\(\int \frac {1}{\sqrt {-1+x^2} (\sqrt {x}+\sqrt {-1+x^2})^2} \, dx\) [2888]

Optimal result

Integrand size = 27, antiderivative size = 313 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\frac {-2 \sqrt {x}+4 x^{3/2}+2 \sqrt {-1+x^2}-4 x \sqrt {-1+x^2}}{5 \left (-1-x+x^2\right )}+\frac {2}{5} \sqrt {\frac {2}{-1+\sqrt {5}}} \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {x}\right )-\frac {4}{5} \sqrt {\frac {2}{-5+5 \sqrt {5}}} \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {x}\right )-\frac {1}{5} \sqrt {-\frac {22}{5}+2 \sqrt {5}} \arctan \left (\frac {\sqrt {-1+x^2}}{\sqrt {2+\sqrt {5}} (1+x)}\right )-\frac {2}{5} \sqrt {\frac {2}{1+\sqrt {5}}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )-\frac {4}{5} \sqrt {\frac {2}{5+5 \sqrt {5}}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )+\frac {1}{5} \sqrt {\frac {22}{5}+2 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {-1+x^2}}{\sqrt {-2+\sqrt {5}} (1+x)}\right ) \]

[Out]

(-2*x^(1/2)+4*x^(3/2)+2*(x^2-1)^(1/2)-4*x*(x^2-1)^(1/2))/(5*x^2-5*x-5)+2/5*2^(1/2)/(5^(1/2)-1)^(1/2)*arctan(2^
(1/2)/(5^(1/2)-1)^(1/2)*x^(1/2))-4/5*2^(1/2)/(-5+5*5^(1/2))^(1/2)*arctan(2^(1/2)/(5^(1/2)-1)^(1/2)*x^(1/2))-1/
25*(-110+50*5^(1/2))^(1/2)*arctan((x^2-1)^(1/2)/(2+5^(1/2))^(1/2)/(1+x))-2/5*2^(1/2)/(5^(1/2)+1)^(1/2)*arctanh
(2^(1/2)/(5^(1/2)+1)^(1/2)*x^(1/2))-4/5*2^(1/2)/(5+5*5^(1/2))^(1/2)*arctanh(2^(1/2)/(5^(1/2)+1)^(1/2)*x^(1/2))
+1/25*(110+50*5^(1/2))^(1/2)*arctanh((x^2-1)^(1/2)/(-2+5^(1/2))^(1/2)/(1+x))

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.17, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6874, 750, 840, 1180, 213, 209, 1032, 1048, 739, 212, 210, 999} \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=-\frac {2}{5} \sqrt {\frac {1}{5} \left (5 \sqrt {5}-2\right )} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )+\sqrt {\frac {2}{5 \left (\sqrt {5}-1\right )}} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (\sqrt {5}-1\right )} \sqrt {x^2-1}}\right )+\frac {1}{5} \sqrt {\frac {2}{5} \left (5 \sqrt {5}-11\right )} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x}\right )-\frac {2}{5} \sqrt {\frac {1}{5} \left (2+5 \sqrt {5}\right )} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )+\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {x^2-1}}\right )-\frac {1}{5} \sqrt {\frac {2}{5} \left (11+5 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )-\frac {2 \sqrt {x^2-1} (1-2 x)}{5 \left (-x^2+x+1\right )}+\frac {2 \sqrt {x} (1-2 x)}{5 \left (-x^2+x+1\right )} \]

[In]

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

[Out]

(2*(1 - 2*x)*Sqrt[x])/(5*(1 + x - x^2)) - (2*(1 - 2*x)*Sqrt[-1 + x^2])/(5*(1 + x - x^2)) + (Sqrt[(2*(-11 + 5*S
qrt[5]))/5]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[x]])/5 + Sqrt[2/(5*(-1 + Sqrt[5]))]*ArcTan[(2 - (1 - Sqrt[5])*x
)/(Sqrt[2*(-1 + Sqrt[5])]*Sqrt[-1 + x^2])] - (2*Sqrt[(-2 + 5*Sqrt[5])/5]*ArcTan[(2 - (1 - Sqrt[5])*x)/(Sqrt[2*
(-1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5 - (Sqrt[(2*(11 + 5*Sqrt[5]))/5]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]])/5
+ Sqrt[2/(5*(1 + Sqrt[5]))]*ArcTanh[(2 - (1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])] - (2*Sqrt[(2
+ 5*Sqrt[5])/5]*ArcTanh[(2 - (1 + Sqrt[5])*x)/(Sqrt[2*(1 + Sqrt[5])]*Sqrt[-1 + x^2])])/5

