\(\int \sqrt {-1+x} \sqrt {1+x} \, dx\) [843]

Optimal result

Integrand size = 15, antiderivative size = 26 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\frac {\text {arccosh}(x)}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {38, 54} \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {1}{2} \sqrt {x-1} x \sqrt {x+1}-\frac {\text {arccosh}(x)}{2} \]

[In]

Int[Sqrt[-1 + x]*Sqrt[1 + x],x]

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 - ArcCosh[x]/2

Rule 38

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

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\frac {1}{2} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx \\ & = \frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\frac {1}{2} \cosh ^{-1}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {1}{2} \sqrt {-1+x} x \sqrt {1+x}-\text {arctanh}\left (\frac {\sqrt {1+x}}{\sqrt {-1+x}}\right ) \]

[In]

Integrate[Sqrt[-1 + x]*Sqrt[1 + x],x]

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 - ArcTanh[Sqrt[1 + x]/Sqrt[-1 + x]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(18)=36\).

Time = 1.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77

method	result	size
risch	\(\frac {x \sqrt {-1+x}\, \sqrt {1+x}}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (-1+x \right ) \left (1+x \right )}}{2 \sqrt {-1+x}\, \sqrt {1+x}}\)	\(46\)
default	\(\frac {\sqrt {-1+x}\, \left (1+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {-1+x}\, \sqrt {1+x}}{2}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (-1+x \right ) \left (1+x \right )}}{2 \sqrt {-1+x}\, \sqrt {1+x}}\)	\(57\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x + \frac {1}{2} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.72 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.04 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\begin {cases} - \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} - \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {\sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} + \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {i \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {1}{2} \, \sqrt {x^{2} - 1} x - \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \]

[In]

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

[Out]

1/2*sqrt(x^2 - 1)*x - 1/2*log(2*x + 2*sqrt(x^2 - 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (18) = 36\).

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} {\left (x - 2\right )} + \sqrt {x + 1} \sqrt {x - 1} + \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \sqrt {-1+x} \sqrt {1+x} \, dx=\frac {x\,\sqrt {x-1}\,\sqrt {x+1}}{2}-\frac {\ln \left (x+\sqrt {x-1}\,\sqrt {x+1}\right )}{2} \]

[In]

int((x - 1)^(1/2)*(x + 1)^(1/2),x)

[Out]

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