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

Optimal. Leaf size=41 \[ 2 \sqrt {x+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {800, 618, 206} \[ 2 \sqrt {x+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + x] + (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 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2+\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {(-1+x) (1+x)^2}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (x-\frac {1}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-4 \operatorname {Subst}\left (\int \frac {1}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}+8 \operatorname {Subst}\left (\int \frac {1}{5-x^2} \, dx,x,1+2 \sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}+\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 = 0.07, size = 41, normalized size = 1.00 \[ 2 \sqrt {x+1}+\frac {8 \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}+1}{\sqrt {5}}\right )}{\sqrt {5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 101, normalized size = 2.46 \[ \frac {4}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \sqrt {x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [A]  time = 0.19, size = 51, normalized size = 1.24 \[ -\frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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) + 2

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maple [A]  time = 0.00, size = 34, normalized size = 0.83 \[ \frac {8 \sqrt {5}\, \arctanh \left (\frac {\left (1+2 \sqrt {1+\sqrt {x +1}}\right ) \sqrt {5}}{5}\right )}{5}+2 \sqrt {x +1}+2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.42, size = 51, normalized size = 1.24 \[ -\frac {4}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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) + 2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 21.08, size = 112, normalized size = 2.73 \[ 2 \sqrt {x + 1} - 16 \left (\begin {cases} - \frac {\sqrt {5} \operatorname {acoth}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} > \frac {5}{4} \\- \frac {\sqrt {5} \operatorname {atanh}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} < \frac {5}{4} \end {cases}\right ) + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1) - 16*Piecewise((-sqrt(5)*acoth(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) + 1/2)/5)/10, (sqrt(sqrt(x + 1)
+ 1) + 1/2)**2 > 5/4), (-sqrt(5)*atanh(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) + 1/2)/5)/10, (sqrt(sqrt(x + 1) + 1) +
1/2)**2 < 5/4)) + 2

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