Optimal. Leaf size=103 \[ \frac{2 \sqrt{2+\sqrt{3}} (t+1) \sqrt{\frac{t^2-t+1}{\left (t+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{t-\sqrt{3}+1}{t+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{t+1}{\left (t+\sqrt{3}+1\right )^2}} \sqrt{t^3+1}} \]
[Out]
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Rubi [A] time = 0.0399457, 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 \[ \frac{2 \sqrt{2+\sqrt{3}} (t+1) \sqrt{\frac{t^2-t+1}{\left (t+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{t-\sqrt{3}+1}{t+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{t+1}{\left (t+\sqrt{3}+1\right )^2}} \sqrt{t^3+1}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[1 + t^3],t]
[Out]
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Rubi in Sympy [A] time = 0.809349, size = 95, normalized size = 0.92 \[ \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{t^{2} - t + 1}{\left (t + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (t + 1\right ) F\left (\operatorname{asin}{\left (\frac{t - \sqrt{3} + 1}{t + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{t + 1}{\left (t + 1 + \sqrt{3}\right )^{2}}} \sqrt{t^{3} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(t**3+1)**(1/2),t)
[Out]
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Mathematica [A] time = 0.068797, size = 88, normalized size = 0.85 \[ \frac{2 \sqrt [6]{-1} \sqrt{-\sqrt [6]{-1} \left (t+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} t^2+\sqrt [3]{-1} t+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (t+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt{t^3+1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/Sqrt[1 + t^3],t]
[Out]
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Maple [A] time = 0.096, size = 116, normalized size = 1.1 \[ 2\,{\frac{3/2-i/2\sqrt{3}}{\sqrt{{t}^{3}+1}}\sqrt{{\frac{1+t}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{t-1/2-i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{t-1/2+i/2\sqrt{3}}{-3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{1+t}{3/2-i/2\sqrt{3}}}},\sqrt{{\frac{-3/2+i/2\sqrt{3}}{-3/2-i/2\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(t^3+1)^(1/2),t)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{t^{3} + 1}}\,{d t} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(t^3 + 1),t, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{t^{3} + 1}}, t\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(t^3 + 1),t, algorithm="fricas")
[Out]
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Sympy [A] time = 0.852905, size = 27, normalized size = 0.26 \[ \frac{t \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{t^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(t**3+1)**(1/2),t)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{t^{3} + 1}}\,{d t} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(t^3 + 1),t, algorithm="giac")
[Out]