### 3.229 $$\int \frac{\sqrt{-1+x}}{1+x} \, dx$$

Optimal. Leaf size=31 $2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )$

[Out]

2*Sqrt[-1 + x] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]]

________________________________________________________________________________________

Rubi [A]  time = 0.0080735, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.231, Rules used = {50, 63, 203} $2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[-1 + x]/(1 + x),x]

[Out]

2*Sqrt[-1 + x] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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} \, dx &=2 \sqrt{-1+x}-2 \int \frac{1}{\sqrt{-1+x} (1+x)} \, dx\\ &=2 \sqrt{-1+x}-4 \operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\sqrt{-1+x}\right )\\ &=2 \sqrt{-1+x}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{-1+x}}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0068531, size = 31, normalized size = 1. $2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[-1 + x]/(1 + x),x]

[Out]

2*Sqrt[-1 + x] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*} -2\,\arctan \left ( 1/2\,\sqrt{-1+x}\sqrt{2} \right ) \sqrt{2}+2\,\sqrt{-1+x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.40636, size = 32, normalized size = 1.03 \begin{align*} -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.87988, size = 81, normalized size = 2.61 \begin{align*} -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.21423, size = 76, normalized size = 2.45 \begin{align*} \begin{cases} 2 \sqrt{x - 1} + 2 \sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 i \sqrt{1 - x} + \sqrt{2} i \log{\left (x + 1 \right )} - 2 \sqrt{2} i \log{\left (\sqrt{\frac{1}{2} - \frac{x}{2}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*sqrt(x - 1) + 2*sqrt(2)*asin(sqrt(2)/sqrt(x + 1)), Abs(x + 1)/2 > 1), (2*I*sqrt(1 - x) + sqrt(2)*
I*log(x + 1) - 2*sqrt(2)*I*log(sqrt(1/2 - x/2) + 1), True))

________________________________________________________________________________________

Giac [A]  time = 1.04958, size = 32, normalized size = 1.03 \begin{align*} -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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