Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {15} x}{2 \sqrt {1+4 x^2}}\right )}{2 \sqrt {15}} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, 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 = {385, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {15} x}{2 \sqrt {4 x^2+1}}\right )}{2 \sqrt {15}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 385
Rubi steps
\begin {align*} \int \frac {1}{\left (4+x^2\right ) \sqrt {1+4 x^2}} \, dx &=\text {Subst}\left (\int \frac {1}{4-15 x^2} \, dx,x,\frac {x}{\sqrt {1+4 x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {15} x}{2 \sqrt {1+4 x^2}}\right )}{2 \sqrt {15}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 40, normalized size = 1.29 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {8+2 x^2-x \sqrt {1+4 x^2}}{2 \sqrt {15}}\right )}{2 \sqrt {15}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 22, normalized size = 0.71
method | result | size |
default | \(\frac {\arctanh \left (\frac {x \sqrt {15}}{2 \sqrt {4 x^{2}+1}}\right ) \sqrt {15}}{30}\) | \(22\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {31 \RootOf \left (\textit {\_Z}^{2}-15\right ) x^{2}+60 \sqrt {4 x^{2}+1}\, x +4 \RootOf \left (\textit {\_Z}^{2}-15\right )}{x^{2}+4}\right )}{60}\) | \(50\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (21) = 42\).
time = 0.39, size = 54, normalized size = 1.74 \begin {gather*} \frac {1}{60} \, \sqrt {15} \log \left (\frac {961 \, x^{2} + 8 \, \sqrt {15} {\left (31 \, x^{2} + 4\right )} + 4 \, \sqrt {4 \, x^{2} + 1} {\left (31 \, \sqrt {15} x + 120 \, x\right )} + 124}{x^{2} + 4}\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 {1}{\left (x^{2} + 4\right ) \sqrt {4 x^{2} + 1}}\, dx \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. 57 vs.
\(2 (21) = 42\).
time = 0.54, size = 57, normalized size = 1.84 \begin {gather*} -\frac {1}{60} \, \sqrt {15} \log \left (\frac {{\left (2 \, x - \sqrt {4 \, x^{2} + 1}\right )}^{2} - 8 \, \sqrt {15} + 31}{{\left (2 \, x - \sqrt {4 \, x^{2} + 1}\right )}^{2} + 8 \, \sqrt {15} + 31}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 61, normalized size = 1.97 \begin {gather*} -\frac {\sqrt {15}\,\left (\ln \left (x-2{}\mathrm {i}\right )-\ln \left (x+\frac {\sqrt {15}\,\sqrt {x^2+\frac {1}{4}}}{4}-\frac {1}{8}{}\mathrm {i}\right )\right )}{60}+\frac {\sqrt {15}\,\left (\ln \left (x+2{}\mathrm {i}\right )-\ln \left (x-\frac {\sqrt {15}\,\sqrt {x^2+\frac {1}{4}}}{4}+\frac {1}{8}{}\mathrm {i}\right )\right )}{60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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