Integrand size = 11, antiderivative size = 124 \[ \int \frac {1}{1+a-b x^3} \, dx=\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )}{\sqrt {3} (1+a)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{1+a}-\sqrt [3]{b} x\right )}{3 (1+a)^{2/3} \sqrt [3]{b}}+\frac {\log \left ((1+a)^{2/3}+\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \]
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Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {206, 31, 648, 631, 210, 642} \[ \int \frac {1}{1+a-b x^3} \, dx=\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}+1}{\sqrt {3}}\right )}{\sqrt {3} (a+1)^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+1}-\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt [3]{1+a}-\sqrt [3]{b} x} \, dx}{3 (1+a)^{2/3}}+\frac {\int \frac {2 \sqrt [3]{1+a}+\sqrt [3]{b} x}{(1+a)^{2/3}+\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 (1+a)^{2/3}} \\ & = -\frac {\log \left (\sqrt [3]{1+a}-\sqrt [3]{b} x\right )}{3 (1+a)^{2/3} \sqrt [3]{b}}+\frac {\int \frac {1}{(1+a)^{2/3}+\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{1+a}}+\frac {\int \frac {\sqrt [3]{1+a} \sqrt [3]{b}+2 b^{2/3} x}{(1+a)^{2/3}+\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 (1+a)^{2/3} \sqrt [3]{b}} \\ & = -\frac {\log \left (\sqrt [3]{1+a}-\sqrt [3]{b} x\right )}{3 (1+a)^{2/3} \sqrt [3]{b}}+\frac {\log \left ((1+a)^{2/3}+\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}\right )}{(1+a)^{2/3} \sqrt [3]{b}} \\ & = \frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )}{\sqrt {3} (1+a)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{1+a}-\sqrt [3]{b} x\right )}{3 (1+a)^{2/3} \sqrt [3]{b}}+\frac {\log \left ((1+a)^{2/3}+\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00 \[ \int \frac {1}{1+a-b x^3} \, dx=\frac {(-1)^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{1+a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )+\log \left ((1+a)^{2/3}-\sqrt [3]{-1} \sqrt [3]{1+a} \sqrt [3]{b} x+(-1)^{2/3} b^{2/3} x^2\right )\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.24
method | result | size |
risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-a -1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\) | \(30\) |
default | \(-\frac {\ln \left (x -\left (\frac {1+a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (x^{2}+\left (\frac {1+a}{b}\right )^{\frac {1}{3}} x +\left (\frac {1+a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1+a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}\) | \(106\) |
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (90) = 180\).
Time = 0.27 (sec) , antiderivative size = 467, normalized size of antiderivative = 3.77 \[ \int \frac {1}{1+a-b x^3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + 1\right )} b \sqrt {\frac {\left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, {\left (a + 1\right )} b x^{3} + 3 \, \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )} x + a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a + 1\right )} b x^{2} - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x + \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right )} \sqrt {\frac {\left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} + 2 \, a + 1}{b x^{3} - a - 1}\right ) + \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x^{2} + \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right ) - 2 \, \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} + 2 \, a + 1\right )} b}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a + 1\right )} b \sqrt {-\frac {\left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right )} \sqrt {-\frac {\left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}}}{a^{2} + 2 \, a + 1}\right ) + \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x^{2} + \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right ) - 2 \, \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x - \left (-{\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} + 2 \, a + 1\right )} b}\right ] \]
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Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {1}{1+a-b x^3} \, dx=- \operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log {\left (- 3 t a - 3 t + x \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1+a-b x^3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x - \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.06 \[ \int \frac {1}{1+a-b x^3} \, dx=\frac {{\left (a b^{2} + b^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b + \sqrt {3} b} + \frac {{\left (a b^{2} + b^{2}\right )}^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b + b\right )}} - \frac {\left (\frac {a + 1}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a + 1\right )}} \]
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Time = 0.44 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.33 \[ \int \frac {1}{1+a-b x^3} \, dx=\frac {\ln \left (3\,b^2\,x+\frac {9\,a\,b^2+9\,b^2}{3\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}\right )}{3\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}+\frac {\ln \left (3\,b^2\,x+\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}-\frac {\ln \left (3\,b^2\,x-\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-b\right )}^{1/3}\,{\left (a+1\right )}^{2/3}} \]
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