3.32 \(\int \frac{(-a)^{2/3} C+2 C x^2}{a-8 x^3} \, dx\)

Optimal. Leaf size=47 \[ \frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right ) \]

[Out]

(C*ArcTan[(1 - (4*x)/(-a)^(1/3))/Sqrt[3]])/(2*Sqrt[3]) - (C*Log[(-a)^(1/3) + 2*x
])/4

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Rubi [A]  time = 0.101419, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right ) \]

Antiderivative was successfully verified.

[In]  Int[((-a)^(2/3)*C + 2*C*x^2)/(a - 8*x^3),x]

[Out]

(C*ArcTan[(1 - (4*x)/(-a)^(1/3))/Sqrt[3]])/(2*Sqrt[3]) - (C*Log[(-a)^(1/3) + 2*x
])/4

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Rubi in Sympy [A]  time = 10.8287, size = 44, normalized size = 0.94 \[ - \frac{C \log{\left (2 x + \sqrt [3]{- a} \right )}}{4} + \frac{\sqrt{3} C \operatorname{atan}{\left (\sqrt{3} \left (- \frac{4 x}{3 \sqrt [3]{- a}} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((-a)**(2/3)*C+2*C*x**2)/(-8*x**3+a),x)

[Out]

-C*log(2*x + (-a)**(1/3))/4 + sqrt(3)*C*atan(sqrt(3)*(-4*x/(3*(-a)**(1/3)) + 1/3
))/6

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Mathematica [B]  time = 0.0688808, size = 106, normalized size = 2.26 \[ \frac{C \left (-a^{2/3} \log \left (8 x^3-a\right )+(-a)^{2/3} \log \left (a^{2/3}+2 \sqrt [3]{a} x+4 x^2\right )-2 (-a)^{2/3} \log \left (\sqrt [3]{a}-2 x\right )+2 \sqrt{3} (-a)^{2/3} \tan ^{-1}\left (\frac{\frac{4 x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )}{12 a^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[((-a)^(2/3)*C + 2*C*x^2)/(a - 8*x^3),x]

[Out]

(C*(2*Sqrt[3]*(-a)^(2/3)*ArcTan[(1 + (4*x)/a^(1/3))/Sqrt[3]] - 2*(-a)^(2/3)*Log[
a^(1/3) - 2*x] + (-a)^(2/3)*Log[a^(2/3) + 2*a^(1/3)*x + 4*x^2] - a^(2/3)*Log[-a
+ 8*x^3]))/(12*a^(2/3))

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Maple [B]  time = 0.011, size = 110, normalized size = 2.3 \[ -{\frac{C{8}^{{\frac{2}{3}}}}{24} \left ( -a \right ) ^{{\frac{2}{3}}}\ln \left ( x-{\frac{{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}} \right ){a}^{-{\frac{2}{3}}}}+{\frac{C{8}^{{\frac{2}{3}}}}{48} \left ( -a \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+{\frac{x{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}}+{\frac{\sqrt [3]{8}}{8}{a}^{{\frac{2}{3}}}} \right ){a}^{-{\frac{2}{3}}}}+{\frac{C{8}^{{\frac{2}{3}}}\sqrt{3}}{24} \left ( -a \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{8}x}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( 8\,{x}^{3}-a \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((-a)^(2/3)*C+2*C*x^2)/(-8*x^3+a),x)

[Out]

-1/24*C*(-a)^(2/3)*8^(2/3)/a^(2/3)*ln(x-1/8*8^(2/3)*a^(1/3))+1/48*C*(-a)^(2/3)*8
^(2/3)/a^(2/3)*ln(x^2+1/8*x*8^(2/3)*a^(1/3)+1/8*8^(1/3)*a^(2/3))+1/24*C*(-a)^(2/
3)*8^(2/3)/a^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2*8^(1/3)/a^(1/3)*x+1))-1/12*C*ln
(8*x^3-a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*C*x^2 + C*(-a)^(2/3))/(8*x^3 - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231279, size = 68, normalized size = 1.45 \[ -\frac{1}{12} \, \sqrt{3}{\left (\sqrt{3} C \log \left (2 \, \left (-a\right )^{\frac{2}{3}} x - a\right ) - 2 \, C \arctan \left (\frac{4 \, \sqrt{3} \left (-a\right )^{\frac{2}{3}} x + \sqrt{3} a}{3 \, a}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*C*x^2 + C*(-a)^(2/3))/(8*x^3 - a),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*(sqrt(3)*C*log(2*(-a)^(2/3)*x - a) - 2*C*arctan(1/3*(4*sqrt(3)*(-a
)^(2/3)*x + sqrt(3)*a)/a))

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Sympy [A]  time = 0.847554, size = 95, normalized size = 2.02 \[ - C \left (\frac{\log{\left (- \frac{a}{2 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{4} + \frac{\sqrt{3} i \log{\left (\frac{a}{4 \left (- a\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{4 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{12} - \frac{\sqrt{3} i \log{\left (\frac{a}{4 \left (- a\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{4 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-a)**(2/3)*C+2*C*x**2)/(-8*x**3+a),x)

[Out]

-C*(log(-a/(2*(-a)**(2/3)) + x)/4 + sqrt(3)*I*log(a/(4*(-a)**(2/3)) - sqrt(3)*I*
a/(4*(-a)**(2/3)) + x)/12 - sqrt(3)*I*log(a/(4*(-a)**(2/3)) + sqrt(3)*I*a/(4*(-a
)**(2/3)) + x)/12)

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GIAC/XCAS [A]  time = 0.242872, size = 130, normalized size = 2.77 \[ \frac{\sqrt{3}{\left (\sqrt{3} a i - a\right )} C \arctan \left (\frac{\sqrt{3}{\left (4 \, x + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{12 \, a} - \frac{{\left (\sqrt{3} a i + 3 \, a\right )} C{\rm ln}\left (x^{2} + \frac{1}{2} \, a^{\frac{1}{3}} x + \frac{1}{4} \, a^{\frac{2}{3}}\right )}{24 \, a} - \frac{{\left (2 \, C \left (-a\right )^{\frac{2}{3}} + C a^{\frac{2}{3}}\right )}{\rm ln}\left ({\left | x - \frac{1}{2} \, a^{\frac{1}{3}} \right |}\right )}{12 \, a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*C*x^2 + C*(-a)^(2/3))/(8*x^3 - a),x, algorithm="giac")

[Out]

1/12*sqrt(3)*(sqrt(3)*a*i - a)*C*arctan(1/3*sqrt(3)*(4*x + a^(1/3))/a^(1/3))/a -
1/24*(sqrt(3)*a*i + 3*a)*C*ln(x^2 + 1/2*a^(1/3)*x + 1/4*a^(2/3))/a - 1/12*(2*C*
(-a)^(2/3) + C*a^(2/3))*ln(abs(x - 1/2*a^(1/3)))/a^(2/3)