Optimal. Leaf size=103 \[ \frac{2 \sqrt{2+\sqrt{3}} (t+1) \sqrt{\frac{t^2-t+1}{\left (t+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{t-\sqrt{3}+1}{t+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{t+1}{\left (t+\sqrt{3}+1\right )^2}} \sqrt{t^3+1}} \]
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Rubi [A] time = 0.0084795, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {218} \[ \frac{2 \sqrt{2+\sqrt{3}} (t+1) \sqrt{\frac{t^2-t+1}{\left (t+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{t-\sqrt{3}+1}{t+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{t+1}{\left (t+\sqrt{3}+1\right )^2}} \sqrt{t^3+1}} \]
Antiderivative was successfully verified.
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Rule 218
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+t^3}} \, dt &=\frac{2 \sqrt{2+\sqrt{3}} (1+t) \sqrt{\frac{1-t+t^2}{\left (1+\sqrt{3}+t\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+t}{1+\sqrt{3}+t}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1+t}{\left (1+\sqrt{3}+t\right )^2}} \sqrt{1+t^3}}\\ \end{align*}
Mathematica [C] time = 0.0018868, size = 17, normalized size = 0.17 \[ t \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{1}{2},\frac{4}{3},-t^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 116, normalized size = 1.1 \begin{align*} 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{t}^{3}+1}}\sqrt{{\frac{1+t}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{t-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{t-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+t}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{t^{3} + 1}}\,{d t} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{t^{3} + 1}}, t\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.530859, size = 27, normalized size = 0.26 \begin{align*} \frac{t \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{t^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{t^{3} + 1}}\,{d t} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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