Optimal. Leaf size=36 \[ 2 \sqrt [4]{x^2+1}-\tan ^{-1}\left (\sqrt [4]{x^2+1}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2+1}\right ) \]
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Rubi [A] time = 0.02, 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 = {266, 50, 63, 212, 206, 203} \begin {gather*} 2 \sqrt [4]{x^2+1}-\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 50
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x} \, dx,x,x^2\right )\\ &=2 \sqrt [4]{1+x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^2\right )\\ &=2 \sqrt [4]{1+x^2}+2 \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^2}\right )\\ &=2 \sqrt [4]{1+x^2}-\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 )\\ &=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 = 0.01, size = 36, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{x^2+1}-\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 = 0.04, size = 36, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{x^2+1}-\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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fricas [A] time = 0.40, 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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giac [A] time = 0.34, 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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maple [C] time = 0.24, size = 45, normalized size = 1.25 \begin {gather*} -\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 \relax (2)+\frac {\pi }{2}+2 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )}{8 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, 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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sympy [C] time = 0.81, 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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