\(\int \frac {1+a x^2}{(-1+a x^2) \sqrt {x+\sqrt {1+x^2}}} \, dx\) [1886]

Optimal result

Integrand size = 32, antiderivative size = 130 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}+\text {RootSum}\left [a-4 \text {$\#$1}^4-2 a \text {$\#$1}^4+a \text {$\#$1}^8\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3-a \text {$\#$1}^3+a \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(263\) vs. \(2(130)=260\).

Time = 0.56 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.02, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6857, 2142, 14, 2144, 1642, 842, 840, 1180, 214, 211} \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {\sqrt {x^2+1}+x}}{\sqrt {1-\sqrt {a+1}}}\right )}{\sqrt [4]{a} \sqrt {1-\sqrt {a+1}}}-\frac {2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {a+1}+1}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+1}+1}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {\sqrt {x^2+1}+x}}{\sqrt {1-\sqrt {a+1}}}\right )}{\sqrt [4]{a} \sqrt {1-\sqrt {a+1}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt {a+1}+1}}\right )}{\sqrt [4]{a} \sqrt {\sqrt {a+1}+1}}+\sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}} \]

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 211

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

Rule 214

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

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 2142

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

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}{\sqrt {x+\sqrt {1+x^2}}}+\frac {2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx \\ & = 2 \int \frac {1}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx+\int \frac {1}{\sqrt {x+\sqrt {1+x^2}}} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x^{5/2}} \, dx,x,x+\sqrt {1+x^2}\right )+2 \int \left (-\frac {1}{2 \left (1-\sqrt {a} x\right ) \sqrt {x+\sqrt {1+x^2}}}-\frac {1}{2 \left (1+\sqrt {a} x\right ) \sqrt {x+\sqrt {1+x^2}}}\right ) \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\int \frac {1}{\left (1-\sqrt {a} x\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx-\int \frac {1}{\left (1+\sqrt {a} x\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (\sqrt {a}+2 x-\sqrt {a} x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\text {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (-\sqrt {a}+2 x+\sqrt {a} x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right ) \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\text {Subst}\left (\int \left (-\frac {1}{\sqrt {a} x^{3/2}}+\frac {2 \left (\sqrt {a}+x\right )}{\sqrt {a} x^{3/2} \left (\sqrt {a}+2 x-\sqrt {a} x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\text {Subst}\left (\int \left (\frac {1}{\sqrt {a} x^{3/2}}+\frac {2 \left (\sqrt {a}-x\right )}{\sqrt {a} x^{3/2} \left (-\sqrt {a}+2 x+\sqrt {a} x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right ) \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {a}+x}{x^{3/2} \left (\sqrt {a}+2 x-\sqrt {a} x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )}{\sqrt {a}}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {a}-x}{x^{3/2} \left (-\sqrt {a}+2 x+\sqrt {a} x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )}{\sqrt {a}} \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\frac {2 \text {Subst}\left (\int \frac {-\sqrt {a}+a x}{\sqrt {x} \left (\sqrt {a}+2 x-\sqrt {a} x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )}{a}+\frac {2 \text {Subst}\left (\int \frac {-\sqrt {a}-a x}{\sqrt {x} \left (-\sqrt {a}+2 x+\sqrt {a} x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )}{a} \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\frac {4 \text {Subst}\left (\int \frac {-\sqrt {a}+a x^2}{\sqrt {a}+2 x^2-\sqrt {a} x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{a}+\frac {4 \text {Subst}\left (\int \frac {-\sqrt {a}-a x^2}{-\sqrt {a}+2 x^2+\sqrt {a} x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )}{a} \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-2 \text {Subst}\left (\int \frac {1}{1-\sqrt {1+a}-\sqrt {a} x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{1+\sqrt {1+a}-\sqrt {a} x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{1-\sqrt {1+a}+\sqrt {a} x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-2 \text {Subst}\left (\int \frac {1}{1+\sqrt {1+a}+\sqrt {a} x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right ) \\ & = -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}-\frac {2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x+\sqrt {1+x^2}}}{\sqrt {1-\sqrt {1+a}}}\right )}{\sqrt [4]{a} \sqrt {1-\sqrt {1+a}}}-\frac {2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {1+a}}}\right )}{\sqrt [4]{a} \sqrt {1+\sqrt {1+a}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x+\sqrt {1+x^2}}}{\sqrt {1-\sqrt {1+a}}}\right )}{\sqrt [4]{a} \sqrt {1-\sqrt {1+a}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x+\sqrt {1+x^2}}}{\sqrt {1+\sqrt {1+a}}}\right )}{\sqrt [4]{a} \sqrt {1+\sqrt {1+a}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}+\text {RootSum}\left [a-4 \text {$\#$1}^4-2 a \text {$\#$1}^4+a \text {$\#$1}^8\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3-a \text {$\#$1}^3+a \text {$\#$1}^7}\&\right ] \]

[In]

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

[Out]

