Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {15} x}{2 \sqrt {4 x^2+1}}\right )}{2 \sqrt {15}} \]
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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 = {377, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {15} x}{2 \sqrt {4 x^2+1}}\right )}{2 \sqrt {15}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 377
Rubi steps
\begin {align*} \int \frac {1}{\left (4+x^2\right ) \sqrt {1+4 x^2}} \, dx &=\operatorname {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 = 0.01, size = 31, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {15} x}{2 \sqrt {4 x^2+1}}\right )}{2 \sqrt {15}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 48, normalized size = 1.55 \[ \frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {15}}-\frac {\sqrt {4 x^2+1} x}{2 \sqrt {15}}+\frac {4}{\sqrt {15}}\right )}{2 \sqrt {15}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 54, normalized size = 1.74 \[ \frac {1}{60} \, \sqrt {15} \log \left (\frac {961 \, x^{2} + 8 \, \sqrt {15} {\left (31 \, x^{2} + 4\right )} + 4 \, \sqrt {4 \, x^{2} + 1} {\left (31 \, \sqrt {15} x + 120 \, x\right )} + 124}{x^{2} + 4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 57, normalized size = 1.84 \[ -\frac {1}{60} \, \sqrt {15} \log \left (\frac {{\left (2 \, x - \sqrt {4 \, x^{2} + 1}\right )}^{2} - 8 \, \sqrt {15} + 31}{{\left (2 \, x - \sqrt {4 \, x^{2} + 1}\right )}^{2} + 8 \, \sqrt {15} + 31}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, 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 \[ \int \frac {1}{\sqrt {4 \, x^{2} + 1} {\left (x^{2} + 4\right )}}\,{d x} \]
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 \[ -\frac {\sqrt {15}\,\left (\ln \left (x-2{}\mathrm {i}\right )-\ln \left (x+\frac {\sqrt {15}\,\sqrt {x^2+\frac {1}{4}}}{4}-\frac {1}{8}{}\mathrm {i}\right )\right )}{60}+\frac {\sqrt {15}\,\left (\ln \left (x+2{}\mathrm {i}\right )-\ln \left (x-\frac {\sqrt {15}\,\sqrt {x^2+\frac {1}{4}}}{4}+\frac {1}{8}{}\mathrm {i}\right )\right )}{60} \]
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 \[ \int \frac {1}{\left (x^{2} + 4\right ) \sqrt {4 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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