Rule 209

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

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 213

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

Rule 739

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

Rule 750

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(b + 2*c
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 840

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 999

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

Rule 1032

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((g*c)*((-b
)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(2*a*f)) - h*(b*c*d + a
*b*f))*x), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + f
*x^2)^q*Simp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*((-b)*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*
(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((g*c)*((-b)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*
(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1)))*x - c*f*(b^2*(g*f
) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}
, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1
])

Rule 1048

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

Rule 1180

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt {x}}{\left (-1-x+x^2\right )^2}+\frac {2 x}{\sqrt {-1+x^2} \left (-1-x+x^2\right )^2}+\frac {1}{\sqrt {-1+x^2} \left (-1-x+x^2\right )}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt {x}}{\left (-1-x+x^2\right )^2} \, dx\right )+2 \int \frac {x}{\sqrt {-1+x^2} \left (-1-x+x^2\right )^2} \, dx+\int \frac {1}{\sqrt {-1+x^2} \left (-1-x+x^2\right )} \, dx \\ & = \frac {2 (1-2 x) \sqrt {x}}{5 \left (1+x-x^2\right )}-\frac {2 (1-2 x) \sqrt {-1+x^2}}{5 \left (1+x-x^2\right )}-\frac {2}{5} \int \frac {-\frac {1}{2}-x}{\sqrt {x} \left (-1-x+x^2\right )} \, dx+\frac {2}{5} \int \frac {-3-x}{\sqrt {-1+x^2} \left (-1-x+x^2\right )} \, dx+\frac {2 \int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {5}} \\ & = \frac {2 (1-2 x) \sqrt {x}}{5 \left (1+x-x^2\right )}-\frac {2 (1-2 x) \sqrt {-1+x^2}}{5 \left (1+x-x^2\right )}-\frac {4}{5} \text {Subst}\left (\int \frac {-\frac {1}{2}-x^2}{-1-x^2+x^4} \, dx,x,\sqrt {x}\right )-\frac {2 \text {Subst}\left (\int \frac {1}{-4+\left (-1-\sqrt {5}\right )^2-x^2} \, dx,x,\frac {-2-\left (-1-\sqrt {5}\right ) x}{\sqrt {-1+x^2}}\right )}{\sqrt {5}}+\frac {2 \text {Subst}\left (\int \frac {1}{-4+\left (-1+\sqrt {5}\right )^2-x^2} \, dx,x,\frac {-2-\left (-1+\sqrt {5}\right ) x}{\sqrt {-1+x^2}}\right )}{\sqrt {5}}-\frac {1}{25} \left (2 \left (5-7 \sqrt {5}\right )\right ) \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \sqrt {-1+x^2}} \, dx-\frac {1}{25} \left (2 \left (5+7 \sqrt {5}\right )\right ) \int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \sqrt {-1+x^2}} \, dx \\ & = \frac {2 (1-2 x) \sqrt {x}}{5 \left (1+x-x^2\right )}-\frac {2 (1-2 x) \sqrt {-1+x^2}}{5 \left (1+x-x^2\right )}+\sqrt {\frac {2}{5 \left (-1+\sqrt {5}\right )}} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (-1+\sqrt {5}\right )} \sqrt {-1+x^2}}\right )+\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {-1+x^2}}\right )+\frac {1}{25} \left (2 \left (5-7 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-4+\left (-1+\sqrt {5}\right )^2-x^2} \, dx,x,\frac {-2-\left (-1+\sqrt {5}\right ) x}{\sqrt {-1+x^2}}\right )+\frac {1}{25} \left (2 \left (5-2 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{25} \left (2 \left (5+2 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{25} \left (2 \left (5+7 \sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-4+\left (-1-\sqrt {5}\right )^2-x^2} \, dx,x,\frac {-2-\left (-1-\sqrt {5}\right ) x}{\sqrt {-1+x^2}}\right ) \\ & = \frac {2 (1-2 x) \sqrt {x}}{5 \left (1+x-x^2\right )}-\frac {2 (1-2 x) \sqrt {-1+x^2}}{5 \left (1+x-x^2\right )}+\frac {1}{5} \sqrt {\frac {2}{5} \left (-11+5 \sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {x}\right )+\sqrt {\frac {2}{5 \left (-1+\sqrt {5}\right )}} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (-1+\sqrt {5}\right )} \sqrt {-1+x^2}}\right )-\frac {2}{5} \sqrt {\frac {1}{5} \left (-2+5 \sqrt {5}\right )} \arctan \left (\frac {2-\left (1-\sqrt {5}\right ) x}{\sqrt {2 \left (-1+\sqrt {5}\right )} \sqrt {-1+x^2}}\right )-\frac {1}{5} \sqrt {\frac {2}{5} \left (11+5 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )+\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {-1+x^2}}\right )-\frac {2}{5} \sqrt {\frac {1}{5} \left (2+5 \sqrt {5}\right )} \text {arctanh}\left (\frac {2-\left (1+\sqrt {5}\right ) x}{\sqrt {2 \left (1+\sqrt {5}\right )} \sqrt {-1+x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 6.48 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\frac {1}{25} \left (-\frac {10 (-1+2 x) \left (-\sqrt {x}+\sqrt {-1+x^2}\right )}{-1-x+x^2}+\sqrt {-110+50 \sqrt {5}} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}\right )-\sqrt {-110+50 \sqrt {5}} \arctan \left (\frac {\sqrt {-2+\sqrt {5}} \sqrt {-1+x^2}}{1+x}\right )-\sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}\right )+\sqrt {110+50 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {5}} \sqrt {-1+x^2}}{1+x}\right )\right ) \]

