\(\int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx\) [1853]

Optimal result

Integrand size = 17, antiderivative size = 127 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=-\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{4} \arctan \left (\frac {1+2 x-2 \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \left (3-2 \sqrt {x+\sqrt {1+x}}\right )}{2+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}}\right )-\frac {3}{2} \text {arctanh}\left (1-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}\right ) \]

[Out]

-(x+(1+x)^(1/2))^(1/2)/x-1/4*arctan((1+2*x-2*(x+(1+x)^(1/2))^(1/2)+(1+x)^(1/2)*(3-2*(x+(1+x)^(1/2))^(1/2)))/(2
+2*(1+x)^(1/2)-2*(x+(1+x)^(1/2))^(1/2)))+3/2*arctanh(-1+(1+x)^(1/2)-(x+(1+x)^(1/2))^(1/2))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1028, 1047, 738, 212, 210} \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=-\frac {1}{4} \arctan \left (\frac {\sqrt {x+1}+3}{2 \sqrt {x+\sqrt {x+1}}}\right )+\frac {3}{4} \text {arctanh}\left (\frac {1-3 \sqrt {x+1}}{2 \sqrt {x+\sqrt {x+1}}}\right )-\frac {\sqrt {x+\sqrt {x+1}}}{x} \]

[In]

Int[Sqrt[x + Sqrt[1 + x]]/x^2,x]

[Out]

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

Rule 210

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

Rule 212

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

Rule 738

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

Rule 1028

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(a*h - g*c*x)*(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/(2*a*c*(p + 1))), x] + Dist[2/(4*a*c*(p + 1)), Int[(a
+ c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[g*c*d*(2*p + 3) - a*(h*e*q) + (g*c*e*(2*p + q + 3) - a*(2*h*f*
q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ
[p, -1] && GtQ[q, 0]

Rule 1047

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(-a)*c, 2]}, Dist[h/2 + c*(g/(2*q)), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - c*(g/
(2*q)), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f
, 0] && PosQ[(-a)*c]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {\sqrt {x+\sqrt {1+x}}}{x}+\text {Subst}\left (\int \frac {\frac {1}{2}+x}{\left (-1+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {\sqrt {x+\sqrt {1+x}}}{x}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ & = -\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-\sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-\frac {3}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+3 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right ) \\ & = -\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{4} \arctan \left (\frac {3+\sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )+\frac {3}{4} \text {arctanh}\left (\frac {1-3 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=-\frac {\sqrt {x+\sqrt {1+x}}}{x}-\frac {1}{2} \arctan \left (1+\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}\right )-\frac {3}{2} \text {arctanh}\left (1-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(97)=194\).

Time = 0.05 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.35

method	result	size
derivativedivides	\(-\frac {\left (\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2\right )^{\frac {3}{2}}}{2 \left (-1+\sqrt {1+x}\right )}+\frac {3 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}}{4}+\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}\right )}{2}-\frac {3 \,\operatorname {arctanh}\left (\frac {-1+3 \sqrt {1+x}}{2 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}}\right )}{4}+\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}}{4}-\frac {\left (\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2\right )^{\frac {3}{2}}}{2 \left (1+\sqrt {1+x}\right )}-\frac {\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}-\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}\right )}{2}+\frac {\arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}\right )}{4}+\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}\)	\(298\)
default	\(-\frac {\left (\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2\right )^{\frac {3}{2}}}{2 \left (-1+\sqrt {1+x}\right )}+\frac {3 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}}{4}+\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}\right )}{2}-\frac {3 \,\operatorname {arctanh}\left (\frac {-1+3 \sqrt {1+x}}{2 \sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}}\right )}{4}+\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {\left (-1+\sqrt {1+x}\right )^{2}+3 \sqrt {1+x}-2}}{4}-\frac {\left (\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2\right )^{\frac {3}{2}}}{2 \left (1+\sqrt {1+x}\right )}-\frac {\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}-\frac {\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}\right )}{2}+\frac {\arctan \left (\frac {-3-\sqrt {1+x}}{2 \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}\right )}{4}+\frac {\left (2 \sqrt {1+x}+1\right ) \sqrt {\left (1+\sqrt {1+x}\right )^{2}-\sqrt {1+x}-2}}{4}\)	\(298\)

[In]

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

[Out]

