\(\int \frac {1}{(-1+x^2) \sqrt {x+\sqrt {1+x^2}}} \, dx\) [2208]

Optimal result

Integrand size = 23, antiderivative size = 163 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-\sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6857, 2144, 1642, 842, 840, 1180, 210, 212, 213, 209} \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {\arctan \left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\arctan \left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}}-\frac {\text {arctanh}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{\sqrt {\sqrt {2}-1}} \]

[In]

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

[Out]

-(ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[1 + Sqrt[2]]) + ArcTan[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sq
rt[1 + x^2]]]/Sqrt[-1 + Sqrt[2]] - ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[1 + Sqrt[2]] + Arc
Tanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]/Sqrt[-1 + Sqrt[2]]

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 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 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

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 1642

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]

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(
m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt
[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \]

[In]

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

[Out]

-(Sqrt[-1 + Sqrt[2]]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]) + Sqrt[1 + Sqrt[2]]*ArcTan[Sqrt[1 + S
qrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] - Sqrt[-1 + Sqrt[2]]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]] + Sq
rt[1 + Sqrt[2]]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]]

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (115) = 230\).

Time = 0.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \log \left (-\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} + 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} + 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \frac {1}{2} \, \sqrt {-\sqrt {2} - 1} \log \left ({\left (\sqrt {2} - 1\right )} \sqrt {-\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \frac {1}{2} \, \sqrt {-\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} - 1\right )} \sqrt {-\sqrt {2} - 1} + \sqrt {x + \sqrt {x^{2} + 1}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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