[In]

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

[Out]

((-10*(-1 + 2*x)*(-Sqrt[x] + Sqrt[-1 + x^2]))/(-1 - x + x^2) + Sqrt[-110 + 50*Sqrt[5]]*ArcTan[Sqrt[(1 + Sqrt[5
])/2]*Sqrt[x]] - Sqrt[-110 + 50*Sqrt[5]]*ArcTan[(Sqrt[-2 + Sqrt[5]]*Sqrt[-1 + x^2])/(1 + x)] - Sqrt[110 + 50*S
qrt[5]]*ArcTanh[Sqrt[(-1 + Sqrt[5])/2]*Sqrt[x]] + Sqrt[110 + 50*Sqrt[5]]*ArcTanh[(Sqrt[2 + Sqrt[5]]*Sqrt[-1 +
x^2])/(1 + x)])/25

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1205\) vs. \(2(220)=440\).

Time = 0.27 (sec) , antiderivative size = 1206, normalized size of antiderivative = 3.85

method	result	size
default	\(\text {Expression too large to display}\)	\(1206\)

[In]

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

[Out]

-6/25*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))^(1/2)/(4
*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^(1/2))-6/25*5^(1/2)/(-2+2*5^(1/2))^(1/2)
*arctan(2*(1-5^(1/2)+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/
2)+1)*(x+1/2*5^(1/2)-1/2)+2-2*5^(1/2))^(1/2))+(2/5+2/5*5^(1/2))*(-1/4/(1/2+1/2*5^(1/2))/(x-1/2*5^(1/2)-1/2)*((
x-1/2*5^(1/2)-1/2)^2+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+1/2+1/2*5^(1/2))^(1/2)+1/4*(5^(1/2)+1)/(1/2+1/2*5^(1/2))/
(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))^(1/2)/(4*(x-1/2*5^(1/2
)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^(1/2)))+(2/5-2/5*5^(1/2))*(-1/4/(1/2-1/2*5^(1/2))/(x+1
/2*5^(1/2)-1/2)*((x+1/2*5^(1/2)-1/2)^2+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+1/2-1/2*5^(1/2))^(1/2)-1/4*(-5^(1/2)+1
)/(1/2-1/2*5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2))/(-2+2*5^(1/2))^
(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+2-2*5^(1/2))^(1/2)))+1/5*(5^(1/2)+1)^2*(-1/4
/(1/2+1/2*5^(1/2))/(x-1/2*5^(1/2)-1/2)*((x-1/2*5^(1/2)-1/2)^2+(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+1/2+1/2*5^(1/2))
^(1/2)+1/4*(5^(1/2)+1)/(1/2+1/2*5^(1/2))/(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2)+1)*(x-1/2*5^(1/2)-1
/2))/(2+2*5^(1/2))^(1/2)/(4*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5^(1/2))^(1/2)))+1/5*(
5^(1/2)-1)^2*(-1/4/(1/2-1/2*5^(1/2))/(x+1/2*5^(1/2)-1/2)*((x+1/2*5^(1/2)-1/2)^2+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/
2)+1/2-1/2*5^(1/2))^(1/2)-1/4*(-5^(1/2)+1)/(1/2-1/2*5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)+(-5^(1/2
)+1)*(x+1/2*5^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+2-2
*5^(1/2))^(1/2)))+1/5/(1/2+1/2*5^(1/2))/(x-1/2*5^(1/2)-1/2)*((x-1/2*5^(1/2)-1/2)^2+(5^(1/2)+1)*(x-1/2*5^(1/2)-
1/2)+1/2+1/2*5^(1/2))^(1/2)-1/5*(5^(1/2)+1)/(1/2+1/2*5^(1/2))/(2+2*5^(1/2))^(1/2)*arctanh(2*(1+5^(1/2)+(5^(1/2
)+1)*(x-1/2*5^(1/2)-1/2))/(2+2*5^(1/2))^(1/2)/(4*(x-1/2*5^(1/2)-1/2)^2+4*(5^(1/2)+1)*(x-1/2*5^(1/2)-1/2)+2+2*5
^(1/2))^(1/2))+1/5/(1/2-1/2*5^(1/2))/(x+1/2*5^(1/2)-1/2)*((x+1/2*5^(1/2)-1/2)^2+(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/
2)+1/2-1/2*5^(1/2))^(1/2)+1/5*(-5^(1/2)+1)/(1/2-1/2*5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*(1-5^(1/2)+(-5^(1/2
)+1)*(x+1/2*5^(1/2)-1/2))/(-2+2*5^(1/2))^(1/2)/(4*(x+1/2*5^(1/2)-1/2)^2+4*(-5^(1/2)+1)*(x+1/2*5^(1/2)-1/2)+2-2
*5^(1/2))^(1/2))+2/5*x^(1/2)/(x-1/2*5^(1/2)-1/2)-8/25*(5/2+5^(1/2))/(2+2*5^(1/2))^(1/2)*arctanh(2*x^(1/2)/(2+2
*5^(1/2))^(1/2))+2/5*x^(1/2)/(x+1/2*5^(1/2)-1/2)-8/25*(-5/2+5^(1/2))/(-2+2*5^(1/2))^(1/2)*arctan(2*x^(1/2)/(-2
+2*5^(1/2))^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (217) = 434\).

