3.7.52 \(\int \frac {x}{x+\sqrt {1+\sqrt {1+x}}} \, dx\)

Optimal. Leaf size=51 \[ x-4 \sqrt {\sqrt {x+1}+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}}{\sqrt {5}}+\frac {1}{\sqrt {5}}\right )}{\sqrt {5}} \]

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Rubi [A]  time = 0.31, antiderivative size = 67, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1628, 618, 206} \begin {gather*} \left (\sqrt {x+1}+1\right )^2-4 \sqrt {\sqrt {x+1}+1}-2 \sqrt {x+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(x + Sqrt[1 + Sqrt[1 + x]]),x]

[Out]

-2*Sqrt[1 + x] - 4*Sqrt[1 + Sqrt[1 + x]] + (1 + Sqrt[1 + x])^2 + (8*ArcTanh[(1 + 2*Sqrt[1 + Sqrt[1 + x]])/Sqrt
[5]])/Sqrt[5]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 48, normalized size = 0.94 \begin {gather*} x-4 \sqrt {\sqrt {x+1}+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - 4*Sqrt[1 + Sqrt[1 + x]] + (8*ArcTanh[(1 + 2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

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IntegrateAlgebraic [A]  time = 0.09, size = 52, normalized size = 1.02 \begin {gather*} 1+x-4 \sqrt {1+\sqrt {1+x}}+\frac {8 \tanh ^{-1}\left (\frac {1}{\sqrt {5}}+\frac {2 \sqrt {1+\sqrt {1+x}}}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1 + x - 4*Sqrt[1 + Sqrt[1 + x]] + (8*ArcTanh[1/Sqrt[5] + (2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]

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fricas [B]  time = 0.46, size = 106, normalized size = 2.08 \begin {gather*} \frac {4}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + x - 4 \, \sqrt {\sqrt {x + 1} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*sqrt(5)*log((2*x^2 - sqrt(5)*(3*x + 1) - (sqrt(5)*(x + 2) - 5*x)*sqrt(x + 1) + (sqrt(5)*(x + 2) + (sqrt(5)
*(2*x - 1) - 5)*sqrt(x + 1) - 5*x)*sqrt(sqrt(x + 1) + 1) + 3*x + 3)/(x^2 - x - 1)) + x - 4*sqrt(sqrt(x + 1) +
1)

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giac [A]  time = 0.18, size = 71, normalized size = 1.39 \begin {gather*} {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) - 2 \, \sqrt {x + 1} - 4 \, \sqrt {\sqrt {x + 1} + 1} - 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x + 1) + 1)^2 - 4/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) +
1) + 1)) - 2*sqrt(x + 1) - 4*sqrt(sqrt(x + 1) + 1) - 2

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maple [A]  time = 0.05, size = 54, normalized size = 1.06

method	result	size
derivativedivides	\(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\frac {8 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\)	\(54\)
default	\(\left (1+\sqrt {1+x}\right )^{2}-2-2 \sqrt {1+x}-4 \sqrt {1+\sqrt {1+x}}+\frac {8 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x+(1+(1+x)^(1/2))^(1/2)),x,method=_RETURNVERBOSE)

[Out]

(1+(1+x)^(1/2))^2-2-2*(1+x)^(1/2)-4*(1+(1+x)^(1/2))^(1/2)+8/5*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1/2)+1)*
5^(1/2))

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maxima [A]  time = 0.43, size = 71, normalized size = 1.39 \begin {gather*} {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) - 2 \, \sqrt {x + 1} - 4 \, \sqrt {\sqrt {x + 1} + 1} - 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x + 1) + 1)^2 - 4/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) +
1) + 1)) - 2*sqrt(x + 1) - 4*sqrt(sqrt(x + 1) + 1) - 2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{x+\sqrt {\sqrt {x+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(x + ((x + 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}{x + \sqrt {\sqrt {x + 1} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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