Optimal. Leaf size=27 \[ 10 \tanh ^{-1}\left (\frac {x}{\sqrt {-4+x^2}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right ) \]
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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 = {223, 212}
\begin {gather*} 10 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
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 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-4+x^2}}\right )+\text {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 [A]
time = 0.02, size = 52, normalized size = 1.93 \begin {gather*} 10 \tanh ^{-1}\left (\frac {x}{\sqrt {-4+x^2}}\right )-\frac {1}{2} \log \left (1-\frac {x}{\sqrt {-1+x^2}}\right )+\frac {1}{2} \log \left (1+\frac {x}{\sqrt {-1+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 24, normalized size = 0.89
method | result | size |
default | \(\ln \left (x +\sqrt {x^{2}-1}\right )+10 \ln \left (x +\sqrt {x^{2}-4}\right )\) | \(24\) |
meijerg | \(\frac {10 \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \arcsin \left (\frac {x}{2}\right )}{\sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}}+\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \arcsin \left (x \right )}{\sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.93, size = 31, normalized size = 1.15 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) + 10 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (-x + \sqrt {x^{2} - 1}\right ) - 10 \, \log \left (-x + \sqrt {x^{2} - 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 8, normalized size = 0.30 \begin {gather*} 10 \operatorname {acosh}{\left (\frac {x}{2} \right )} + \operatorname {acosh}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (23) = 46\).
time = 0.69, size = 51, normalized size = 1.89 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 1} x + 5 \, \sqrt {x^{2} - 4} x + \frac {1}{2} \, \log \left ({\left | -x + \sqrt {x^{2} - 1} \right |}\right ) + 20 \, \log \left ({\left | -x + \sqrt {x^{2} - 4} \right |}\right ) \end {gather*}
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 \begin {gather*} \ln \left (x+\sqrt {x^2-1}\right )+10\,\ln \left (x+\sqrt {x^2-4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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