Optimal. Leaf size=23 \[ \tan ^{-1}\left (\sqrt [4]{x^2+1}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 63, 298, 203, 206} \begin {gather*} \tan ^{-1}\left (\sqrt [4]{x^2+1}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [4]{1+x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^2\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^2}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=\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 = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt [4]{x^2+1}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} \tan ^{-1}\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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fricas [A] time = 0.46, size = 33, normalized size = 1.43 \begin {gather*} \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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giac [A] time = 0.37, size = 33, normalized size = 1.43 \begin {gather*} \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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maple [C] time = 1.33, size = 59, normalized size = 2.57
method | result | size |
meijerg | \(\frac {\Gamma \left (\frac {3}{4}\right ) \sqrt {2}\, \left (\frac {\left (-3 \ln \relax (2)-\frac {\pi }{2}+2 \ln \relax (x )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}-\frac {\hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{2}\right ) \pi \sqrt {2}\, x^{2}}{4 \Gamma \left (\frac {3}{4}\right )}\right )}{4 \pi }\) | \(59\) |
trager | \(-\frac {\ln \left (-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}+2 \sqrt {x^{2}+1}+x^{2}+2 \left (x^{2}+1\right )^{\frac {1}{4}}+2}{x^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}+1}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{2}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{2}+1\right )^{\frac {1}{4}}}{x^{2}}\right )}{2}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 33, normalized size = 1.43 \begin {gather*} \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 = 19, normalized size = 0.83 \begin {gather*} \mathrm {atan}\left ({\left (x^2+1\right )}^{1/4}\right )-\mathrm {atanh}\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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sympy [C] time = 0.74, size = 32, normalized size = 1.39 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{2 \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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