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

Optimal. Leaf size=434 \[ \frac {\sqrt {\sqrt {x^2+1}+x} x^2}{2 \left (x^2-1\right )}-\frac {\sqrt {x^2+1} \sqrt {\sqrt {x^2+1}+x} x}{2 \left (x^2-1\right )}+\frac {\left (\sqrt {2 \left (1+\sqrt {2}\right )} x^2-4 \sqrt {1+\sqrt {2}} x^2+\sqrt {2 \left (1+5 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )}{8 (x-1) (x+1)}+\frac {\left (\sqrt {2 \left (\sqrt {2}-1\right )} x^2+4 \sqrt {\sqrt {2}-1} x^2-\sqrt {2 \left (5 \sqrt {2}-1\right )}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )}{8 (x-1) (x+1)}+\frac {\left (\sqrt {2 \left (1+\sqrt {2}\right )} x^2-4 \sqrt {1+\sqrt {2}} x^2+\sqrt {2 \left (1+5 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {2}-1}}\right )}{8 (x-1) (x+1)}+\frac {\left (\sqrt {2 \left (\sqrt {2}-1\right )} x^2+4 \sqrt {\sqrt {2}-1} x^2-\sqrt {2 \left (5 \sqrt {2}-1\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {1+\sqrt {2}}}\right )}{8 (x-1) (x+1)} \]

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Rubi [A]  time = 0.80, antiderivative size = 463, normalized size of antiderivative = 1.07, number of steps used = 34, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6742, 2119, 1648, 12, 707, 1093, 204, 206, 207, 203, 6725, 1628, 828, 826, 1166} \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (-\left (\sqrt {x^2+1}+x\right )^2-2 \left (\sqrt {x^2+1}+x\right )+1\right )}+\frac {\sqrt {\sqrt {x^2+1}+x}}{2 \left (-\left (\sqrt {x^2+1}+x\right )^2+2 \left (\sqrt {x^2+1}+x\right )+1\right )}+\frac {\tan ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )-\frac {\tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {\sqrt {2}-1}}+\frac {\tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )-\frac {\tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}\right )}{2 \sqrt {\sqrt {2}-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

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

Rule 204

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

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 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 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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}, 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]

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 828

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 1093

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

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]

Rule 1628

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 1648

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[Po
lynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)*(f*(b*c*d
- b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x))/((p + 1)*(b^2 - 4*a*c)*(c*
d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x +
c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Q + f*(b*c*d*e*(2*p - m + 2) + b^2*e
^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d
- b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a,
b, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] &&  !
(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 2119

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 6725

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]

Rule 6742

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

Rubi steps

\begin {align*} \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \left (\frac {1}{4 (1-x)^2 \sqrt {x+\sqrt {1+x^2}}}+\frac {1}{4 (1+x)^2 \sqrt {x+\sqrt {1+x^2}}}+\frac {1}{2 \left (1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx\\ &=\frac {1}{4} \int \frac {1}{(1-x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x)^2 \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{2} \int \frac {1}{\left (1-x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=\frac {1}{2} \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+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1+2 x-x^2\right )^2} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+2 x+x^2\right )^2} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}-\frac {1}{16} \operatorname {Subst}\left (\int -\frac {4}{\sqrt {x} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{16} \operatorname {Subst}\left (\int -\frac {4}{\sqrt {x} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{4} \int \frac {1}{(1-x) \sqrt {x+\sqrt {1+x^2}}} \, dx+\frac {1}{4} \int \frac {1}{(1+x) \sqrt {x+\sqrt {1+x^2}}} \, dx\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{4} \operatorname {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 )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{4} \operatorname {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 )\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {1}{4} \operatorname {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 )+\frac {1}{4} \operatorname {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 )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x}{x^{3/2} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x}{x^{3/2} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{4 \sqrt {2}}\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x}{\sqrt {x} \left (1+2 x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1-x}{\sqrt {x} \left (-1+2 x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\operatorname {Subst}\left (\int \frac {-1+x^2}{1+2 x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\operatorname {Subst}\left (\int \frac {-1-x^2}{-1+2 x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1-2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}+\frac {\sqrt {x+\sqrt {1+x^2}}}{2 \left (1+2 \left (x+\sqrt {1+x^2}\right )-\left (x+\sqrt {1+x^2}\right )^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{2 \sqrt {-1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{2 \sqrt {-1+\sqrt {2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ \end {align*}

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Mathematica [A]  time = 0.84, size = 249, normalized size = 0.57 \begin {gather*} \frac {1}{8} \left (-\frac {4 x \left (\sqrt {x^2+1}+x\right )^{3/2}}{\left (x^2-1\right ) \left (2 x^2+2 \sqrt {x^2+1} x+1\right )}+\left (\sqrt {2}-3\right ) \sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}}\right )+\sqrt {2 \left (\sqrt {2}-1\right )} \left (3+\sqrt {2}\right ) \tan ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}}\right )-\left (\sqrt {2}-3\right ) \sqrt {2 \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {1}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+x}}\right )-\sqrt {2 \left (\sqrt {2}-1\right )} \left (3+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+x}}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-4*x*(x + Sqrt[1 + x^2])^(3/2))/((-1 + x^2)*(1 + 2*x^2 + 2*x*Sqrt[1 + x^2])) + (-3 + Sqrt[2])*Sqrt[2*(1 + Sq
rt[2])]*ArcTan[1/(Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]])] + Sqrt[2*(-1 + Sqrt[2])]*(3 + Sqrt[2])*ArcTan[1
/(Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]])] - (-3 + Sqrt[2])*Sqrt[2*(1 + Sqrt[2])]*ArcTanh[1/(Sqrt[-1 + Sqrt
[2]]*Sqrt[x + Sqrt[1 + x^2]])] - Sqrt[2*(-1 + Sqrt[2])]*(3 + Sqrt[2])*ArcTanh[1/(Sqrt[1 + Sqrt[2]]*Sqrt[x + Sq
rt[1 + x^2]])])/8

