3.18 \(\int \frac{c+d x}{c^3+d^3 x^3} \, dx\)

Optimal. Leaf size=29 \[ -\frac{2 \tan ^{-1}\left (\frac{c-2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]

[Out]

(-2*ArcTan[(c - 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)

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Rubi [A]  time = 0.0467412, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 \tan ^{-1}\left (\frac{c-2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTan[(c - 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)

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Rubi in Sympy [A]  time = 5.16106, size = 31, normalized size = 1.07 \[ - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{c}{3} - \frac{2 d x}{3}\right )}{c} \right )}}{3 c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(3)*atan(sqrt(3)*(c/3 - 2*d*x/3)/c)/(3*c*d)

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Mathematica [A]  time = 0.0186217, size = 31, normalized size = 1.07 \[ \frac{2 \tan ^{-1}\left (\frac{2 d x-c}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTan[(-c + 2*d*x)/(Sqrt[3]*c)])/(Sqrt[3]*c*d)

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Maple [A]  time = 0.009, size = 35, normalized size = 1.2 \[{\frac{2\,\sqrt{3}}{3\,cd}\arctan \left ({\frac{ \left ( 2\,{d}^{2}x-cd \right ) \sqrt{3}}{3\,cd}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(d^3*x^3+c^3),x)

[Out]

2/3*3^(1/2)/c/d*arctan(1/3*(2*d^2*x-c*d)*3^(1/2)/c/d)

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Maxima [A]  time = 1.52522, size = 46, normalized size = 1.59 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d^{2} x - c d\right )}}{3 \, c d}\right )}{3 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*d^2*x - c*d)/(c*d))/(c*d)

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Fricas [A]  time = 0.217404, size = 38, normalized size = 1.31 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d x - c\right )}}{3 \, c}\right )}{3 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*d*x - c)/c)/(c*d)

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Sympy [A]  time = 0.200274, size = 54, normalized size = 1.86 \[ \frac{- \frac{\sqrt{3} i \log{\left (x + \frac{- c - \sqrt{3} i c}{2 d} \right )}}{3} + \frac{\sqrt{3} i \log{\left (x + \frac{- c + \sqrt{3} i c}{2 d} \right )}}{3}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-sqrt(3)*I*log(x + (-c - sqrt(3)*I*c)/(2*d))/3 + sqrt(3)*I*log(x + (-c + sqrt(3
)*I*c)/(2*d))/3)/(c*d)

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GIAC/XCAS [A]  time = 0.21067, size = 38, normalized size = 1.31 \[ \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d x - c\right )}}{3 \, c}\right )}{3 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*d*x - c)/c)/(c*d)