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

Optimal. Leaf size=104 \[ -\frac {2}{15} \sqrt {\sqrt {x^2+1}+1} \left (3 x^2+1\right )+\sqrt {x^2+1} \left (\frac {8 x}{15 \sqrt {\sqrt {x^2+1}+1}}-\frac {2}{15} \sqrt {\sqrt {x^2+1}+1}\right )+\frac {2 x \left (3 x^2+11\right )}{15 \sqrt {\sqrt {x^2+1}+1}} \]

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Rubi [F]  time = 0.17, 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 {\sqrt {1+\sqrt {1+x^2}}}{x+\sqrt {1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 72, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {\sqrt {x^2+1}+1} \left (-3 x^3+\left (3 \sqrt {x^2+1}+1\right ) x^2-\left (\sqrt {x^2+1}+1\right ) x+7 \left (\sqrt {x^2+1}-1\right )\right )}{15 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + Sqrt[1 + x^2]]*(-3*x^3 + 7*(-1 + Sqrt[1 + x^2]) - x*(1 + Sqrt[1 + x^2]) + x^2*(1 + 3*Sqrt[1 + x^2]
)))/(15*x)

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IntegrateAlgebraic [A]  time = 0.40, size = 104, normalized size = 1.00 \begin {gather*} \frac {2 x \left (11+3 x^2\right )}{15 \sqrt {1+\sqrt {1+x^2}}}-\frac {2}{15} \left (1+3 x^2\right ) \sqrt {1+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (\frac {8 x}{15 \sqrt {1+\sqrt {1+x^2}}}-\frac {2}{15} \sqrt {1+\sqrt {1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(11 + 3*x^2))/(15*Sqrt[1 + Sqrt[1 + x^2]]) - (2*(1 + 3*x^2)*Sqrt[1 + Sqrt[1 + x^2]])/15 + Sqrt[1 + x^2]*(
(8*x)/(15*Sqrt[1 + Sqrt[1 + x^2]]) - (2*Sqrt[1 + Sqrt[1 + x^2]])/15)

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fricas [A]  time = 0.62, size = 48, normalized size = 0.46 \begin {gather*} -\frac {2 \, {\left (3 \, x^{3} - x^{2} - {\left (3 \, x^{2} - x + 7\right )} \sqrt {x^{2} + 1} + x + 7\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{15 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(3*x^3 - x^2 - (3*x^2 - x + 7)*sqrt(x^2 + 1) + x + 7)*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 {\sqrt {\sqrt {x^{2} + 1} + 1}}{x + \sqrt {x^{2} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x^{2}+1}}}{x +\sqrt {x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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