Optimal. Leaf size=27 \[ 10 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {217, 206} \[ 10 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rubi steps
\begin {align*} \int \left (\frac {10}{\sqrt {-4+x^2}}+\frac {1}{\sqrt {-1+x^2}}\right ) \, dx &=10 \int \frac {1}{\sqrt {-4+x^2}} \, dx+\int \frac {1}{\sqrt {-1+x^2}} \, dx\\ &=10 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-4+x^2}}\right )+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=10 \tanh ^{-1}\left (\frac {x}{\sqrt {-4+x^2}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 71, normalized size = 2.63 \[ -5 \log \left (1-\frac {x}{\sqrt {x^2-4}}\right )+5 \log \left (\frac {x}{\sqrt {x^2-4}}+1\right )-\frac {1}{2} \log \left (1-\frac {x}{\sqrt {x^2-1}}\right )+\frac {1}{2} \log \left (\frac {x}{\sqrt {x^2-1}}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 29, normalized size = 1.07 \[ -\log \left (-x + \sqrt {x^{2} - 1}\right ) - 10 \, \log \left (-x + \sqrt {x^{2} - 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 31, normalized size = 1.15 \[ -\log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) - 10 \, \log \left ({\left | -x + \sqrt {x^{2} - 4} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ 10 \ln \left (x +\sqrt {x^{2}-4}\right )+\ln \left (x +\sqrt {x^{2}-1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 31, normalized size = 1.15 \[ \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) + 10 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 23, normalized size = 0.85 \[ \ln \left (x+\sqrt {x^2-1}\right )+10\,\ln \left (x+\sqrt {x^2-4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 8, normalized size = 0.30 \[ 10 \operatorname {acosh}{\left (\frac {x}{2} \right )} + \operatorname {acosh}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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