Integrand size = 21, antiderivative size = 29 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2 \arctan \left (\frac {c+2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1600, 631, 210} \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2 \arctan \left (\frac {c+2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]
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Rule 210
Rule 631
Rule 1600
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{c^2+c d x+d^2 x^2} \, dx \\ & = -\frac {2 \text {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*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2 \arctan \left (\frac {c+2 d x}{\sqrt {3} c}\right )}{\sqrt {3} c d} \]
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Time = 1.50 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {2 d \sqrt {3}\, x}{3 c}+\frac {\sqrt {3}}{3}\right )}{3 d c}\) | \(29\) |
default | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x \,d^{2}+c d \right ) \sqrt {3}}{3 c d}\right )}{3 c d}\) | \(34\) |
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none
Time = 0.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d x + c\right )}}{3 \, c}\right )}{3 \, c d} \]
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Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {- \frac {\sqrt {3} i \log {\left (x + \frac {c - \sqrt {3} i c}{2 d} \right )}}{3} + \frac {\sqrt {3} i \log {\left (x + \frac {c + \sqrt {3} i c}{2 d} \right )}}{3}}{c d} \]
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none
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d^{2} x + c d\right )}}{3 \, c d}\right )}{3 \, c d} \]
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none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, d x + c\right )}}{3 \, c}\right )}{3 \, c d} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {c-d x}{c^3-d^3 x^3} \, dx=\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}+\frac {2\,\sqrt {3}\,d\,x}{3\,c}\right )}{3\,c\,d} \]
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