Optimal. Leaf size=16 \[ \sinh ^{-1}\left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {56, 221}
\begin {gather*} \sinh ^{-1}\left (\frac {\sqrt {2 x+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+2 x} \sqrt {3+2 x}} \, dx &=\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {4+2 x^2}} \, dx,x,\sqrt {1+2 x}\right )\\ &=\sinh ^{-1}\left (\frac {\sqrt {1+2 x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 20, normalized size = 1.25 \begin {gather*} \tanh ^{-1}\left (\frac {\sqrt {3+2 x}}{\sqrt {1+2 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(13)=26\).
time = 0.08, size = 57, normalized size = 3.56
method | result | size |
default | \(\frac {\sqrt {\left (1+2 x \right ) \left (3+2 x \right )}\, \ln \left (\frac {\left (4+4 x \right ) \sqrt {4}}{4}+\sqrt {4 x^{2}+8 x +3}\right ) \sqrt {4}}{4 \sqrt {1+2 x}\, \sqrt {3+2 x}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 22, normalized size = 1.38 \begin {gather*} \frac {1}{2} \, \log \left (8 \, x + 4 \, \sqrt {4 \, x^{2} + 8 \, x + 3} + 8\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 23, normalized size = 1.44 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {2 \, x + 3} \sqrt {2 \, x + 1} - 2 \, x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 27, normalized size = 1.69 \begin {gather*} \begin {cases} \operatorname {acosh}{\left (\sqrt {x + \frac {3}{2}} \right )} & \text {for}\: \left |{x + \frac {3}{2}}\right | > 1 \\- i \operatorname {asin}{\left (\sqrt {x + \frac {3}{2}} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.42, size = 20, normalized size = 1.25 \begin {gather*} -\log \left (\sqrt {2 \, x + 3} - \sqrt {2 \, x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 28, normalized size = 1.75 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {\sqrt {3}-\sqrt {2\,x+3}}{\sqrt {2\,x+1}-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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