Integrand size = 9, antiderivative size = 103 \[ \int \frac {1}{\sqrt {1+t^3}} \, dt=\frac {2 \sqrt {2+\sqrt {3}} (1+t) \sqrt {\frac {1-t+t^2}{\left (1+\sqrt {3}+t\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+t}{1+\sqrt {3}+t}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+t}{\left (1+\sqrt {3}+t\right )^2}} \sqrt {1+t^3}} \]
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Time = 0.01 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {224} \[ \int \frac {1}{\sqrt {1+t^3}} \, dt=\frac {2 \sqrt {2+\sqrt {3}} (t+1) \sqrt {\frac {t^2-t+1}{\left (t+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {t-\sqrt {3}+1}{t+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {t+1}{\left (t+\sqrt {3}+1\right )^2}} \sqrt {t^3+1}} \]
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Rule 224
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {2+\sqrt {3}} (1+t) \sqrt {\frac {1-t+t^2}{\left (1+\sqrt {3}+t\right )^2}} \operatorname {EllipticF}\left (\arcsin \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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\sqrt {1+t^3}} \, dt=t \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-t^3\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.14
method | result | size |
meijerg | \(t {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-t^{3}\right )\) | \(14\) |
default | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+t}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {t -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {t -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+t}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {t^{3}+1}}\) | \(116\) |
elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+t}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {t -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {t -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+t}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {t^{3}+1}}\) | \(116\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.06 \[ \int \frac {1}{\sqrt {1+t^3}} \, dt=2 \, {\rm weierstrassPInverse}\left (0, -4, t\right ) \]
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Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt {1+t^3}} \, dt=\frac {t \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {t^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
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\[ \int \frac {1}{\sqrt {1+t^3}} \, dt=\int { \frac {1}{\sqrt {t^{3} + 1}} \,d t } \]
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\[ \int \frac {1}{\sqrt {1+t^3}} \, dt=\int { \frac {1}{\sqrt {t^{3} + 1}} \,d t } \]
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Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {1+t^3}} \, dt=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {t-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {t+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-t+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {t+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {t^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,t-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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