Integrand size = 11, antiderivative size = 123 \[ \int \frac {x}{a+b x^6} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {281, 206, 31, 648, 631, 210, 642} \[ \int \frac {x}{a+b x^6} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 281
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{6 a^{2/3}}+\frac {\text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{6 a^{2/3}} \\ & = \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac {\text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{4 \sqrt [3]{a}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 a^{2/3} \sqrt [3]{b}} \\ & = \frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 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^2}{\sqrt [3]{a}}\right )}{2 a^{2/3} \sqrt [3]{b}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{2/3} \sqrt [3]{b}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.25 \[ \int \frac {x}{a+b x^6} \, dx=-\frac {2 \sqrt {3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 a^{2/3} \sqrt [3]{b}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.22
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,a^{2}-1\right )}{\sum }\textit {\_R} \ln \left (a \textit {\_R} +x^{2}\right )\right )}{6}\) | \(27\) |
default | \(\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\) | \(97\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.54 \[ \int \frac {x}{a+b x^6} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{6} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{2} - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{4} + \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{6} + a}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{12 \, a^{2} b}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{2} - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{4} - \left (a^{2} b\right )^{\frac {2}{3}} x^{2} + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{12 \, a^{2} b}\right ] \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {x}{a+b x^6} \, dx=\operatorname {RootSum} {\left (216 t^{3} a^{2} b - 1, \left ( t \mapsto t \log {\left (6 t a + x^{2} \right )} \right )\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.85 \[ \int \frac {x}{a+b x^6} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int \frac {x}{a+b x^6} \, dx=-\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, a} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, a b} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, a b} \]
[In]
[Out]
Time = 5.62 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {x}{a+b x^6} \, dx=\frac {\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{6\,a^{2/3}\,b^{1/3}}+\frac {\ln \left (6\,b^4\,x^2+3\,a^{1/3}\,b^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12\,a^{2/3}\,b^{1/3}}-\frac {\ln \left (6\,b^4\,x^2-3\,a^{1/3}\,b^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12\,a^{2/3}\,b^{1/3}} \]
[In]
[Out]