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

Optimal. Leaf size=91 \[ \frac {2 \sqrt {x-\sqrt {x^2+1}} (-x-2)+2 \sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{\sqrt {x^2+1}-x-1}-2 \tan ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right ) \]

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Rubi [A]  time = 0.44, antiderivative size = 83, normalized size of antiderivative = 0.91, number of steps used = 16, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6742, 2119, 457, 329, 298, 203, 206, 2120, 463, 12, 321, 212} \begin {gather*} \frac {\sqrt {x-\sqrt {x^2+1}}}{x}+2 \sqrt {x-\sqrt {x^2+1}}-\frac {1}{x \sqrt {x-\sqrt {x^2+1}}}-2 \tan ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 463

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

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 2120

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

Rule 6742

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

Rubi steps

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

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Mathematica [C]  time = 14.17, size = 1129, normalized size = 12.41 \begin {gather*} -\frac {159120 \left (x^2-\sqrt {x^2+1} x+1\right ) \left (\frac {16}{585} \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};\left (x-\sqrt {x^2+1}\right )^2\right ) \left (\left (x-\sqrt {x^2+1}\right )^3+x-\sqrt {x^2+1}\right )^2+\frac {681 \left (x-\sqrt {x^2+1}\right )^6-1483 \left (x-\sqrt {x^2+1}\right )^4-6769 \left (x-\sqrt {x^2+1}\right )^2+5 \left (\left (x-\sqrt {x^2+1}\right )^8-248 \left (x-\sqrt {x^2+1}\right )^6+102 \left (x-\sqrt {x^2+1}\right )^4+1208 \left (x-\sqrt {x^2+1}\right )^2+729\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\left (x-\sqrt {x^2+1}\right )^2\right )-3645}{640 \left (x-\sqrt {x^2+1}\right )^4}\right ) \left (x-\sqrt {x^2+1}\right )^{23/2}}{x \left (1-\frac {x}{\sqrt {x^2+1}}\right ) \left (\left (x-\sqrt {x^2+1}\right )^2+1\right ) \left (1989 \left (-217600 x^{12}+217600 \sqrt {x^2+1} x^{11}-540032 x^{10}+431232 \sqrt {x^2+1} x^9-565792 x^8+377376 \sqrt {x^2+1} x^7-331584 x^6+183200 \sqrt {x^2+1} x^5-99050 x^4+36170 \sqrt {x^2+1} x^3-8563 x^2+10 \left (-1024 x^{13}+1024 \sqrt {x^2+1} x^{12}+28672 x^{11}-29184 \sqrt {x^2+1} x^{10}+75840 x^9-61120 \sqrt {x^2+1} x^8+79168 x^7-52320 \sqrt {x^2+1} x^6+44684 x^5-24300 \sqrt {x^2+1} x^4+13240 x^3-4978 \sqrt {x^2+1} x^2+1335 x-182 \sqrt {x^2+1}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\left (x-\sqrt {x^2+1}\right )^2\right ) x+687 \sqrt {x^2+1} x+140\right )+4352 x \left (-106496 x^{15}+106496 \sqrt {x^2+1} x^{14}-409600 x^{13}+356352 \sqrt {x^2+1} x^{12}-632320 x^{11}+467456 \sqrt {x^2+1} x^{10}-499200 x^9+303360 \sqrt {x^2+1} x^8-212160 x^7+100800 \sqrt {x^2+1} x^6-46592 x^5+15904 \sqrt {x^2+1} x^4-4550 x^3+938 \sqrt {x^2+1} x^2-130 x+9 \sqrt {x^2+1}\right ) \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};\left (x-\sqrt {x^2+1}\right )^2\right )+40960 x \left (-32768 x^{17}+32768 \sqrt {x^2+1} x^{16}-147456 x^{15}+131072 \sqrt {x^2+1} x^{14}-274432 x^{13}+212992 \sqrt {x^2+1} x^{12}-272384 x^{11}+180224 \sqrt {x^2+1} x^{10}-154880 x^9+84480 \sqrt {x^2+1} x^8-50304 x^7+21504 \sqrt {x^2+1} x^6-8736 x^5+2688 \sqrt {x^2+1} x^4-688 x^3+128 \sqrt {x^2+1} x^2-16 x+\sqrt {x^2+1}\right ) \, _4F_3\left (\frac {9}{4},3,3,3;2,2,\frac {21}{4};\left (x-\sqrt {x^2+1}\right )^2\right )\right )}-\frac {\left (\left (x-\sqrt {x^2+1}\right )^2-1\right )^2 \left (-\frac {2 \left (x-\sqrt {x^2+1}\right )^{3/2}}{\left (x-\sqrt {x^2+1}\right )^2-1}+\tan ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right )-\tanh ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right )\right )}{4 x^2 \left (1-\frac {x}{\sqrt {x^2+1}}\right ) \left (1-\frac {\left (x-\sqrt {x^2+1}\right )^2-1}{2 \left (x-\sqrt {x^2+1}\right )^2}\right ) \left (x-\sqrt {x^2+1}\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*((-1 + (x - Sqrt[1 + x^2])^2)^2*((-2*(x - Sqrt[1 + x^2])^(3/2))/(-1 + (x - Sqrt[1 + x^2])^2) + ArcTan[Sqr
