Integrand size = 14, antiderivative size = 631 \[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\frac {x}{c}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} c^{4/3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} c^{4/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} c^{4/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} c^{4/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}} \] Output:
x/c+1/6*(b-(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctan(1/3*(1-2*2^(1/3)*c^(1/3 )*x/(b-(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)/c^(4/3)/(b-(-4* a*c+b^2)^(1/2))^(2/3)+1/6*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctan(1/3*( 1-2*2^(1/3)*c^(1/3)*x/(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*2^(2/3)*3^(1/ 2)/c^(4/3)/(b+(-4*a*c+b^2)^(1/2))^(2/3)-1/6*(b-(-2*a*c+b^2)/(-4*a*c+b^2)^( 1/2))*ln((b-(-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)*2^(2/3)/c^(4/3)/( b-(-4*a*c+b^2)^(1/2))^(2/3)-1/6*(b+(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*ln((b+ (-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)*2^(2/3)/c^(4/3)/(b+(-4*a*c+b^ 2)^(1/2))^(2/3)+1/12*(b-(-2*a*c+b^2)/(-4*a*c+b^2)^(1/2))*ln((b-(-4*a*c+b^2 )^(1/2))^(2/3)-2^(1/3)*c^(1/3)*(b-(-4*a*c+b^2)^(1/2))^(1/3)*x+2^(2/3)*c^(2 /3)*x^2)*2^(2/3)/c^(4/3)/(b-(-4*a*c+b^2)^(1/2))^(2/3)+1/12*(b+(-2*a*c+b^2) /(-4*a*c+b^2)^(1/2))*ln((b+(-4*a*c+b^2)^(1/2))^(2/3)-2^(1/3)*c^(1/3)*(b+(- 4*a*c+b^2)^(1/2))^(1/3)*x+2^(2/3)*c^(2/3)*x^2)*2^(2/3)/c^(4/3)/(b+(-4*a*c+ b^2)^(1/2))^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.11 \[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\frac {x}{c}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {a \log (x-\text {$\#$1})+b \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}^2+2 c \text {$\#$1}^5}\&\right ]}{3 c} \] Input:
Integrate[(c + a/x^6 + b/x^3)^(-1),x]
Output:
x/c - RootSum[a + b*#1^3 + c*#1^6 & , (a*Log[x - #1] + b*Log[x - #1]*#1^3) /(b*#1^2 + 2*c*#1^5) & ]/(3*c)
Time = 1.04 (sec) , antiderivative size = 521, normalized size of antiderivative = 0.83, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {1679, 1703, 1752, 750, 16, 27, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\frac {a}{x^6}+\frac {b}{x^3}+c} \, dx\) |
| \(\Big \downarrow \) 1679 |
| \(\displaystyle \int \frac {x^6}{a+b x^3+c x^6}dx\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle \frac {x}{c}-\frac {\int \frac {b x^3+a}{c x^6+b x^3+a}dx}{c}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^3+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \int \frac {1}{c x^3+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{c}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{c}-\frac {\frac {1}{2} \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )+\frac {1}{2} \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{c}\) |
Input:
Int[(c + a/x^6 + b/x^3)^(-1),x]
Output:
x/c - (((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*((2^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^( 2/3)) + (2*2^(2/3)*(-1/2*(Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - S qrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/c^(1/3) - Log[(b - Sqrt[b^2 - 4*a*c])^( 2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x ^2]/(4*c^(1/3))))/(3*(b - Sqrt[b^2 - 4*a*c])^(2/3))))/2 + ((b + (b^2 - 2*a *c)/Sqrt[b^2 - 4*a*c])*((2^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/ 3)*c^(1/3)*x])/(3*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + (2*2^(2/3)*(-1/ 2*(Sqrt[3]*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3) )/Sqrt[3]])/c^(1/3) - Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)* (b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2]/(4*c^(1/3))))/(3*(b + Sqrt[b^2 - 4*a*c])^(2/3))))/2)/c
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^( 2*n*p)*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[d^(2*n - 1)*(d*x)^(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^( p + 1)/(c*(m + 2*n*p + 1))), x] - Simp[d^(2*n)/(c*(m + 2*n*p + 1)) Int[(d *x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x ^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0 ] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{3} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 c}\) | \(59\) |
| risch | \(\frac {x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{3} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 c}\) | \(59\) |
Input:
int(1/(c+a/x^6+b/x^3),x,method=_RETURNVERBOSE)
Output:
x/c+1/3/c*sum((-_R^3*b-a)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3 *b+a))
Leaf count of result is larger than twice the leaf count of optimal. 2882 vs. \(2 (495) = 990\).
