3.145 \(\int \frac{1}{\sqrt [3]{c-3 d x^2} \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=204 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c-3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}} \]

[Out]

ArcTan[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) + ArcTan[(c^(1/6)*(c^(1/
3) - 2^(1/3)*(c - 3*d*x^2)^(1/3)))/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - Ar
cTanh[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6)*Sqrt[d]) + (Sqrt[3
]*ArcTanh[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c - 3*d*x^2)^(1/3)))]
)/(2*2^(2/3)*c^(5/6)*Sqrt[d])

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Rubi [A]  time = 0.0975356, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c-3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]

[Out]

ArcTan[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) + ArcTan[(c^(1/6)*(c^(1/
3) - 2^(1/3)*(c - 3*d*x^2)^(1/3)))/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - Ar
cTanh[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6)*Sqrt[d]) + (Sqrt[3
]*ArcTanh[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c - 3*d*x^2)^(1/3)))]
)/(2*2^(2/3)*c^(5/6)*Sqrt[d])

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Rubi in Sympy [A]  time = 85.0538, size = 410, normalized size = 2.01 \[ \frac{\sqrt [3]{2} \sqrt{3} \sqrt [3]{1 - \frac{3 d x^{2}}{c}} \log{\left (\sqrt [3]{2} \sqrt [3]{1 - \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}} + \left (1 + \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )^{\frac{2}{3}} \right )}}{8 \sqrt{c} \sqrt{d} \sqrt [3]{c - 3 d x^{2}}} - \frac{\sqrt [3]{2} \sqrt{3} \sqrt [3]{1 - \frac{3 d x^{2}}{c}} \log{\left (\left (1 - \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )^{\frac{2}{3}} + \sqrt [3]{2} \sqrt [3]{1 + \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}} \right )}}{8 \sqrt{c} \sqrt{d} \sqrt [3]{c - 3 d x^{2}}} - \frac{\sqrt [3]{2} \sqrt [3]{1 - \frac{3 d x^{2}}{c}} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \left (1 + \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )^{\frac{2}{3}}}{3 \sqrt [3]{1 - \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}}} \right )}}{4 \sqrt{c} \sqrt{d} \sqrt [3]{c - 3 d x^{2}}} - \frac{\sqrt [3]{2} \sqrt [3]{1 - \frac{3 d x^{2}}{c}} \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \sqrt{3} \left (1 - \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )^{\frac{2}{3}}}{3 \sqrt [3]{1 + \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}}} - \frac{\sqrt{3}}{3} \right )}}{4 \sqrt{c} \sqrt{d} \sqrt [3]{c - 3 d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3*d*x**2+c)**(1/3)/(d*x**2+c),x)

[Out]

2**(1/3)*sqrt(3)*(1 - 3*d*x**2/c)**(1/3)*log(2**(1/3)*(1 - sqrt(3)*sqrt(d)*x/sqr
t(c))**(1/3) + (1 + sqrt(3)*sqrt(d)*x/sqrt(c))**(2/3))/(8*sqrt(c)*sqrt(d)*(c - 3
*d*x**2)**(1/3)) - 2**(1/3)*sqrt(3)*(1 - 3*d*x**2/c)**(1/3)*log((1 - sqrt(3)*sqr
t(d)*x/sqrt(c))**(2/3) + 2**(1/3)*(1 + sqrt(3)*sqrt(d)*x/sqrt(c))**(1/3))/(8*sqr
t(c)*sqrt(d)*(c - 3*d*x**2)**(1/3)) - 2**(1/3)*(1 - 3*d*x**2/c)**(1/3)*atan(sqrt
(3)/3 - 2**(2/3)*sqrt(3)*(1 + sqrt(3)*sqrt(d)*x/sqrt(c))**(2/3)/(3*(1 - sqrt(3)*
sqrt(d)*x/sqrt(c))**(1/3)))/(4*sqrt(c)*sqrt(d)*(c - 3*d*x**2)**(1/3)) - 2**(1/3)
*(1 - 3*d*x**2/c)**(1/3)*atan(2**(2/3)*sqrt(3)*(1 - sqrt(3)*sqrt(d)*x/sqrt(c))**
(2/3)/(3*(1 + sqrt(3)*sqrt(d)*x/sqrt(c))**(1/3)) - sqrt(3)/3)/(4*sqrt(c)*sqrt(d)
*(c - 3*d*x**2)**(1/3))

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Mathematica [C]  time = 0.235873, size = 156, normalized size = 0.76 \[ \frac{3 c x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )}{\sqrt [3]{c-3 d x^2} \left (c+d x^2\right ) \left (2 d x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )\right )+3 c F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]

[Out]

(3*c*x*AppellF1[1/2, 1/3, 1, 3/2, (3*d*x^2)/c, -((d*x^2)/c)])/((c - 3*d*x^2)^(1/
3)*(c + d*x^2)*(3*c*AppellF1[1/2, 1/3, 1, 3/2, (3*d*x^2)/c, -((d*x^2)/c)] + 2*d*
x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (3*d*x^2)/c, -((d*x^2)/c)] + AppellF1[3/2, 4/3,
1, 5/2, (3*d*x^2)/c, -((d*x^2)/c)])))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c}{\frac{1}{\sqrt [3]{-3\,d{x}^{2}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x)

[Out]

int(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + c\right )}{\left (-3 \, d x^{2} + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)),x, algorithm="maxima")

[Out]

integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{c - 3 d x^{2}} \left (c + d x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*d*x**2+c)**(1/3)/(d*x**2+c),x)

[Out]

Integral(1/((c - 3*d*x**2)**(1/3)*(c + d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + c\right )}{\left (-3 \, d x^{2} + c\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)),x, algorithm="giac")

[Out]

integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)), x)