3.175 \(\int \frac{1}{\sqrt{1+t^3}} \, dt\)

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]

(2*Sqrt[2 + Sqrt[3]]*(1 + t)*Sqrt[(1 - t + t^2)/(1 + Sqrt[3] + t)^2]*EllipticF[A
rcSin[(1 - Sqrt[3] + t)/(1 + Sqrt[3] + t)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 +
t)/(1 + Sqrt[3] + t)^2]*Sqrt[1 + t^3])

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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]

(2*Sqrt[2 + Sqrt[3]]*(1 + t)*Sqrt[(1 - t + t^2)/(1 + Sqrt[3] + t)^2]*EllipticF[A
rcSin[(1 - Sqrt[3] + t)/(1 + Sqrt[3] + t)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 +
t)/(1 + Sqrt[3] + t)^2]*Sqrt[1 + t^3])

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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]

2*3**(3/4)*sqrt((t**2 - t + 1)/(t + 1 + sqrt(3))**2)*sqrt(sqrt(3) + 2)*(t + 1)*e
lliptic_f(asin((t - sqrt(3) + 1)/(t + 1 + sqrt(3))), -7 - 4*sqrt(3))/(3*sqrt((t
+ 1)/(t + 1 + sqrt(3))**2)*sqrt(t**3 + 1))

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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]

(2*(-1)^(1/6)*Sqrt[-((-1)^(1/6)*((-1)^(2/3) + t))]*Sqrt[1 + (-1)^(1/3)*t + (-1)^
(2/3)*t^2]*EllipticF[ArcSin[Sqrt[-((-1)^(5/6)*(1 + t))]/3^(1/4)], (-1)^(1/3)])/(
3^(1/4)*Sqrt[1 + t^3])

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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]

2*(3/2-1/2*I*3^(1/2))*((1+t)/(3/2-1/2*I*3^(1/2)))^(1/2)*((t-1/2-1/2*I*3^(1/2))/(
-3/2-1/2*I*3^(1/2)))^(1/2)*((t-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(t
^3+1)^(1/2)*EllipticF(((1+t)/(3/2-1/2*I*3^(1/2)))^(1/2),((-3/2+1/2*I*3^(1/2))/(-
3/2-1/2*I*3^(1/2)))^(1/2))

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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]

integrate(1/sqrt(t^3 + 1), t)

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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]

integral(1/sqrt(t^3 + 1), t)

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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]

t*gamma(1/3)*hyper((1/3, 1/2), (4/3,), t**3*exp_polar(I*pi))/(3*gamma(4/3))

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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]

integrate(1/sqrt(t^3 + 1), t)