Integrand size = 14, antiderivative size = 558 \[ \int \frac {1}{a+b x^3+c x^6} \, dx=-\frac {2^{2/3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}} \] Output:
-1/3*2^(2/3)*c^(2/3)*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))*3^(1/2)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/3 )+1/3*2^(2/3)*c^(2/3)*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))*3^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/ 3)+1/3*2^(2/3)*c^(2/3)*ln((b-(-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)/ (-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/3)-1/3*2^(2/3)*c^(2/3)*ln((b+ (-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c +b^2)^(1/2))^(2/3)-1/6*c^(2/3)*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)/(-4*a*c+b ^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/3)+1/6*c^(2/3)*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)/(-4*a*c+b^2)^(1/2)/(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 = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.08 \[ \int \frac {1}{a+b x^3+c x^6} \, dx=\frac {1}{3} \text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{b \text {$\#$1}^2+2 c \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(a + b*x^3 + c*x^6)^(-1),x]
Output:
RootSum[a + b*#1^3 + c*#1^6 & , Log[x - #1]/(b*#1^2 + 2*c*#1^5) & ]/3
Time = 0.77 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.87, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {1685, 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.
| \(\displaystyle \int \frac {1}{a+b x^3+c x^6} \, dx\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {c \int \frac {1}{c x^3+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{c x^3+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \int \frac {1}{\sqrt [3]{c} x+\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2^{2/3} \int \frac {2 \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}-\frac {\int -\frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {\int \frac {\sqrt [3]{c} \left (2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x\right )}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{b-\sqrt {b^2-4 a c}} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {3 \sqrt [3]{\sqrt {b^2-4 a c}+b} \int \frac {1}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx}{2 \sqrt [3]{2}}+\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (\frac {1}{4} \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-4 \sqrt [3]{c} x}{2 c^{2/3} x^2-2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\sqrt [3]{2} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {c \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (\frac {2\ 2^{2/3} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}-\frac {\log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{4 \sqrt [3]{c}}\right )}{3 \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}\right )}{\sqrt {b^2-4 a c}}\) |
Input:
Int[(a + b*x^3 + c*x^6)^(-1),x]
Output:
(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))))/Sqrt[b^2 - 4*a*c] - (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))))/Sqrt[b^2 - 4*a*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_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ c/q Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.07
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}\right )}{3}\) | \(40\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}\right )}{3}\) | \(40\) |
Input:
int(1/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/3*sum(1/(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. 2206 vs. \(2 (421) = 842\).
Time = 0.13 (sec) , antiderivative size = 2206, normalized size of antiderivative = 3.95 \[ \int \frac {1}{a+b x^3+c x^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(c*x^6+b*x^3+a),x, algorithm="fricas")
Output:
-1/6*(1/2)^(1/3)*(sqrt(-3) + 1)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2* c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b )/(a^2*b^2 - 4*a^3*c))^(1/3)*log(-4*(b^2*c - 2*a*c^2)*x - (1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2 + sqrt(-3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2) - (a^2*b ^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 + sqrt(-3)*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4 *b*c^2))*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a ^6*b^2*c^2 - 64*a^7*c^3)))*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4 *a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^ 2*b^2 - 4*a^3*c))^(1/3)) + 1/6*(1/2)^(1/3)*(sqrt(-3) - 1)*(((a^2*b^2 - 4*a ^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6* b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)*log(-4*(b^2*c - 2*a *c^2)*x - (1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2 - sqrt(-3)*(b^4 - 6*a*b ^2*c + 8*a^2*c^2) - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2 - sqrt(-3)*(a^2* b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2))*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4 *b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)))*(((a^2*b^2 - 4*a^3*c) *sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c ^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)) - 1/6*(1/2)^(1/3)*(sqrt (-3) + 1)*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b ^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c) )^(1/3)*log(-4*(b^2*c - 2*a*c^2)*x - (1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a...
Time = 2.66 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.28 \[ \int \frac {1}{a+b x^3+c x^6} \, dx=\operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a^{5} c^{3} - 34992 a^{4} b^{2} c^{2} + 8748 a^{3} b^{4} c - 729 a^{2} b^{6}\right ) + t^{3} \cdot \left (432 a^{2} b c^{2} - 216 a b^{3} c + 27 b^{5}\right ) + c^{2}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{4} b c^{2} + 648 t^{4} a^{3} b^{3} c - 81 t^{4} a^{2} b^{5} + 12 t a^{2} c^{2} - 15 t a b^{2} c + 3 t b^{4}}{2 a c^{2} - b^{2} c} \right )} \right )\right )} \] Input:
integrate(1/(c*x**6+b*x**3+a),x)
Output:
RootSum(_t**6*(46656*a**5*c**3 - 34992*a**4*b**2*c**2 + 8748*a**3*b**4*c - 729*a**2*b**6) + _t**3*(432*a**2*b*c**2 - 216*a*b**3*c + 27*b**5) + c**2, Lambda(_t, _t*log(x + (-1296*_t**4*a**4*b*c**2 + 648*_t**4*a**3*b**3*c - 81*_t**4*a**2*b**5 + 12*_t*a**2*c**2 - 15*_t*a*b**2*c + 3*_t*b**4)/(2*a*c* *2 - b**2*c))))
\[ \int \frac {1}{a+b x^3+c x^6} \, dx=\int { \frac {1}{c x^{6} + b x^{3} + a} \,d x } \] Input:
integrate(1/(c*x^6+b*x^3+a),x, algorithm="maxima")
Output:
integrate(1/(c*x^6 + b*x^3 + a), x)
\[ \int \frac {1}{a+b x^3+c x^6} \, dx=\int { \frac {1}{c x^{6} + b x^{3} + a} \,d x } \] Input:
integrate(1/(c*x^6+b*x^3+a),x, algorithm="giac")
Output:
integrate(1/(c*x^6 + b*x^3 + a), x)
Time = 26.98 (sec) , antiderivative size = 2597, normalized size of antiderivative = 4.65 \[ \int \frac {1}{a+b x^3+c x^6} \, dx=\text {Too large to display} \] Input:
int(1/(a + b*x^3 + c*x^6),x)
Output:
log(6*c^5*x + (2^(2/3)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^ 2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/ 3)*(36*a*c^5 - 9*b^2*c^4 + (9*2^(1/3)*b*c^3*(x + (2^(2/3)*a*(-(b^5 + b^2*( -(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2) ^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3))/2)*(4*a*c - b^2)^2*(-(b^5 + b^2*( -(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2) ^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(2/3))/2))/6)*((b^5 + b^2*(-(4*a*c - b^2 )^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(5 4*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) + log(6*c ^5*x + (2^(2/3)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a *b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3)*(36* a*c^5 - 9*b^2*c^4 + (9*2^(1/3)*b*c^3*(x + (2^(2/3)*a*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/ 2))/(a^2*(4*a*c - b^2)^3))^(1/3))/2)*(4*a*c - b^2)^2*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/ 2))/(a^2*(4*a*c - b^2)^3))^(2/3))/2))/6)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1 /2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^2* b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) + log(6*c^5*x - (2^(2/3)*(3^(1/2)*1i - 1)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b *c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3...
\[ \int \frac {1}{a+b x^3+c x^6} \, dx=\int \frac {1}{c \,x^{6}+b \,x^{3}+a}d x \] Input:
int(1/(c*x^6+b*x^3+a),x)
Output:
int(1/(a + b*x**3 + c*x**6),x)