3.14.6 \(\int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx\)

Optimal. Leaf size=56 \[ \frac {\sqrt {x-1}}{8 (x+1)}-\frac {\sqrt {x-1}}{2 (x+1)^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {47, 51, 63, 203} \begin {gather*} \frac {\sqrt {x-1}}{8 (x+1)}-\frac {\sqrt {x-1}}{2 (x+1)^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + x]/(1 + x)^3,x]

[Out]

-Sqrt[-1 + x]/(2*(1 + x)^2) + Sqrt[-1 + x]/(8*(1 + x)) + ArcTan[Sqrt[-1 + x]/Sqrt[2]]/(8*Sqrt[2])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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])

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {1}{4} \int \frac {1}{\sqrt {-1+x} (1+x)^2} \, dx\\ &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {1}{16} \int \frac {1}{\sqrt {-1+x} (1+x)} \, dx\\ &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 28, normalized size = 0.50 \begin {gather*} \frac {1}{12} (x-1)^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {1-x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + x]/(1 + x)^3,x]

[Out]

((-1 + x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (1 - x)/2])/12

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.06, size = 43, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x-1} (x-3)}{8 (x+1)^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 + x]/(1 + x)^3,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.12, size = 46, normalized size = 0.82 \begin {gather*} \frac {\sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + 2 \, \sqrt {x - 1} {\left (x - 3\right )}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^(1/2)/(1+x)^3,x, algorithm="fricas")

[Out]

1/16*(sqrt(2)*(x^2 + 2*x + 1)*arctan(1/2*sqrt(2)*sqrt(x - 1)) + 2*sqrt(x - 1)*(x - 3))/(x^2 + 2*x + 1)

________________________________________________________________________________________

giac [A]  time = 1.04, size = 37, normalized size = 0.66 \begin {gather*} \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + \frac {{\left (x - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x - 1}}{8 \, {\left (x + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^(1/2)/(1+x)^3,x, algorithm="giac")

[Out]

1/16*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x - 1)) + 1/8*((x - 1)^(3/2) - 2*sqrt(x - 1))/(x + 1)^2

________________________________________________________________________________________

maple [A]  time = 0.01, size = 40, normalized size = 0.71 \begin {gather*} \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x -1}\, \sqrt {2}}{2}\right )}{16}+\frac {\frac {\left (x -1\right )^{\frac {3}{2}}}{8}-\frac {\sqrt {x -1}}{4}}{\left (x +1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-1)^(1/2)/(x+1)^3,x)

[Out]

2*(1/16*(x-1)^(3/2)-1/8*(x-1)^(1/2))/(x+1)^2+1/16*2^(1/2)*arctan(1/2*(x-1)^(1/2)*2^(1/2))

________________________________________________________________________________________

maxima [A]  time = 3.03, size = 43, normalized size = 0.77 \begin {gather*} \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + \frac {{\left (x - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x - 1}}{8 \, {\left ({\left (x - 1\right )}^{2} + 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^(1/2)/(1+x)^3,x, algorithm="maxima")

[Out]

1/16*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x - 1)) + 1/8*((x - 1)^(3/2) - 2*sqrt(x - 1))/((x - 1)^2 + 4*x)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 45, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-1}}{2}\right )}{16}-\frac {\frac {\sqrt {x-1}}{4}-\frac {{\left (x-1\right )}^{3/2}}{8}}{4\,x+{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)^(1/2)/(x + 1)^3,x)

[Out]

(2^(1/2)*atan((2^(1/2)*(x - 1)^(1/2))/2))/16 - ((x - 1)^(1/2)/4 - (x - 1)^(3/2)/8)/(4*x + (x - 1)^2)

________________________________________________________________________________________

sympy [A]  time = 2.61, size = 167, normalized size = 2.98 \begin {gather*} \begin {cases} \frac {\sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{16} - \frac {i}{8 \sqrt {-1 + \frac {2}{x + 1}} \sqrt {x + 1}} + \frac {3 i}{4 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} - \frac {i}{\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {5}{2}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{16} + \frac {1}{8 \sqrt {1 - \frac {2}{x + 1}} \sqrt {x + 1}} - \frac {3}{4 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} + \frac {1}{\sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)**(1/2)/(1+x)**3,x)

[Out]

Piecewise((sqrt(2)*I*acosh(sqrt(2)/sqrt(x + 1))/16 - I/(8*sqrt(-1 + 2/(x + 1))*sqrt(x + 1)) + 3*I/(4*sqrt(-1 +
2/(x + 1))*(x + 1)**(3/2)) - I/(sqrt(-1 + 2/(x + 1))*(x + 1)**(5/2)), 2/Abs(x + 1) > 1), (-sqrt(2)*asin(sqrt(
2)/sqrt(x + 1))/16 + 1/(8*sqrt(1 - 2/(x + 1))*sqrt(x + 1)) - 3/(4*sqrt(1 - 2/(x + 1))*(x + 1)**(3/2)) + 1/(sqr
t(1 - 2/(x + 1))*(x + 1)**(5/2)), True))

________________________________________________________________________________________