-1/2/(-1+(1+x)^(1/2))*((-1+(1+x)^(1/2))^2+3*(1+x)^(1/2)-2)^(3/2)+3/4*((-1+(1+x)^(1/2))^2+3*(1+x)^(1/2)-2)^(1/2
)+1/2*ln(1/2+(1+x)^(1/2)+((-1+(1+x)^(1/2))^2+3*(1+x)^(1/2)-2)^(1/2))-3/4*arctanh(1/2*(-1+3*(1+x)^(1/2))/((-1+(
1+x)^(1/2))^2+3*(1+x)^(1/2)-2)^(1/2))+1/4*(2*(1+x)^(1/2)+1)*((-1+(1+x)^(1/2))^2+3*(1+x)^(1/2)-2)^(1/2)-1/2/(1+
(1+x)^(1/2))*((1+(1+x)^(1/2))^2-(1+x)^(1/2)-2)^(3/2)-1/4*((1+(1+x)^(1/2))^2-(1+x)^(1/2)-2)^(1/2)-1/2*ln(1/2+(1
+x)^(1/2)+((1+(1+x)^(1/2))^2-(1+x)^(1/2)-2)^(1/2))+1/4*arctan(1/2*(-3-(1+x)^(1/2))/((1+(1+x)^(1/2))^2-(1+x)^(1
/2)-2)^(1/2))+1/4*(2*(1+x)^(1/2)+1)*((1+(1+x)^(1/2))^2-(1+x)^(1/2)-2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 1.93 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=\frac {x \arctan \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} - 3\right )}}{x - 8}\right ) + 3 \, x \log \left (\frac {2 \, \sqrt {x + \sqrt {x + 1}} {\left (\sqrt {x + 1} + 1\right )} - 3 \, x - 2 \, \sqrt {x + 1} - 2}{x}\right ) - 4 \, \sqrt {x + \sqrt {x + 1}}}{4 \, x} \]

[In]

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

[Out]

1/4*(x*arctan(2*sqrt(x + sqrt(x + 1))*(sqrt(x + 1) - 3)/(x - 8)) + 3*x*log((2*sqrt(x + sqrt(x + 1))*(sqrt(x +
1) + 1) - 3*x - 2*sqrt(x + 1) - 2)/x) - 4*sqrt(x + sqrt(x + 1)))/x

Sympy [F]

\[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=\int \frac {\sqrt {x + \sqrt {x + 1}}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=\int { \frac {\sqrt {x + \sqrt {x + 1}}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=-\frac {2 \, {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{3} - 3 \, {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{2} - \sqrt {x + \sqrt {x + 1}} + \sqrt {x + 1} + 1}{{\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{4} - 2 \, {\left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1}\right )}^{2} + 4 \, \sqrt {x + \sqrt {x + 1}} - 4 \, \sqrt {x + 1}} + \frac {1}{2} \, \arctan \left (\sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} - 1\right ) - \frac {3}{4} \, \log \left ({\left | \sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} + 2 \right |}\right ) + \frac {3}{4} \, \log \left ({\left | \sqrt {x + \sqrt {x + 1}} - \sqrt {x + 1} \right |}\right ) \]

[In]

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

[Out]

-(2*(sqrt(x + sqrt(x + 1)) - sqrt(x + 1))^3 - 3*(sqrt(x + sqrt(x + 1)) - sqrt(x + 1))^2 - sqrt(x + sqrt(x + 1)
) + sqrt(x + 1) + 1)/((sqrt(x + sqrt(x + 1)) - sqrt(x + 1))^4 - 2*(sqrt(x + sqrt(x + 1)) - sqrt(x + 1))^2 + 4*
sqrt(x + sqrt(x + 1)) - 4*sqrt(x + 1)) + 1/2*arctan(sqrt(x + sqrt(x + 1)) - sqrt(x + 1) - 1) - 3/4*log(abs(sqr
t(x + sqrt(x + 1)) - sqrt(x + 1) + 2)) + 3/4*log(abs(sqrt(x + sqrt(x + 1)) - sqrt(x + 1)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x+\sqrt {1+x}}}{x^2} \, dx=\int \frac {\sqrt {x+\sqrt {x+1}}}{x^2} \,d x \]

[In]

int((x + (x + 1)^(1/2))^(1/2)/x^2,x)

[Out]

int((x + (x + 1)^(1/2))^(1/2)/x^2, x)