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

Optimal. Leaf size=147 \[ -16 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3-18 \text {$\#$1}^2+20 \text {$\#$1}+25\& ,\frac {\text {$\#$1} \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2-9 \text {$\#$1}+5}\& \right ]+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}-\frac {7}{4} \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]

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Rubi [A]  time = 0.83, antiderivative size = 217, normalized size of antiderivative = 1.48, number of steps used = 13, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1586, 6725, 742, 640, 621, 206, 984, 724, 204} \begin {gather*} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (4-\sqrt {2}\right ) \sqrt {x+1}+2 \left (1+\sqrt {2}\right )}{2 \sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x+\sqrt {x+1}}}\right )+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\sqrt {2 \left (\sqrt {2}-1\right )} \tanh ^{-1}\left (\frac {\left (4+\sqrt {2}\right ) \sqrt {x+1}+2 \left (1-\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[x + Sqrt[1 + x]])/2 + Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + Sqrt[2*(1 + Sqrt[2])]*ArcTan[(2*(1 + Sqrt[2
]) + (4 - Sqrt[2])*Sqrt[1 + x])/(2*Sqrt[2*(-1 + Sqrt[2])]*Sqrt[x + Sqrt[1 + x]])] + (7*ArcTanh[(1 + 2*Sqrt[1 +
x])/(2*Sqrt[x + Sqrt[1 + x]])])/4 - Sqrt[2*(-1 + Sqrt[2])]*ArcTanh[(2*(1 - Sqrt[2]) + (4 + Sqrt[2])*Sqrt[1 +
x])/(2*Sqrt[2*(1 + Sqrt[2])]*Sqrt[x + Sqrt[1 + x]])]

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 621

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

Rule 640

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

Rule 724

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 742

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

Rule 984

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx &=\int \frac {1+x^2}{(-1+x) \sqrt {1+x} \sqrt {x+\sqrt {1+x}}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1+\left (-1+x^2\right )^2}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x+x^2}}+\frac {2}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+4 \operatorname {Subst}\left (\int \frac {1}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+2 \operatorname {Subst}\left (\int \frac {1}{\left (-2-\sqrt {2} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{\left (-2+\sqrt {2} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {1-\frac {3 x}{2}}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 \operatorname {Subst}\left (\int \frac {1}{8-8 \sqrt {2}-x^2} \, dx,x,\frac {2+2 \sqrt {2}-\left (-4+\sqrt {2}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-4 \operatorname {Subst}\left (\int \frac {1}{8+8 \sqrt {2}-x^2} \, dx,x,\frac {2-2 \sqrt {2}+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \left (1+\sqrt {2}\right )+\left (4-\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {2 \left (1-\sqrt {2}\right )+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )+\frac {7}{2} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \left (1+\sqrt {2}\right )+\left (4-\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )+\frac {7}{4} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {2 \left (1-\sqrt {2}\right )+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 209, normalized size = 1.42 \begin {gather*} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (2 \sqrt {2}-1\right ) \sqrt {x+1}+\sqrt {2}+2}{2 \sqrt {\sqrt {2}-1} \sqrt {x+\sqrt {x+1}}}\right )+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\sqrt {2 \left (\sqrt {2}-1\right )} \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {2}\right ) \sqrt {x+1}\right )-\sqrt {2}+2}{2 \sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[x + Sqrt[1 + x]])/2 + Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + Sqrt[2*(1 + Sqrt[2])]*ArcTan[(2 + Sqrt[2] +
(-1 + 2*Sqrt[2])*Sqrt[1 + x])/(2*Sqrt[-1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x]])] + (7*ArcTanh[(1 + 2*Sqrt[1 + x])/
(2*Sqrt[x + Sqrt[1 + x]])])/4 + Sqrt[2*(-1 + Sqrt[2])]*ArcTanh[(2 - Sqrt[2] - (1 + 2*Sqrt[2])*Sqrt[1 + x])/(2*
Sqrt[1 + Sqrt[2]]*Sqrt[x + Sqrt[1 + x]])]

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IntegrateAlgebraic [A]  time = 0.38, size = 157, normalized size = 1.07 \begin {gather*} \frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-2 \text {RootSum}\left [-1+8 \text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}}{2-3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x + Sqrt[1 + x]]*(-3 + 2*Sqrt[1 + x]))/2 - (7*Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/4 - 2*R
ootSum[-1 + 8*#1 - 6*#1^2 + #1^4 & , (-Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1] + 2*Log[-Sqrt[1 + x] + S
qrt[x + Sqrt[1 + x]] - #1]*#1)/(2 - 3*#1 + #1^3) & ]

