Integrand size = 10, antiderivative size = 139 \[ \int \frac {1}{-1+a+b x^3} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1-a}}}{\sqrt {3}}\right )}{\sqrt {3} (1-a)^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{1-a}-\sqrt [3]{b} x\right )}{3 (1-a)^{2/3} \sqrt [3]{b}}-\frac {\log \left ((1-a)^{2/3}+\sqrt [3]{1-a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1-a)^{2/3} \sqrt [3]{b}} \]
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Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {206, 31, 648, 631, 210, 642} \[ \int \frac {1}{-1+a+b x^3} \, dx=-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1-a}}+1}{\sqrt {3}}\right )}{\sqrt {3} (1-a)^{2/3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{1-a} \sqrt [3]{b} x+(1-a)^{2/3}+b^{2/3} x^2\right )}{6 (1-a)^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{1-a}-\sqrt [3]{b} x\right )}{3 (1-a)^{2/3} \sqrt [3]{b}} \]
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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 = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int \frac {1}{-1+a+b x^3} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{-1+a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{-1+a}+\sqrt [3]{b} x\right )-\log \left ((-1+a)^{2/3}-\sqrt [3]{-1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (-1+a)^{2/3} \sqrt [3]{b}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.77 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.20
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a -1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\) | \(28\) |
default | \(\frac {\ln \left (x +\left (\frac {a -1}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a -1}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a -1}{b}\right )^{\frac {1}{3}} x +\left (\frac {a -1}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a -1}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a -1}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a -1}{b}\right )^{\frac {2}{3}}}\) | \(105\) |
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Time = 0.29 (sec) , antiderivative size = 446, normalized size of antiderivative = 3.21 \[ \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 = 32, normalized size of antiderivative = 0.23 \[ \int \frac {1}{-1+a+b x^3} \, dx=\operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{2} b - 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log {\left (3 t a - 3 t + x \right )} \right )\right )} \]
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Exception generated. \[ \int \frac {1}{-1+a+b x^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02 \[ \int \frac {1}{-1+a+b x^3} \, dx=\frac {{\left (-a b^{2} + b^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a - 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a - 1}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b - \sqrt {3} b} + \frac {{\left (-a b^{2} + b^{2}\right )}^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a - 1}{b}\right )^{\frac {1}{3}} + \left (-\frac {a - 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b - b\right )}} - \frac {\left (-\frac {a - 1}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a - 1}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a - 1\right )}} \]
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Time = 5.76 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99 \[ \int \frac {1}{-1+a+b x^3} \, dx=\frac {\ln \left (a+b^{1/3}\,x\,{\left (a-1\right )}^{2/3}-1\right )}{3\,b^{1/3}\,{\left (a-1\right )}^{2/3}}+\frac {\ln \left (3\,b^2\,x+\frac {\left (9\,a\,b^2-9\,b^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a-1\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a-1\right )}^{2/3}}-\frac {\ln \left (3\,b^2\,x-\frac {\left (9\,a\,b^2-9\,b^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a-1\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a-1\right )}^{2/3}} \]
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