Optimal. Leaf size=32 \[ -2 \sqrt {1-x^2}+2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )-2 \log (x) \]
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Rubi [A] time = 0.20, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6688, 6742, 266, 50, 63, 206} \begin {gather*} -2 \sqrt {1-x^2}+2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )-2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 266
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right )}{x} \, dx &=-\int \frac {\left (\sqrt {1-x}+\sqrt {1+x}\right )^2}{x} \, dx\\ &=-\int \left (\frac {2}{x}+\frac {2 \sqrt {1-x^2}}{x}\right ) \, dx\\ &=-2 \log (x)-2 \int \frac {\sqrt {1-x^2}}{x} \, dx\\ &=-2 \log (x)-\operatorname {Subst}\left (\int \frac {\sqrt {1-x}}{x} \, dx,x,x^2\right )\\ &=-2 \sqrt {1-x^2}-2 \log (x)-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-2 \sqrt {1-x^2}-2 \log (x)+2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-2 \sqrt {1-x^2}+2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )-2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} -2 \sqrt {1-x^2}+2 \tanh ^{-1}\left (\sqrt {1-x^2}\right )-2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.17, size = 75, normalized size = 2.34 \begin {gather*} -2 \sqrt {1-x} \sqrt {x+1}-4 \log \left (\sqrt {1-x}-i \sqrt {x+1}+(1-i)\right )-8 \tanh ^{-1}\left ((-1-i) \sqrt {1-x}-(1-i) \sqrt {x+1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 41, normalized size = 1.28 \begin {gather*} -2 \, \sqrt {x + 1} \sqrt {-x + 1} - 2 \, \log \relax (x) - 2 \, \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\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: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 1.59 \begin {gather*} -2 \ln \relax (x )-\frac {2 \sqrt {x +1}\, \sqrt {-x +1}\, \left (-\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+\sqrt {-x^{2}+1}\right )}{\sqrt {-x^{2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, size = 41, normalized size = 1.28 \begin {gather*} -2 \, \sqrt {-x^{2} + 1} - 2 \, \log \relax (x) + 2 \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 122, normalized size = 3.81 \begin {gather*} 2\,\ln \left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )-2\,\ln \left (\frac {{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}-1\right )-2\,\ln \relax (x)-\frac {16\,{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2\,\left (\frac {2\,{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+\frac {{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}+1\right )} \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 {2}{x}\, dx - \int \frac {2 \sqrt {1 - x} \sqrt {x + 1}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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