Optimal. Leaf size=36 \[ 2 \sqrt [4]{1+x^2}-\text {ArcTan}\left (\sqrt [4]{1+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{1+x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 52, 65,
218, 212, 209} \begin {gather*} -\text {ArcTan}\left (\sqrt [4]{x^2+1}\right )+2 \sqrt [4]{x^2+1}-\tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x} \, dx,x,x^2\right )\\ &=2 \sqrt [4]{1+x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^2\right )\\ &=2 \sqrt [4]{1+x^2}+2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=2 \sqrt [4]{1+x^2}-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^2}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=2 \sqrt [4]{1+x^2}-\tan ^{-1}\left (\sqrt [4]{1+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{1+x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{1+x^2}-\text {ArcTan}\left (\sqrt [4]{1+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.12, size = 45, normalized size = 1.25
method | result | size |
meijerg | \(-\frac {-\Gamma \left (\frac {3}{4}\right ) x^{2} \hypergeom \left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], -x^{2}\right )-4 \left (4-3 \ln \left (2\right )+\frac {\pi }{2}+2 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{8 \Gamma \left (\frac {3}{4}\right )}\) | \(45\) |
trager | \(2 \left (x^{2}+1\right )^{\frac {1}{4}}+\frac {\ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{2}+1}-x^{2}+2 \left (x^{2}+1\right )^{\frac {1}{4}}-2}{x^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}+1}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{2}+1\right )^{\frac {1}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{2}}\right )}{2}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 44, normalized size = 1.22 \begin {gather*} 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} - \arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 44, normalized size = 1.22 \begin {gather*} 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} - \arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\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 = 37, normalized size = 1.03 \begin {gather*} - \frac {\sqrt {x} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 44, normalized size = 1.22 \begin {gather*} 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} - \arctan \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 30, normalized size = 0.83 \begin {gather*} 2\,{\left (x^2+1\right )}^{1/4}-\mathrm {atanh}\left ({\left (x^2+1\right )}^{1/4}\right )-\mathrm {atan}\left ({\left (x^2+1\right )}^{1/4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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