t[x - Sqrt[1 + x^2]]] - ArcTanh[Sqrt[x - Sqrt[1 + x^2]]]))/(x^2*(1 - x/Sqrt[1 + x^2])*(x - Sqrt[1 + x^2])^2*(1
- (-1 + (x - Sqrt[1 + x^2])^2)/(2*(x - Sqrt[1 + x^2])^2))) - (159120*(x - Sqrt[1 + x^2])^(23/2)*(1 + x^2 - x*
Sqrt[1 + x^2])*((-3645 - 6769*(x - Sqrt[1 + x^2])^2 - 1483*(x - Sqrt[1 + x^2])^4 + 681*(x - Sqrt[1 + x^2])^6 +
5*(729 + 1208*(x - Sqrt[1 + x^2])^2 + 102*(x - Sqrt[1 + x^2])^4 - 248*(x - Sqrt[1 + x^2])^6 + (x - Sqrt[1 + x
^2])^8)*Hypergeometric2F1[1/4, 1, 5/4, (x - Sqrt[1 + x^2])^2])/(640*(x - Sqrt[1 + x^2])^4) + (16*(x - Sqrt[1 +
x^2] + (x - Sqrt[1 + x^2])^3)^2*HypergeometricPFQ[{5/4, 2, 2, 2}, {1, 1, 17/4}, (x - Sqrt[1 + x^2])^2])/585))
/(x*(1 - x/Sqrt[1 + x^2])*(1 + (x - Sqrt[1 + x^2])^2)*(1989*(140 - 8563*x^2 - 99050*x^4 - 331584*x^6 - 565792*
x^8 - 540032*x^10 - 217600*x^12 + 687*x*Sqrt[1 + x^2] + 36170*x^3*Sqrt[1 + x^2] + 183200*x^5*Sqrt[1 + x^2] + 3
77376*x^7*Sqrt[1 + x^2] + 431232*x^9*Sqrt[1 + x^2] + 217600*x^11*Sqrt[1 + x^2] + 10*x*(1335*x + 13240*x^3 + 44
684*x^5 + 79168*x^7 + 75840*x^9 + 28672*x^11 - 1024*x^13 - 182*Sqrt[1 + x^2] - 4978*x^2*Sqrt[1 + x^2] - 24300*
x^4*Sqrt[1 + x^2] - 52320*x^6*Sqrt[1 + x^2] - 61120*x^8*Sqrt[1 + x^2] - 29184*x^10*Sqrt[1 + x^2] + 1024*x^12*S
qrt[1 + x^2])*Hypergeometric2F1[1/4, 1, 5/4, (x - Sqrt[1 + x^2])^2]) + 4352*x*(-130*x - 4550*x^3 - 46592*x^5 -
212160*x^7 - 499200*x^9 - 632320*x^11 - 409600*x^13 - 106496*x^15 + 9*Sqrt[1 + x^2] + 938*x^2*Sqrt[1 + x^2] +
15904*x^4*Sqrt[1 + x^2] + 100800*x^6*Sqrt[1 + x^2] + 303360*x^8*Sqrt[1 + x^2] + 467456*x^10*Sqrt[1 + x^2] + 3
56352*x^12*Sqrt[1 + x^2] + 106496*x^14*Sqrt[1 + x^2])*HypergeometricPFQ[{5/4, 2, 2, 2}, {1, 1, 17/4}, (x - Sqr
t[1 + x^2])^2] + 40960*x*(-16*x - 688*x^3 - 8736*x^5 - 50304*x^7 - 154880*x^9 - 272384*x^11 - 274432*x^13 - 14
7456*x^15 - 32768*x^17 + Sqrt[1 + x^2] + 128*x^2*Sqrt[1 + x^2] + 2688*x^4*Sqrt[1 + x^2] + 21504*x^6*Sqrt[1 + x
^2] + 84480*x^8*Sqrt[1 + x^2] + 180224*x^10*Sqrt[1 + x^2] + 212992*x^12*Sqrt[1 + x^2] + 131072*x^14*Sqrt[1 + x
^2] + 32768*x^16*Sqrt[1 + x^2])*HypergeometricPFQ[{9/4, 3, 3, 3}, {2, 2, 21/4}, (x - Sqrt[1 + x^2])^2]))

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IntegrateAlgebraic [A]  time = 0.10, size = 91, normalized size = 1.00 \begin {gather*} \frac {2 (-2-x) \sqrt {x-\sqrt {1+x^2}}+2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x+\sqrt {1+x^2}}-2 \tan ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 - x)*Sqrt[x - Sqrt[1 + x^2]] + 2*Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]])/(-1 - x + Sqrt[1 + x^2]) - 2*Ar
cTan[Sqrt[x - Sqrt[1 + x^2]]]

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fricas [A]  time = 0.60, size = 50, normalized size = 0.55 \begin {gather*} -\frac {2 \, x \arctan \left (\sqrt {x - \sqrt {x^{2} + 1}}\right ) - {\left (3 \, x + \sqrt {x^{2} + 1} + 1\right )} \sqrt {x - \sqrt {x^{2} + 1}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x*arctan(sqrt(x - sqrt(x^2 + 1))) - (3*x + sqrt(x^2 + 1) + 1)*sqrt(x - sqrt(x^2 + 1)))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x -\sqrt {x^{2}+1}}}{1-\sqrt {x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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