Time = 0.19 (sec) , antiderivative size = 2882, normalized size of antiderivative = 4.57 \[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\text {Too large to display} \] Input:
integrate(1/(c+a/x^6+b/x^3),x, algorithm="fricas")
Output:
-1/6*((1/2)^(1/3)*(sqrt(-3)*c + c)*(-(b^3 - 2*a*b*c + (b^2*c^4 - 4*a*c^5)* sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(b^6* c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)))/(b^2*c^4 - 4*a*c^5)) ^(1/3)*log(4*(a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*x - (1/2)^(1/3)*(b^6 - 8*a* b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c^3 + sqrt(-3)*(b^6 - 8*a*b^4*c + 18*a^2*b^ 2*c^2 - 8*a^3*c^3) - (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6 + sqrt(-3)*(b^5 *c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6))*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)))*(-(b^3 - 2*a*b*c + (b^2*c^4 - 4*a*c^5)*sqrt((b^8 - 8*a*b^6 *c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)))/(b^2*c^4 - 4*a*c^5))^(1/3)) - (1/2)^(1/ 3)*(sqrt(-3)*c - c)*(-(b^3 - 2*a*b*c + (b^2*c^4 - 4*a*c^5)*sqrt((b^8 - 8*a *b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(b^6*c^8 - 12*a*b^4* c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)))/(b^2*c^4 - 4*a*c^5))^(1/3)*log(4*(a *b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*x - (1/2)^(1/3)*(b^6 - 8*a*b^4*c + 18*a^2* b^2*c^2 - 8*a^3*c^3 - sqrt(-3)*(b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c ^3) - (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(-3)*(b^5*c^4 - 8*a*b^3* c^5 + 16*a^2*b*c^6))*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c ^3 + 4*a^4*c^4)/(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11))) *(-(b^3 - 2*a*b*c + (b^2*c^4 - 4*a*c^5)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*...
Time = 5.06 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.31 \[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a^{3} c^{7} - 34992 a^{2} b^{2} c^{6} + 8748 a b^{4} c^{5} - 729 b^{6} c^{4}\right ) + t^{3} \cdot \left (864 a^{3} b c^{3} - 864 a^{2} b^{3} c^{2} + 270 a b^{5} c - 27 b^{7}\right ) + a^{4}, \left ( t \mapsto t \log {\left (x + \frac {1296 t^{4} a^{2} b c^{6} - 648 t^{4} a b^{3} c^{5} + 81 t^{4} b^{5} c^{4} - 12 t a^{3} c^{3} + 39 t a^{2} b^{2} c^{2} - 21 t a b^{4} c + 3 t b^{6}}{2 a^{3} c^{2} - 4 a^{2} b^{2} c + a b^{4}} \right )} \right )\right )} + \frac {x}{c} \] Input:
integrate(1/(c+a/x**6+b/x**3),x)
Output:
RootSum(_t**6*(46656*a**3*c**7 - 34992*a**2*b**2*c**6 + 8748*a*b**4*c**5 - 729*b**6*c**4) + _t**3*(864*a**3*b*c**3 - 864*a**2*b**3*c**2 + 270*a*b**5 *c - 27*b**7) + a**4, Lambda(_t, _t*log(x + (1296*_t**4*a**2*b*c**6 - 648* _t**4*a*b**3*c**5 + 81*_t**4*b**5*c**4 - 12*_t*a**3*c**3 + 39*_t*a**2*b**2 *c**2 - 21*_t*a*b**4*c + 3*_t*b**6)/(2*a**3*c**2 - 4*a**2*b**2*c + a*b**4) ))) + x/c
\[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\int { \frac {1}{c + \frac {b}{x^{3}} + \frac {a}{x^{6}}} \,d x } \] Input:
integrate(1/(c+a/x^6+b/x^3),x, algorithm="maxima")
Output:
x/c - integrate((b*x^3 + a)/(c*x^6 + b*x^3 + a), x)/c
\[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\int { \frac {1}{c + \frac {b}{x^{3}} + \frac {a}{x^{6}}} \,d x } \] Input:
integrate(1/(c+a/x^6+b/x^3),x, algorithm="giac")
Output:
integrate(1/(c + b/x^3 + a/x^6), x)
Time = 23.57 (sec) , antiderivative size = 2280, normalized size of antiderivative = 3.61 \[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\text {Too large to display} \] Input:
int(1/(c + a/x^6 + b/x^3),x)
Output:
log((3*a^2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c - (3*2^(2/3)*a*(-(b^4*(-(4*a *c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4* a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(c^ 4*(4*a*c - b^2)^3))^(1/3)*(b^4 + 2*a^2*c^2 - 4*a*b^2*c)*(b*(-(4*a*c - b^2) ^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(4*c*(4*a*c - b^2)))*(-(b^4*(-( 4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(- (4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/ (54*(64*a^3*c^7 - b^6*c^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)))^(1/3) + x/c + log((3*a^2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c + (3*2^(2/3)*a*((b^7 + b^4* (-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4* a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(c^ 4*(4*a*c - b^2)^3))^(1/3)*(b^4 + 2*a^2*c^2 - 4*a*b^2*c)*(b*(-(4*a*c - b^2) ^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c))/(4*c*(4*a*c - b^2)))*((b^7 + b^ 4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-( 4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/( 54*(64*a^3*c^7 - b^6*c^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)))^(1/3) + log((3 *a^2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c + (3*2^(2/3)*a*(3^(1/2)*1i - 1)*(( b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2 *c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^ (1/2))/(c^4*(4*a*c - b^2)^3))^(1/3)*(b^4 + 2*a^2*c^2 - 4*a*b^2*c)*(b*(-...
\[ \int \frac {1}{c+\frac {a}{x^6}+\frac {b}{x^3}} \, dx=\frac {-\left (\int \frac {x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b -\left (\int \frac {1}{c \,x^{6}+b \,x^{3}+a}d x \right ) a +x}{c} \] Input:
int(1/(c+a/x^6+b/x^3),x)
Output:
( - int(x**3/(a + b*x**3 + c*x**6),x)*b - int(1/(a + b*x**3 + c*x**6),x)*a + x)/c