3.16.28 \(\int \frac {1}{\sqrt {-3+b x} \sqrt {1+b x}} \, dx\) [1528]

Optimal. Leaf size=19 \[ \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {-3+b x}\right )}{b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, 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 {1}{2} \sqrt {b x-3}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-3 + b*x]*Sqrt[1 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[-3 + b*x]/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 {-3+b x} \sqrt {1+b x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {4+x^2}} \, dx,x,\sqrt {-3+b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {-3+b x}\right )}{b}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {1+b x}}{\sqrt {-3+b x}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-3 + b*x]*Sqrt[1 + b*x]),x]

[Out]

(2*ArcTanh[Sqrt[1 + b*x]/Sqrt[-3 + b*x]])/b

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(15)=30\).
time = 0.16, size = 66, normalized size = 3.47

method	result	size
default	\(\frac {\sqrt {\left (b x -3\right ) \left (b x +1\right )}\, \ln \left (\frac {b^{2} x -b}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}-2 b x -3}\right )}{\sqrt {b x -3}\, \sqrt {b x +1}\, \sqrt {b^{2}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x-3)^(1/2)/(b*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((b*x-3)*(b*x+1))^(1/2)/(b*x-3)^(1/2)/(b*x+1)^(1/2)*ln((b^2*x-b)/(b^2)^(1/2)+(b^2*x^2-2*b*x-3)^(1/2))/(b^2)^(1
/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
time = 0.29, size = 33, normalized size = 1.74 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 2 \, b x - 3} b - 2 \, b\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*b^2*x + 2*sqrt(b^2*x^2 - 2*b*x - 3)*b - 2*b)/b

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Fricas [A]
time = 0.63, size = 27, normalized size = 1.42 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + 1} \sqrt {b x - 3} + 1\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-b*x + sqrt(b*x + 1)*sqrt(b*x - 3) + 1)/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b x - 3} \sqrt {b x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(b*x - 3)*sqrt(b*x + 1)), x)

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Giac [A]
time = 1.19, size = 23, normalized size = 1.21 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + 1} - \sqrt {b x - 3}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(sqrt(b*x + 1) - sqrt(b*x - 3))/b

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Mupad [B]
time = 0.29, size = 46, normalized size = 2.42 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {b\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{\left (\sqrt {b\,x+1}-1\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)*(b*x - 3)^(1/2)),x)

[Out]

(4*atan((b*(3^(1/2)*1i - (b*x - 3)^(1/2)))/(((b*x + 1)^(1/2) - 1)*(-b^2)^(1/2))))/(-b^2)^(1/2)

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