Optimal. Leaf size=29 \[ -\frac{2 \tan ^{-1}\left (\frac{c-2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]
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Rubi [A] time = 0.0216336, 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, Rules used = {1586, 617, 204} \[ -\frac{2 \tan ^{-1}\left (\frac{c-2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{c+d x}{c^3+d^3 x^3} \, dx &=\int \frac{1}{c^2-c d x+d^2 x^2} \, dx\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 d x}{c}\right )}{c d}\\ &=-\frac{2 \tan ^{-1}\left (\frac{c-2 d x}{\sqrt{3} c}\right )}{\sqrt{3} c d}\\ \end{align*}
Mathematica [A] time = 0.0101002, 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.
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Maple [A] time = 0.067, size = 35, normalized size = 1.2 \begin{align*}{\frac{2\,\sqrt{3}}{3\,cd}\arctan \left ({\frac{ \left ( 2\,{d}^{2}x-cd \right ) \sqrt{3}}{3\,cd}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45929, size = 46, normalized size = 1.59 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d^{2} x - c d\right )}}{3 \, c d}\right )}{3 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.903478, size = 72, normalized size = 2.48 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d x - c\right )}}{3 \, c}\right )}{3 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.156581, size = 54, normalized size = 1.86 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11921, size = 38, normalized size = 1.31 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \, d x - c\right )}}{3 \, c}\right )}{3 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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