Integrand size = 27, antiderivative size = 54 \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{\sqrt {1-a^2}} \]
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Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {95, 214} \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\sqrt {1-a^2}} \]
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Rule 95
Rule 214
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{-1-a-(-1+a) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}}\right ) \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{\sqrt {1-a^2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.63 \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=\frac {2 (-1+a+b x)^{3/2} (1+a+b x)^{3/2} \arctan \left (\frac {\sqrt {1-a^2} \sqrt {\frac {-1+a+b x}{1+a+b x}}}{-1+a}\right )}{\sqrt {1-a^2} (-((-1+a+b x) (1+a+b x)))^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.73 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {\sqrt {-b x -a +1}\, \sqrt {b x +a +1}\, \ln \left (-\frac {2 \left (a b x +a^{2}-\sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-1\right )}{x}\right ) \sqrt {-a^{2}+1}\, \operatorname {csgn}\left (b \right )^{2}}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a -1\right ) \left (1+a \right )}\) | \(114\) |
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none
Time = 0.24 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.33 \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=\left [-\frac {\sqrt {-a^{2} + 1} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} - 4 \, a^{2} + 2}{x^{2}}\right )}{2 \, {\left (a^{2} - 1\right )}}, \frac {\arctan \left (\frac {{\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} \sqrt {b x + a + 1} \sqrt {-b x - a + 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right )}{\sqrt {a^{2} - 1}}\right ] \]
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\[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=\int \frac {1}{x \sqrt {- a - b x + 1} \sqrt {a + b x + 1}}\, dx \]
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Exception generated. \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 3.53 (sec) , antiderivative size = 260, normalized size of antiderivative = 4.81 \[ \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx=\frac {\ln \left (\frac {2\,a\,\left (\sqrt {1-a}-\sqrt {1-b\,x-a}\right )}{\sqrt {a+1}-\sqrt {a+b\,x+1}}-\sqrt {1-a}\,\sqrt {a+1}+\frac {\sqrt {1-a}\,\sqrt {a+1}\,{\left (\sqrt {1-a}-\sqrt {1-b\,x-a}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}\right )-\ln \left (\frac {8\,\left (\sqrt {1-a}-\sqrt {1-b\,x-a}\right )}{\sqrt {a+1}-\sqrt {a+b\,x+1}}-a\,\sqrt {1-a}\,\sqrt {a+1}+a\,{\left (1-a\right )}^{3/2}\,{\left (a+1\right )}^{3/2}-\frac {8\,a^2\,\left (\sqrt {1-a}-\sqrt {1-b\,x-a}\right )}{\sqrt {a+1}-\sqrt {a+b\,x+1}}+a^3\,\sqrt {1-a}\,\sqrt {a+1}\right )}{\sqrt {1-a}\,\sqrt {a+1}} \]
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