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fricas [B]  time = 4.54, size = 378, normalized size = 2.57 \begin {gather*} -2 \, \sqrt {2} \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {\sqrt {5} \sqrt {2} {\left (28 \, x^{2} + \sqrt {2} {\left (7 \, x^{2} + 2 \, x - 9\right )} + 2 \, {\left (\sqrt {2} {\left (x - 13\right )} - 10 \, x + 10\right )} \sqrt {x + 1} - 28 \, x - 48\right )} \sqrt {\sqrt {2} + 1} \sqrt {\sqrt {2} - 1} + 2 \, \sqrt {2} {\left (8 \, \sqrt {2} {\left (x + 2\right )} - {\left (3 \, \sqrt {2} {\left (7 \, x - 11\right )} - 14 \, x + 62\right )} \sqrt {x + 1} + 18 \, x + 6\right )} \sqrt {x + \sqrt {x + 1}} \sqrt {\sqrt {2} + 1}}{2 \, {\left (49 \, x^{2} - 74 \, x - 119\right )}}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} + 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} - 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + \frac {7}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(2)*sqrt(sqrt(2) + 1)*arctan(-1/2*(sqrt(5)*sqrt(2)*(28*x^2 + sqrt(2)*(7*x^2 + 2*x - 9) + 2*(sqrt(2)*(x
- 13) - 10*x + 10)*sqrt(x + 1) - 28*x - 48)*sqrt(sqrt(2) + 1)*sqrt(sqrt(2) - 1) + 2*sqrt(2)*(8*sqrt(2)*(x + 2)
- (3*sqrt(2)*(7*x - 11) - 14*x + 62)*sqrt(x + 1) + 18*x + 6)*sqrt(x + sqrt(x + 1))*sqrt(sqrt(2) + 1))/(49*x^2
- 74*x - 119)) - 1/2*sqrt(2)*sqrt(sqrt(2) - 1)*log(-2*(sqrt(2)*(sqrt(2)*(5*x + 3) + 2*sqrt(x + 1)*(3*sqrt(2)
+ 4) + 6*x + 6)*sqrt(sqrt(2) - 1) + 4*(sqrt(x + 1)*(sqrt(2) + 1) + sqrt(2) + 2)*sqrt(x + sqrt(x + 1)))/(x - 1)
) + 1/2*sqrt(2)*sqrt(sqrt(2) - 1)*log(2*(sqrt(2)*(sqrt(2)*(5*x + 3) + 2*sqrt(x + 1)*(3*sqrt(2) + 4) + 6*x + 6)
*sqrt(sqrt(2) - 1) - 4*(sqrt(x + 1)*(sqrt(2) + 1) + sqrt(2) + 2)*sqrt(x + sqrt(x + 1)))/(x - 1)) + 1/2*sqrt(x
+ sqrt(x + 1))*(2*sqrt(x + 1) - 3) + 7/8*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) + 8*x + 8*sqrt(x + 1)
+ 5)

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giac [B]  time = 0.94, size = 208, normalized size = 1.41 \begin {gather*} \frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (\frac {\sqrt {2} - \sqrt {x + \sqrt {x + 1}} + \sqrt {x + 1}}{\sqrt {\sqrt {2} - 1}}\right ) - 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | 10 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 20 \, \sqrt {2} + 20 \, \sqrt {x + \sqrt {x + 1}} - 20 \, \sqrt {x + 1} + 20 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | -2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 4 \, \sqrt {2} + 4 \, \sqrt {x + \sqrt {x + 1}} - 4 \, \sqrt {x + 1} - 4 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) - \frac {7}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) - 3) + 4*sqrt(1/2*sqrt(2) + 1/2)*arctan((sqrt(2) - sqrt(x + sqrt(x +
1)) + sqrt(x + 1))/sqrt(sqrt(2) - 1)) - 2*sqrt(1/2*sqrt(2) - 1/2)*log(abs(10*sqrt(2)*sqrt(2*sqrt(2) - 2) + 20*
sqrt(2) + 20*sqrt(x + sqrt(x + 1)) - 20*sqrt(x + 1) + 20*sqrt(2*sqrt(2) - 2))) + 2*sqrt(1/2*sqrt(2) - 1/2)*log
(abs(-2*sqrt(2)*sqrt(2*sqrt(2) - 2) + 4*sqrt(2) + 4*sqrt(x + sqrt(x + 1)) - 4*sqrt(x + 1) - 4*sqrt(2*sqrt(2) -
2))) - 7/4*log(-2*sqrt(x + sqrt(x + 1)) + 2*sqrt(x + 1) + 1)

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maple [B]  time = 0.18, size = 217, normalized size = 1.48

method	result	size
derivativedivides	\(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {\sqrt {2}\, \arctanh \left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\)	\(217\)
default	\(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {\sqrt {2}\, \arctanh \left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+x)^(1/2)*(x+(1+x)^(1/2))^(1/2)-3/2*(x+(1+x)^(1/2))^(1/2)+7/4*ln(1/2+(1+x)^(1/2)+(x+(1+x)^(1/2))^(1/2))-2^(1
/2)/(2^(1/2)-1)^(1/2)*arctan(1/2*(2-2*2^(1/2)+(1-2*2^(1/2))*((1+x)^(1/2)+2^(1/2)))/(2^(1/2)-1)^(1/2)/(((1+x)^(
1/2)+2^(1/2))^2+(1-2*2^(1/2))*((1+x)^(1/2)+2^(1/2))+1-2^(1/2))^(1/2))-2^(1/2)/(1+2^(1/2))^(1/2)*arctanh(1/2*(2
+2*2^(1/2)+(1+2*2^(1/2))*((1+x)^(1/2)-2^(1/2)))/(1+2^(1/2))^(1/2)/(((1+x)^(1/2)-2^(1/2))^2+(1+2*2^(1/2))*((1+x
)^(1/2)-2^(1/2))+1+2^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2+1\right )\,\sqrt {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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