Integrand size = 29, antiderivative size = 386 \[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=-\frac {d \arctan \left (\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} (c+d x)}{\sqrt [4]{c} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}\right )}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \arctan \left (\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}+\sqrt {2} (c+d x)}{\sqrt [4]{c} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}\right )}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} (c+d x)}{\sqrt {c} \sqrt {c^3+4 a d^2}+(c+d x)^2}\right )}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}} \] Output:
-1/4*d*arctan((c^(1/4)*(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)-2^(1/2)*(d*x+c) )/c^(1/4)/(-c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2))*2^(1/2)/c^(3/4)/(4*a*d^2+c ^3)^(1/2)/(-c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)+1/4*d*arctan((c^(1/4)*(c^(3 /2)+(4*a*d^2+c^3)^(1/2))^(1/2)+2^(1/2)*(d*x+c))/c^(1/4)/(-c^(3/2)+(4*a*d^2 +c^3)^(1/2))^(1/2))*2^(1/2)/c^(3/4)/(4*a*d^2+c^3)^(1/2)/(-c^(3/2)+(4*a*d^2 +c^3)^(1/2))^(1/2)+1/4*d*arctanh(2^(1/2)*c^(1/4)*(c^(3/2)+(4*a*d^2+c^3)^(1 /2))^(1/2)*(d*x+c)/(c^(1/2)*(4*a*d^2+c^3)^(1/2)+(d*x+c)^2))*2^(1/2)/c^(3/4 )/(4*a*d^2+c^3)^(1/2)/(c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.18 \[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\frac {1}{4} \text {RootSum}\left [4 a c+4 c^2 \text {$\#$1}^2+4 c d \text {$\#$1}^3+d^2 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{2 c^2 \text {$\#$1}+3 c d \text {$\#$1}^2+d^2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-1),x]
Output:
RootSum[4*a*c + 4*c^2*#1^2 + 4*c*d*#1^3 + d^2*#1^4 & , Log[x - #1]/(2*c^2* #1 + 3*c*d*#1^2 + d^2*#1^3) & ]/4
Time = 1.01 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2458, 1407, 27, 1142, 25, 27, 1083, 219, 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.
| \(\displaystyle \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}d\left (\frac {c}{d}+x\right )\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {d \int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-d \left (\frac {c}{d}+x\right )}{d \left (\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {d \int \frac {d \left (\frac {c}{d}+x\right )+\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d \left (\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-d \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\int \frac {d \left (\frac {c}{d}+x\right )+\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\frac {1}{2} d \int -\frac {\sqrt {2} \left (\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )\right )}{d \left (\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {1}{2} d \int \frac {\sqrt {2} \left (\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}\right )}{d \left (\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {1}{2} d \int \frac {\sqrt {2} \left (\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )\right )}{d \left (\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {1}{2} d \int \frac {\sqrt {2} \left (\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}\right )}{d \left (\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {\int \frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\frac {\sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {\int \frac {\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\frac {2 \sqrt {c} \left (c^{3/2}-\sqrt {c^3+4 a d^2}\right )}{d^2}-\left (2 \left (\frac {c}{d}+x\right )-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )^2}d\left (2 \left (\frac {c}{d}+x\right )-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\frac {\int \frac {\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \int \frac {1}{\frac {2 \sqrt {c} \left (c^{3/2}-\sqrt {c^3+4 a d^2}\right )}{d^2}-\left (2 \left (\frac {c}{d}+x\right )+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )^2}d\left (2 \left (\frac {c}{d}+x\right )+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\frac {d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \text {arctanh}\left (\frac {d \left (2 \left (\frac {c}{d}+x\right )-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}{d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}\right )}{\sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\frac {\int \frac {\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\frac {d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \text {arctanh}\left (\frac {d \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}{d}+2 \left (\frac {c}{d}+x\right )\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}\right )}{\sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \text {arctanh}\left (\frac {d \left (2 \left (\frac {c}{d}+x\right )-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}{d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}\right )}{\sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}-\frac {1}{2} d \log \left (\sqrt {c} \sqrt {4 a d^2+c^3}-\sqrt {2} \sqrt [4]{c} d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \left (\frac {c}{d}+x\right )+d^2 \left (\frac {c}{d}+x\right )^2\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\frac {1}{2} d \log \left (\sqrt {c} \sqrt {4 a d^2+c^3}+\sqrt {2} \sqrt [4]{c} d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \left (\frac {c}{d}+x\right )+d^2 \left (\frac {c}{d}+x\right )^2\right )-\frac {d \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}} \text {arctanh}\left (\frac {d \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}{d}+2 \left (\frac {c}{d}+x\right )\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}\right )}{\sqrt {c^{3/2}-\sqrt {4 a d^2+c^3}}}}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}\) |
