3.16.46 \(\int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx\) [1546]

Optimal. Leaf size=20 \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \]

[Out]

-2*arcsinh(1/2*(-b*x)^(1/2)*2^(1/2))/b

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {65, 221} \begin {gather*} -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-(b*x)]*Sqrt[2 - b*x]),x]

[Out]

(-2*ArcSinh[Sqrt[-(b*x)]/Sqrt[2]])/b

Rule 65

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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-b x} \sqrt {2-b x}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,\sqrt {-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x}}{\sqrt {2}}\right )}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(20)=40\).
time = 0.01, size = 48, normalized size = 2.40 \begin {gather*} -\frac {2 \sqrt {x} \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{\sqrt {-b} \sqrt {-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x]*Log[-(Sqrt[-b]*Sqrt[x]) + Sqrt[2 - b*x]])/(Sqrt[-b]*Sqrt[-(b*x)])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(17)=34\).
time = 0.14, size = 64, normalized size = 3.20

method	result	size
meijerg	\(\frac {2 \sqrt {x}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}\, \sqrt {-b x}}\)	\(27\)
default	\(\frac {\sqrt {-b x \left (-b x +2\right )}\, \ln \left (\frac {b^{2} x -b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}-2 b x}\right )}{\sqrt {-b x}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 32, normalized size = 1.60 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 2 \, b x} b - 2 \, b\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.97, size = 27, normalized size = 1.35 \begin {gather*} -\frac {\log \left (-b x + \sqrt {-b x + 2} \sqrt {-b x} + 1\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 51, normalized size = 2.55 \begin {gather*} \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b, Abs(b*x) > 2), (-2*I*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b, True
))

________________________________________________________________________________________

Giac [A]
time = 1.52, size = 23, normalized size = 1.15 \begin {gather*} \frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sqrt(-b*x + 2) - sqrt(-b*x))/b

________________________________________________________________________________________

Mupad [B]
time = 0.28, size = 39, normalized size = 1.95 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\sqrt {-b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*atan((b*(2^(1/2) - (2 - b*x)^(1/2)))/((-b*x)^(1/2)*(-b^2)^(1/2))))/(-b^2)^(1/2)

________________________________________________________________________________________