-1/3*1/(x + Sqrt[1 + x^2])^(3/2) + Sqrt[x + Sqrt[1 + x^2]] + RootSum[a - 4*#1^4 - 2*a*#1^4 + a*#1^8 & , (Log[S
qrt[x + Sqrt[1 + x^2]] - #1] + Log[Sqrt[x + Sqrt[1 + x^2]] - #1]*#1^4)/(-2*#1^3 - a*#1^3 + a*#1^7) & ]

Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.22

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.26 (sec) , antiderivative size = 721, normalized size of antiderivative = 5.55 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=-\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} + \sqrt {-\sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}}} \log \left (8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} - 1\right )} \sqrt {-\sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {-\sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}}} \log \left (-8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} - 1\right )} \sqrt {-\sqrt {\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \sqrt {-\sqrt {-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}}} \log \left (8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} + 1\right )} \sqrt {-\sqrt {-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \sqrt {-\sqrt {-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}}} \log \left (-8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} + 1\right )} \sqrt {-\sqrt {-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} - 1\right )} \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (-8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} - 1\right )} \left (\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} + a + 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) - \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} + 1\right )} \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) + \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} \log \left (-8 \, {\left (a^{3} \sqrt {\frac {a + 1}{a^{6}}} + 1\right )} \left (-\frac {2 \, a^{3} \sqrt {\frac {a + 1}{a^{6}}} - a - 2}{a^{3}}\right )^{\frac {1}{4}} + 8 \, \sqrt {x + \sqrt {x^{2} + 1}}\right ) \]

[In]

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

[Out]

-2/3*(x^2 - sqrt(x^2 + 1)*x - 1)*sqrt(x + sqrt(x^2 + 1)) + sqrt(-sqrt((2*a^3*sqrt((a + 1)/a^6) + a + 2)/a^3))*
log(8*(a^3*sqrt((a + 1)/a^6) - 1)*sqrt(-sqrt((2*a^3*sqrt((a + 1)/a^6) + a + 2)/a^3)) + 8*sqrt(x + sqrt(x^2 + 1
))) - sqrt(-sqrt((2*a^3*sqrt((a + 1)/a^6) + a + 2)/a^3))*log(-8*(a^3*sqrt((a + 1)/a^6) - 1)*sqrt(-sqrt((2*a^3*
sqrt((a + 1)/a^6) + a + 2)/a^3)) + 8*sqrt(x + sqrt(x^2 + 1))) - sqrt(-sqrt(-(2*a^3*sqrt((a + 1)/a^6) - a - 2)/
a^3))*log(8*(a^3*sqrt((a + 1)/a^6) + 1)*sqrt(-sqrt(-(2*a^3*sqrt((a + 1)/a^6) - a - 2)/a^3)) + 8*sqrt(x + sqrt(
x^2 + 1))) + sqrt(-sqrt(-(2*a^3*sqrt((a + 1)/a^6) - a - 2)/a^3))*log(-8*(a^3*sqrt((a + 1)/a^6) + 1)*sqrt(-sqrt
(-(2*a^3*sqrt((a + 1)/a^6) - a - 2)/a^3)) + 8*sqrt(x + sqrt(x^2 + 1))) + ((2*a^3*sqrt((a + 1)/a^6) + a + 2)/a^
3)^(1/4)*log(8*(a^3*sqrt((a + 1)/a^6) - 1)*((2*a^3*sqrt((a + 1)/a^6) + a + 2)/a^3)^(1/4) + 8*sqrt(x + sqrt(x^2
+ 1))) - ((2*a^3*sqrt((a + 1)/a^6) + a + 2)/a^3)^(1/4)*log(-8*(a^3*sqrt((a + 1)/a^6) - 1)*((2*a^3*sqrt((a + 1
)/a^6) + a + 2)/a^3)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) - (-(2*a^3*sqrt((a + 1)/a^6) - a - 2)/a^3)^(1/4)*log(8
*(a^3*sqrt((a + 1)/a^6) + 1)*(-(2*a^3*sqrt((a + 1)/a^6) - a - 2)/a^3)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1))) + (-(
2*a^3*sqrt((a + 1)/a^6) - a - 2)/a^3)^(1/4)*log(-8*(a^3*sqrt((a + 1)/a^6) + 1)*(-(2*a^3*sqrt((a + 1)/a^6) - a
- 2)/a^3)^(1/4) + 8*sqrt(x + sqrt(x^2 + 1)))

Sympy [N/A]

Not integrable

Time = 7.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.21 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {a x^{2} + 1}{\sqrt {x + \sqrt {x^{2} + 1}} \left (a x^{2} - 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.23 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {a x^{2} + 1}{{\left (a x^{2} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.23 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int { \frac {a x^{2} + 1}{{\left (a x^{2} - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.90 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.23 \[ \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {x+\sqrt {1+x^2}}} \, dx=\int \frac {a\,x^2+1}{\sqrt {x+\sqrt {x^2+1}}\,\left (a\,x^2-1\right )} \,d x \]

[In]

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

[Out]

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