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IntegrateAlgebraic [A]  time = 3.50, size = 262, normalized size = 0.60 \begin {gather*} \frac {x^2 \sqrt {x+\sqrt {1+x^2}}}{2 \left (-1+x^2\right )}-\frac {x \sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}}{2 \left (-1+x^2\right )}+\frac {1}{4} \sqrt {\frac {1}{2} \left (-1+5 \sqrt {2}\right )} \tan ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (1+5 \sqrt {2}\right )} \tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )+\frac {1}{4} \sqrt {\frac {1}{2} \left (-1+5 \sqrt {2}\right )} \tanh ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right )-\frac {1}{4} \sqrt {\frac {1}{2} \left (1+5 \sqrt {2}\right )} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[x + Sqrt[1 + x^2]])/(2*(-1 + x^2)) - (x*Sqrt[1 + x^2]*Sqrt[x + Sqrt[1 + x^2]])/(2*(-1 + x^2)) + (Sqr
t[(-1 + 5*Sqrt[2])/2]*ArcTan[Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4 - (Sqrt[(1 + 5*Sqrt[2])/2]*ArcTan[
Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4 + (Sqrt[(-1 + 5*Sqrt[2])/2]*ArcTanh[Sqrt[-1 + Sqrt[2]]*Sqrt[x +
Sqrt[1 + x^2]]])/4 - (Sqrt[(1 + 5*Sqrt[2])/2]*ArcTanh[Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x^2]]])/4

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fricas [A]  time = 0.74, size = 399, normalized size = 0.92 \begin {gather*} \frac {4 \, \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} + 1} \arctan \left (\frac {1}{7} \, \sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} - 1} \sqrt {5 \, \sqrt {2} + 1} {\left (2 \, \sqrt {2} + 1\right )} - \frac {1}{7} \, \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {5 \, \sqrt {2} + 1} {\left (2 \, \sqrt {2} + 1\right )}\right ) - 4 \, \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} - 1} \arctan \left (\frac {1}{7} \, \sqrt {x + \sqrt {2} + \sqrt {x^{2} + 1} + 1} \sqrt {5 \, \sqrt {2} - 1} {\left (2 \, \sqrt {2} - 1\right )} - \frac {1}{7} \, \sqrt {x + \sqrt {x^{2} + 1}} \sqrt {5 \, \sqrt {2} - 1} {\left (2 \, \sqrt {2} - 1\right )}\right ) + \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} - 1} \log \left (\sqrt {5 \, \sqrt {2} - 1} {\left (\sqrt {2} + 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} - 1} \log \left (-\sqrt {5 \, \sqrt {2} - 1} {\left (\sqrt {2} + 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (\sqrt {5 \, \sqrt {2} + 1} {\left (\sqrt {2} - 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {2} {\left (x^{2} - 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (-\sqrt {5 \, \sqrt {2} + 1} {\left (\sqrt {2} - 3\right )} + 7 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + 8 \, {\left (x^{2} - \sqrt {x^{2} + 1} x\right )} \sqrt {x + \sqrt {x^{2} + 1}}}{16 \, {\left (x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) + 1)*arctan(1/7*sqrt(x + sqrt(2) + sqrt(x^2 + 1) - 1)*sqrt(5*sqrt(2)
+ 1)*(2*sqrt(2) + 1) - 1/7*sqrt(x + sqrt(x^2 + 1))*sqrt(5*sqrt(2) + 1)*(2*sqrt(2) + 1)) - 4*sqrt(2)*(x^2 - 1)*
sqrt(5*sqrt(2) - 1)*arctan(1/7*sqrt(x + sqrt(2) + sqrt(x^2 + 1) + 1)*sqrt(5*sqrt(2) - 1)*(2*sqrt(2) - 1) - 1/7
*sqrt(x + sqrt(x^2 + 1))*sqrt(5*sqrt(2) - 1)*(2*sqrt(2) - 1)) + sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) - 1)*log(sqrt
(5*sqrt(2) - 1)*(sqrt(2) + 3) + 7*sqrt(x + sqrt(x^2 + 1))) - sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) - 1)*log(-sqrt(5
*sqrt(2) - 1)*(sqrt(2) + 3) + 7*sqrt(x + sqrt(x^2 + 1))) + sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) + 1)*log(sqrt(5*sq
rt(2) + 1)*(sqrt(2) - 3) + 7*sqrt(x + sqrt(x^2 + 1))) - sqrt(2)*(x^2 - 1)*sqrt(5*sqrt(2) + 1)*log(-sqrt(5*sqrt
(2) + 1)*(sqrt(2) - 3) + 7*sqrt(x + sqrt(x^2 + 1))) + 8*(x^2 - sqrt(x^2 + 1)*x)*sqrt(x + sqrt(x^2 + 1)))/(x^2
- 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{2}-1\right )^{2} \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 1\right )}^{2} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (x^2-1\right )}^2\,\sqrt {x+\sqrt {x^2+1}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \sqrt {x + \sqrt {x^{2} + 1}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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