3.1541 \(\int \frac{1}{\sqrt{-2-b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=29 \[ \frac{\sqrt{b x+2} \log (b x+2)}{b \sqrt{-b x-2}} \]

[Out]

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

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Rubi [A]  time = 0.0042249, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {23, 31} \[ \frac{\sqrt{b x+2} \log (b x+2)}{b \sqrt{-b x-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2-b x} \sqrt{2+b x}} \, dx &=\frac{\sqrt{2+b x} \int \frac{1}{2+b x} \, dx}{\sqrt{-2-b x}}\\ &=\frac{\sqrt{2+b x} \log (2+b x)}{b \sqrt{-2-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0083102, size = 28, normalized size = 0.97 \[ \frac{(b x+2) \log (b x+2)}{b \sqrt{-(b x+2)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 26, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( bx+2 \right ) }{b}\sqrt{bx+2}{\frac{1}{\sqrt{-bx-2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.928029, size = 22, normalized size = 0.76 \begin{align*} \sqrt{-\frac{1}{b^{2}}} \log \left (x + \frac{2}{b}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-1/b^2)*log(x + 2/b)

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Fricas [A]  time = 2.04833, size = 4, normalized size = 0.14 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [C]  time = 1.87655, size = 53, normalized size = 1.83 \begin{align*} \begin{cases} - \frac{i \log{\left (x + \frac{2}{b} \right )}}{b} & \text{for}\: \left |{x + \frac{2}{b}}\right | < 1 \\\frac{i \log{\left (\frac{1}{x + \frac{2}{b}} \right )}}{b} & \text{for}\: \frac{1}{\left |{x + \frac{2}{b}}\right |} < 1 \\\frac{i{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x + \frac{2}{b}} \right )}}{b} - \frac{i{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x + \frac{2}{b}} \right )}}{b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*log(x + 2/b)/b, Abs(x + 2/b) < 1), (I*log(1/(x + 2/b))/b, 1/Abs(x + 2/b) < 1), (I*meijerg(((), (
1, 1)), ((0, 0), ()), x + 2/b)/b - I*meijerg(((1, 1), ()), ((), (0, 0)), x + 2/b)/b, True))

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Giac [C]  time = 1.05501, size = 16, normalized size = 0.55 \begin{align*} -\frac{i \, \log \left ({\left | b x + 2 \right |}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(abs(b*x + 2))/b