Optimal. Leaf size=31 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {1-x^2}}\right )}{2 \sqrt {5}} \]
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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, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {5} x}{2 \sqrt {1-x^2}}\right )}{2 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 385
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x^2} \left (4+x^2\right )} \, dx &=\text {Subst}\left (\int \frac {1}{4+5 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {1-x^2}}\right )}{2 \sqrt {5}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 42, normalized size = 1.35 \begin {gather*} -\frac {i \tanh ^{-1}\left (\frac {4+x^2+i x \sqrt {1-x^2}}{2 \sqrt {5}}\right )}{2 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 29, normalized size = 0.94
method | result | size |
default | \(-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {-x^{2}+1}\, x}{2 x^{2}-2}\right )}{10}\) | \(29\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+5\right ) \ln \left (\frac {9 \RootOf \left (\textit {\_Z}^{2}+5\right ) x^{2}+20 x \sqrt {-x^{2}+1}-4 \RootOf \left (\textit {\_Z}^{2}+5\right )}{x^{2}+4}\right )}{20}\) | \(50\) |
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 [A]
time = 0.38, size = 23, normalized size = 0.74 \begin {gather*} -\frac {1}{10} \, \sqrt {5} \arctan \left (\frac {2 \, \sqrt {5} \sqrt {-x^{2} + 1}}{5 \, x}\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}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 4\right )}\, 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. 51 vs.
\(2 (21) = 42\).
time = 0.47, size = 51, normalized size = 1.65 \begin {gather*} \frac {1}{20} \, \sqrt {5} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {5} x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 79, normalized size = 2.55 \begin {gather*} \frac {\sqrt {5}\,\ln \left (\frac {\frac {\sqrt {5}\,\left (-1+x\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{20}-\frac {\sqrt {5}\,\ln \left (\frac {\frac {\sqrt {5}\,\left (1+x\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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