^{2} + 1}}{\sqrt {\sqrt {x^{2} + 1} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(x^2 + 1) + 1)^(3/2) - 2*sqrt(sqrt(x^2 + 1) + 1) - integrate(sqrt(x^2 + 1)/sqrt(sqrt(x^2 + 1) + 1), x
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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