3.16.47 \(\int \frac {1}{\sqrt {-1-b x} \sqrt {2-b x}} \, dx\) [1547]

Optimal. Leaf size=22 \[ -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-1-b x}}{\sqrt {3}}\right )}{b} \]

[Out]

-2*arcsinh(1/3*(-b*x-1)^(1/2)*3^(1/2))/b

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {65, 221} \begin {gather*} -\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-b x-1}}{\sqrt {3}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcSinh[Sqrt[-1 - b*x]/Sqrt[3]])/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 {-1-b x} \sqrt {2-b x}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {3+x^2}} \, dx,x,\sqrt {-1-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\frac {\sqrt {-1-b x}}{\sqrt {3}}\right )}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 27, normalized size = 1.23 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2-b x}}{\sqrt {-1-b x}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded in comparison} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded in comparison

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(19)=38\).
time = 0.16, size = 70, normalized size = 3.18

method	result	size
default	\(\frac {\sqrt {\left (-b x -1\right ) \left (-b x +2\right )}\, \ln \left (\frac {-\frac {1}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}-b x -2}\right )}{\sqrt {-b x -1}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 33, normalized size = 1.50 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - b x - 2} b - b\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.29, size = 30, normalized size = 1.36 \begin {gather*} -\frac {\log \left (-2 \, b x + 2 \, \sqrt {-b x + 2} \sqrt {-b x - 1} + 1\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 27, normalized size = 1.23 \begin {gather*} \frac {2 \ln \left (\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)^(1/2)/(-b*x+2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.28, size = 46, normalized size = 2.09 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {-b\,x-1}-\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {2-b\,x}\right )\,\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)^(1/2)*(2 - b*x)^(1/2)),x)

[Out]

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

________________________________________________________________________________________