Optimal. Leaf size=49 \[ \frac {\sqrt {1+2 x^2} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt {2} \sqrt {1+x^2} \sqrt {\frac {1+2 x^2}{1+x^2}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {429}
\begin {gather*} \frac {\sqrt {2 x^2+1} F(\text {ArcTan}(x)|-1)}{\sqrt {2} \sqrt {x^2+1} \sqrt {\frac {2 x^2+1}{x^2+1}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 429
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^2} \sqrt {2+4 x^2}} \, dx &=\frac {\sqrt {1+2 x^2} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt {2} \sqrt {1+x^2} \sqrt {\frac {1+2 x^2}{1+x^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 17, normalized size = 0.35 \begin {gather*} -\frac {i F\left (\left .i \sinh ^{-1}(x)\right |2\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 15, normalized size = 0.31
method | result | size |
default | \(-\frac {i \EllipticF \left (i x , \sqrt {2}\right ) \sqrt {2}}{2}\) | \(15\) |
elliptic | \(-\frac {i \sqrt {\left (x^{2}+1\right ) \left (2 x^{2}+1\right )}\, \EllipticF \left (i x , \sqrt {2}\right )}{\sqrt {4 x^{4}+6 x^{2}+2}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 10, normalized size = 0.20 \begin {gather*} -\frac {1}{2} i \, \sqrt {2} {\rm ellipticF}\left (i \, x, 2\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*} \frac {\sqrt {2} \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {2 x^{2} + 1}}\, dx}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2+1}\,\sqrt {4\,x^2+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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