Input:
Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-1),x]
Output:
(-((d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*ArcTanh[(d*(-((Sqrt[2]*c^(1/4)*S qrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]])/d) + 2*(c/d + x)))/(Sqrt[2]*c^(1/4)*Sq rt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]])])/Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] - Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2) + Sq rt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/2)/(2*Sqrt[2]*c^(3/4)*Sqr t[c^3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]) + (-((d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*ArcTanh[(d*((Sqrt[2]*c^(1/4)*Sqrt[c^(3/2) + Sqrt[c ^3 + 4*a*d^2]])/d + 2*(c/d + x)))/(Sqrt[2]*c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]])])/Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*Log[Sqrt[c]*Sqrt [c^3 + 4*a*d^2] + Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c /d + x) + d^2*(c/d + x)^2])/2)/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt [c^(3/2) + Sqrt[c^3 + 4*a*d^2]])
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.17
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{2} \textit {\_Z}^{4}+4 c d \,\textit {\_Z}^{3}+4 c^{2} \textit {\_Z}^{2}+4 a c \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} d^{2}+3 \textit {\_R}^{2} c d +2 \textit {\_R} \,c^{2}}\right )}{4}\) | \(64\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{2} \textit {\_Z}^{4}+4 c d \,\textit {\_Z}^{3}+4 c^{2} \textit {\_Z}^{2}+4 a c \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} d^{2}+3 \textit {\_R}^{2} c d +2 \textit {\_R} \,c^{2}}\right )}{4}\) | \(64\) |
Input:
int(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x,method=_RETURNVERBOSE)
Output:
1/4*sum(1/(_R^3*d^2+3*_R^2*c*d+2*_R*c^2)*ln(x-_R),_R=RootOf(_Z^4*d^2+4*_Z^ 3*c*d+4*_Z^2*c^2+4*a*c))
Leaf count of result is larger than twice the leaf count of optimal. 905 vs. \(2 (295) = 590\).
Time = 0.08 (sec) , antiderivative size = 905, normalized size of antiderivative = 2.34 \[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x, algorithm="fricas")
Output:
1/8*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3 *c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d + (2*a*c*d^2 + (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqr t(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^ 4)) + 1)/(a*c^3 + 4*a^2*d^2))) - 1/8*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^ 2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))*log( d^2*x + c*d - (2*a*c*d^2 + (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^ 2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^ 9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))) + 1/8*sqrt ((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4) ) - 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d + (2*a*c*d^2 - (a*c^7 + 4*a^2* c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt((2*(a*c ^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/( a*c^3 + 4*a^2*d^2))) - 1/8*sqrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2))*log(d^2*x + c*d - (2*a*c*d^2 - (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^ 6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2)))
Time = 0.63 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.23 \[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (16384 a^{3} c^{3} d^{2} + 4096 a^{2} c^{6}\right ) + 128 t^{2} a c^{3} + 1, \left ( t \mapsto t \log {\left (x + \frac {- 1024 t^{3} a^{2} c^{4} d^{2} - 256 t^{3} a c^{7} + 16 t a c d^{2} - 4 t c^{4} + c d}{d^{2}} \right )} \right )\right )} \] Input:
integrate(1/(d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c),x)
Output:
RootSum(_t**4*(16384*a**3*c**3*d**2 + 4096*a**2*c**6) + 128*_t**2*a*c**3 + 1, Lambda(_t, _t*log(x + (-1024*_t**3*a**2*c**4*d**2 - 256*_t**3*a*c**7 + 16*_t*a*c*d**2 - 4*_t*c**4 + c*d)/d**2)))
\[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int { \frac {1}{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c} \,d x } \] Input:
integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x, algorithm="maxima")
Output:
integrate(1/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)
Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (295) = 590\).