Time = 0.27 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=-\frac {\sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} - 4 \, x + 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} + 4 \, \sqrt {x}\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (-\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} - 4 \, x + 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {10 \, \sqrt {5} + 22} \log \left (-\sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )} + 4 \, \sqrt {x}\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} - 4 \, x - 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} + 4 \, \sqrt {x}\right ) - \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} - 4 \, x - 2 \, \sqrt {5} + 4 \, \sqrt {x^{2} - 1} + 2\right ) + \sqrt {5} {\left (x^{2} - x - 1\right )} \sqrt {-10 \, \sqrt {5} + 22} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-10 \, \sqrt {5} + 22} + 4 \, \sqrt {x}\right ) + 40 \, x^{2} + 20 \, \sqrt {x^{2} - 1} {\left (2 \, x - 1\right )} - 20 \, {\left (2 \, x - 1\right )} \sqrt {x} - 40 \, x - 40}{50 \, {\left (x^{2} - x - 1\right )}} \]

[In]

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

[Out]

-1/50*(sqrt(5)*(x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(sqrt(10*sqrt(5) + 22)*(sqrt(5) - 3) - 4*x + 2*sqrt(5) +
4*sqrt(x^2 - 1) + 2) - sqrt(5)*(x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(sqrt(10*sqrt(5) + 22)*(sqrt(5) - 3) +
4*sqrt(x)) - sqrt(5)*(x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(-sqrt(10*sqrt(5) + 22)*(sqrt(5) - 3) - 4*x + 2*sq
rt(5) + 4*sqrt(x^2 - 1) + 2) + sqrt(5)*(x^2 - x - 1)*sqrt(10*sqrt(5) + 22)*log(-sqrt(10*sqrt(5) + 22)*(sqrt(5)
- 3) + 4*sqrt(x)) + sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log((sqrt(5) + 3)*sqrt(-10*sqrt(5) + 22) - 4
*x - 2*sqrt(5) + 4*sqrt(x^2 - 1) + 2) - sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log((sqrt(5) + 3)*sqrt(-1
0*sqrt(5) + 22) + 4*sqrt(x)) - sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log(-(sqrt(5) + 3)*sqrt(-10*sqrt(5
) + 22) - 4*x - 2*sqrt(5) + 4*sqrt(x^2 - 1) + 2) + sqrt(5)*(x^2 - x - 1)*sqrt(-10*sqrt(5) + 22)*log(-(sqrt(5)
+ 3)*sqrt(-10*sqrt(5) + 22) + 4*sqrt(x)) + 40*x^2 + 20*sqrt(x^2 - 1)*(2*x - 1) - 20*(2*x - 1)*sqrt(x) - 40*x -
40)/(x^2 - x - 1)

