3.16.41 \(\int \frac {1}{\sqrt {-2-b x} \sqrt {2+b x}} \, dx\) [1541]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {23, 31} \begin {gather*} \frac {\sqrt {b x+2} \log (b x+2)}{b \sqrt {-b x-2}} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.97 \begin {gather*} \frac {(2+b x) \log (2+b x)}{b \sqrt {-(2+b x)^2}} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.37, size = 153, normalized size = 5.28 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (\text {Log}\left [\frac {b}{2+b x}\right ]-\text {Log}\left [\frac {2+b x}{b}\right ]\right )}{b},\text {Abs}\left [\frac {2}{b}+x\right ]<1\text {\&\&}\text {Abs}\left [\frac {b}{2+b x}\right ]<1\right \},\left \{-\frac {I \text {Log}\left [\frac {2}{b}+x\right ]}{b},\text {Abs}\left [\frac {2}{b}+x\right ]<1\right \},\left \{\frac {I \text {Log}\left [\frac {b}{2+b x}\right ]}{b},\text {Abs}\left [\frac {b}{2+b x}\right ]<1\right \}\right \},-\frac {I \text {meijerg}\left [\left \{\left \{1,1\right \},\left \{\right \}\right \},\left \{\left \{\right \},\left \{0,0\right \}\right \},\frac {2}{b}+x\right ]}{b}+\frac {I \text {meijerg}\left [\left \{\left \{\right \},\left \{1,1\right \}\right \},\left \{\left \{0,0\right \},\left \{\right \}\right \},\frac {2}{b}+x\right ]}{b}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{I (Log[b / (2 + b x)] - Log[(2 + b x) / b]) / b, Abs[2 / b + x] < 1 && Abs[b / (2 + b x)] < 1}, {-
I Log[2 / b + x] / b, Abs[2 / b + x] < 1}, {I Log[b / (2 + b x)] / b, Abs[b / (2 + b x)] < 1}}, -I meijerg[{{1
, 1}, {}}, {{}, {0, 0}}, 2 / b + x] / b + I meijerg[{{}, {1, 1}}, {{0, 0}, {}}, 2 / b + x] / b]

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Maple [A]
time = 0.24, size = 26, normalized size = 0.90

method	result	size
default	\(\frac {\ln \left (b x +2\right ) \sqrt {b x +2}}{b \sqrt {-b x -2}}\)	\(26\)
meijerg	\(\frac {\sqrt {\mathrm {signum}\left (\frac {b x}{2}+1\right )}\, \ln \left (\frac {b x}{2}+1\right )}{\sqrt {-\mathrm {signum}\left (\frac {b x}{2}+1\right )}\, b}\)	\(32\)
risch	\(-\frac {i \sqrt {\frac {-b x -2}{b x +2}}\, \sqrt {b x +2}\, \ln \left (b x +2\right )}{\sqrt {-b x -2}\, b}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.55 \begin {gather*} \sqrt {-\frac {1}{b^{2}}} \log \left (x + \frac {2}{b}\right ) \end {gather*}

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 = 0.29, size = 1, normalized size = 0.03 \begin {gather*} 0 \end {gather*}

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] Result contains complex when optimal does not.
time = 1.14, size = 87, normalized size = 3.00 \begin {gather*} \begin {cases} \frac {i \log {\left (\frac {1}{x + \frac {2}{b}} \right )}}{b} - \frac {i \log {\left (x + \frac {2}{b} \right )}}{b} & \text {for}\: \frac {1}{\left |{x + \frac {2}{b}}\right |} < 1 \wedge \left |{x + \frac {2}{b}}\right | < 1 \\- \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 {gather*}

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(1/(x + 2/b))/b - I*log(x + 2/b)/b, (Abs(x + 2/b) < 1) & (1/Abs(x + 2/b) < 1)), (-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] Result contains complex when optimal does not.
time = 0.00, size = 17, normalized size = 0.59 \begin {gather*} -\frac {\mathrm {sign}\left (b\right ) \mathrm {sign}\left (x\right ) \mathrm {i} \ln \left |-b x-2\right |}{b} \end {gather*}

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]

-I*log(abs(-b*x - 2))*sgn(b)*sgn(x)/b

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Mupad [B]
time = 0.07, size = 47, normalized size = 1.62 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\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 + 2)^(1/2)*(- b*x - 2)^(1/2)),x)

[Out]

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

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