Time = 0.11 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.56 \[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=-\frac {\log \left (x + \sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{4 \, {\left (d^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{3} - 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}\right )}} + \frac {\log \left (x - \sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{4 \, {\left (d^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{3} + 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} + 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}\right )}} - \frac {\log \left (x + \sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{4 \, {\left (d^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{3} - 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}\right )}} + \frac {\log \left (x - \sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} + \frac {c}{d}\right )}{4 \, {\left (d^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{3} + 3 \, c d {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}^{2} + 2 \, c^{2} {\left (\sqrt {\frac {c^{2} d^{2} - 2 \, \sqrt {-a c} d^{3}}{d^{4}}} - \frac {c}{d}\right )}\right )}} \] Input:
integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x, algorithm="giac")
Output:
-1/4*log(x + sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)/(d^2*(sqrt((c^2 *d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)^3 - 3*c*d*(sqrt((c^2*d^2 + 2*sqrt(-a* c)*d^3)/d^4) + c/d)^2 + 2*c^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/ d)) + 1/4*log(x - sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)/(d^2*(sqrt ((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)^3 + 3*c*d*(sqrt((c^2*d^2 + 2*sqr t(-a*c)*d^3)/d^4) - c/d)^2 + 2*c^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)) - 1/4*log(x + sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)/(d^2* (sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)^3 - 3*c*d*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)^2 + 2*c^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3) /d^4) + c/d)) + 1/4*log(x - sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)/ (d^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) - c/d)^3 + 3*c*d*(sqrt((c^2*d ^2 - 2*sqrt(-a*c)*d^3)/d^4) - c/d)^2 + 2*c^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c) *d^3)/d^4) - c/d))
Time = 23.83 (sec) , antiderivative size = 1551, normalized size of antiderivative = 4.02 \[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\text {Too large to display} \] Input:
int(1/(4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3),x)
Output:
atan(((-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/ 2)*(((256*a*c^4*d^5 + 256*a*c^3*d^6*x)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(6 4*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2) - 64*a*c*d^6)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2) + 4*c*d^5 + 4*d^6*x)*1i + ( -(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2)*(((2 56*a*c^4*d^5 + 256*a*c^3*d^6*x)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2* c^6 + 4*a^3*c^3*d^2)))^(1/2) + 64*a*c*d^6)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3 )/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2) + 4*c*d^5 + 4*d^6*x)*1i)/((-(2*d*( -a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2)*(((256*a*c^ 4*d^5 + 256*a*c^3*d^6*x)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4 *a^3*c^3*d^2)))^(1/2) - 64*a*c*d^6)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*( a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2) + 4*c*d^5 + 4*d^6*x) - (-(2*d*(-a^3*c^3)^ (1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2)*(((256*a*c^4*d^5 + 25 6*a*c^3*d^6*x)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d ^2)))^(1/2) + 64*a*c*d^6)*(-(2*d*(-a^3*c^3)^(1/2) + a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2) + 4*c*d^5 + 4*d^6*x)))*(-(2*d*(-a^3*c^3)^(1/2) + a* c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2)*2i + atan((((2*d*(-a^3*c^3)^(1/ 2) - a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)))^(1/2)*(((256*a*c^4*d^5 + 256*a *c^3*d^6*x)*((2*d*(-a^3*c^3)^(1/2) - a*c^3)/(64*(a^2*c^6 + 4*a^3*c^3*d^2)) )^(1/2) - 64*a*c*d^6)*((2*d*(-a^3*c^3)^(1/2) - a*c^3)/(64*(a^2*c^6 + 4*...
\[ \int \frac {1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int \frac {1}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \] Input:
int(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x)
Output:
int(1/(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4),x)