Sympy [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (\sqrt {x} + \sqrt {x^{2} - 1}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\int { \frac {1}{\sqrt {x^{2} - 1} {\left (\sqrt {x^{2} - 1} + \sqrt {x}\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 1.45 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {x}+\sqrt {-1+x^2}\right )^2} \, dx=\frac {2}{5} \, \sqrt {\frac {1}{10}} \sqrt {5 \, \sqrt {5} - 11} \arctan \left (\frac {2 \, x + \sqrt {5} - 2 \, \sqrt {x^{2} - 1} - 1}{\sqrt {2 \, \sqrt {5} - 2}}\right ) + \frac {1}{5} \, \sqrt {\frac {1}{10}} \sqrt {5 \, \sqrt {5} + 11} \log \left ({\left | -153040 \, x + 22956 \, \sqrt {5} \sqrt {50 \, \sqrt {5} + 110} + 76520 \, \sqrt {5} + 153040 \, \sqrt {x^{2} - 1} - 38260 \, \sqrt {50 \, \sqrt {5} + 110} + 76520 \right |}\right ) - \frac {1}{5} \, \sqrt {\frac {1}{10}} \sqrt {5 \, \sqrt {5} + 11} \log \left ({\left | -153040 \, x - 22956 \, \sqrt {5} \sqrt {50 \, \sqrt {5} + 110} + 76520 \, \sqrt {5} + 153040 \, \sqrt {x^{2} - 1} + 38260 \, \sqrt {50 \, \sqrt {5} + 110} + 76520 \right |}\right ) + \frac {1}{25} \, \sqrt {50 \, \sqrt {5} - 110} \arctan \left (\frac {\sqrt {x}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{50} \, \sqrt {50 \, \sqrt {5} + 110} \log \left (\sqrt {x} + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) + \frac {1}{50} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | \sqrt {x} - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {4 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{3} + 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 3 \, x - 3 \, \sqrt {x^{2} - 1} - 2\right )}}{5 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{4} - 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{3} - 2 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 2 \, x + 2 \, \sqrt {x^{2} - 1} + 1\right )}} + \frac {2 \, {\left (2 \, x^{\frac {3}{2}} - \sqrt {x}\right )}}{5 \, {\left (x^{2} - x - 1\right )}} \]

[In]

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

[Out]

2/5*sqrt(1/10)*sqrt(5*sqrt(5) - 11)*arctan((2*x + sqrt(5) - 2*sqrt(x^2 - 1) - 1)/sqrt(2*sqrt(5) - 2)) + 1/5*sq
rt(1/10)*sqrt(5*sqrt(5) + 11)*log(abs(-153040*x + 22956*sqrt(5)*sqrt(50*sqrt(5) + 110) + 76520*sqrt(5) + 15304
0*sqrt(x^2 - 1) - 38260*sqrt(50*sqrt(5) + 110) + 76520)) - 1/5*sqrt(1/10)*sqrt(5*sqrt(5) + 11)*log(abs(-153040
*x - 22956*sqrt(5)*sqrt(50*sqrt(5) + 110) + 76520*sqrt(5) + 153040*sqrt(x^2 - 1) + 38260*sqrt(50*sqrt(5) + 110
) + 76520)) + 1/25*sqrt(50*sqrt(5) - 110)*arctan(sqrt(x)/sqrt(1/2*sqrt(5) - 1/2)) - 1/50*sqrt(50*sqrt(5) + 110
)*log(sqrt(x) + sqrt(1/2*sqrt(5) + 1/2)) + 1/50*sqrt(50*sqrt(5) + 110)*log(abs(sqrt(x) - sqrt(1/2*sqrt(5) + 1/
2))) + 4/5*((x - sqrt(x^2 - 1))^3 + 2*(x - sqrt(x^2 - 1))^2 + 3*x - 3*sqrt(x^2 - 1) - 2)/((x - sqrt(x^2 - 1))^
4 - 2*(x - sqrt(x^2 - 1))^3 - 2*(x - sqrt(x^2 - 1))^2 - 2*x + 2*sqrt(x^2 - 1) + 1) + 2/5*(2*x^(3/2) - sqrt(x))
/(x^2 - x - 1)

Mupad [F(-1)]

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

[In]

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

[Out]

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