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

Optimal. Leaf size=92 \[ 4 \sqrt {\sqrt {x+1}+1}-\frac {2}{5} \left (3 \sqrt {5}-5\right ) \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}}{\sqrt {5}-1}\right )-\frac {2}{5} \left (5+3 \sqrt {5}\right ) \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}}{1+\sqrt {5}}\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 114, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {824, 826, 1166, 207} \begin {gather*} 4 \sqrt {\sqrt {x+1}+1}-2 \sqrt {\frac {2}{5} \left (7+3 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )-2 \sqrt {\frac {2}{5} \left (7-3 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\sqrt {x+1}+1}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

4*Sqrt[1 + Sqrt[1 + x]] - 2*Sqrt[(2*(7 + 3*Sqrt[5]))/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*Sqrt[1 + Sqrt[1 + x]]] -
2*Sqrt[(2*(7 - 3*Sqrt[5]))/5]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*Sqrt[1 + Sqrt[1 + x]]]

Rule 207

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

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, 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] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {1+x}}{-1-x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}+2 \operatorname {Subst}\left (\int \frac {1+2 x}{\sqrt {1+x} \left (-1-x+x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}+4 \operatorname {Subst}\left (\int \frac {-1+2 x^2}{1-3 x^2+x^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}+\frac {1}{5} \left (4 \left (5-2 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (4 \left (5+2 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}-2 \sqrt {\frac {2}{5} \left (7+3 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{5} \sqrt {70-30 \sqrt {5}} \tanh ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 125, normalized size = 1.36 \begin {gather*} \frac {1}{5} \left (20 \sqrt {\sqrt {x+1}+1}+\sqrt {6-2 \sqrt {5}} \left (\sqrt {5}-5\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{3-\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )-\sqrt {2 \left (3+\sqrt {5}\right )} \left (5+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*Sqrt[1 + Sqrt[1 + x]] + Sqrt[6 - 2*Sqrt[5]]*(-5 + Sqrt[5])*ArcTanh[Sqrt[2/(3 - Sqrt[5])]*Sqrt[1 + Sqrt[1 +
x]]] - Sqrt[2*(3 + Sqrt[5])]*(5 + Sqrt[5])*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*Sqrt[1 + Sqrt[1 + x]]])/5

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IntegrateAlgebraic [A]  time = 0.15, size = 92, normalized size = 1.00 \begin {gather*} 4 \sqrt {1+\sqrt {1+x}}-\frac {2}{5} \left (5+3 \sqrt {5}\right ) \tanh ^{-1}\left (\frac {1}{2} \left (-1+\sqrt {5}\right ) \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{5} \left (-5+3 \sqrt {5}\right ) \tanh ^{-1}\left (\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt {1+\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*Sqrt[1 + Sqrt[1 + x]] - (2*(5 + 3*Sqrt[5])*ArcTanh[((-1 + Sqrt[5])*Sqrt[1 + Sqrt[1 + x]])/2])/5 - (2*(-5 + 3
*Sqrt[5])*ArcTanh[((1 + Sqrt[5])*Sqrt[1 + Sqrt[1 + x]])/2])/5

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fricas [B]  time = 0.47, size = 234, normalized size = 2.54 \begin {gather*} \frac {3}{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 ) + \frac {3}{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 ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*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)) + 3/5*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)) + 4*sqrt(sqrt(x + 1) + 1) - log(sqrt(x + 1) + sqrt(s
qrt(x + 1) + 1)) + log(sqrt(x + 1) - sqrt(sqrt(x + 1) + 1))

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giac [A]  time = 0.56, size = 136, normalized size = 1.48 \begin {gather*} \frac {3}{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 ) + \frac {3}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}\right ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + \log \left ({\left | \sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 97, normalized size = 1.05

method	result	size
derivativedivides	\(4 \sqrt {1+\sqrt {1+x}}-\ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}+\ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\)	\(97\)
default	\(4 \sqrt {1+\sqrt {1+x}}-\ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}+\ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(1+(1+x)^(1/2))^(1/2)-ln((1+x)^(1/2)+(1+(1+x)^(1/2))^(1/2))-6/5*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1/2)
+1)*5^(1/2))+ln((1+x)^(1/2)-(1+(1+x)^(1/2))^(1/2))-6/5*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 = 1.06, size = 132, normalized size = 1.43 \begin {gather*} \frac {3}{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 ) + \frac {3}{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 ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 12.35, size = 260, normalized size = 2.83 \begin {gather*} 4 \sqrt {\sqrt {x + 1} + 1} + 12 \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 ) + 12 \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 ) + \log {\left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right )} - \log {\left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(sqrt(x + 1) + 1) + 12*Piecewise((-sqrt(5)*acoth(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) - 1/2)/5)/10, (sqrt(sq
rt(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)) + 12*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)) + log(sqrt(x + 1) - sqrt(sqrt(x + 1) + 1)) - log(sqrt(x + 1) + sqrt(sqrt(x + 1)
+ 1))

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