Optimal. Leaf size=54 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\sqrt {1-a^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {93, 208} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{\sqrt {1-a^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx &=2 \operatorname {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*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 1.70 \begin {gather*} \frac {2 \sqrt {a+b x-1} \sqrt {a+b x+1} \tanh ^{-1}\left (\sqrt {\frac {a+1}{a-1}} \sqrt {\frac {a+b x-1}{a+b x+1}}\right )}{\sqrt {-a-1} \sqrt {1-a} \sqrt {-((a+b x-1) (a+b x+1))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 64, normalized size = 1.19 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {-a-b x+1}}{\sqrt {\frac {a}{a+1}-\frac {1}{a+1}} \sqrt {a+b x+1}}\right )}{\sqrt {\frac {a-1}{a+1}} (a+1)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 180, normalized size = 3.33 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 114, normalized size = 2.11 \begin {gather*} \frac {\sqrt {-b x -a +1}\, \sqrt {b x +a +1}\, \sqrt {-a^{2}+1}\, \mathrm {csgn}\relax (b )^{2} \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 {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a -1\right ) \left (a +1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.22, size = 260, normalized size = 4.81 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- a - b x + 1} \sqrt {a + b x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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