Integrand size = 29, antiderivative size = 565 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=-\frac {(c+d x) \left (c^3-4 a d^2-c (c+d x)^2\right )}{16 a c d \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac {d \left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} (c+d x)}{\sqrt [4]{c} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \left (c^3+12 a d^2+c^{3/2} \sqrt {c^3+4 a d^2}\right ) \arctan \left (\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}+\sqrt {2} (c+d x)}{\sqrt [4]{c} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {-c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {d \left (c^3+12 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} (c+d x)}{\sqrt {c} \sqrt {c^3+4 a d^2}+(c+d x)^2}\right )}{32 \sqrt {2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}} \] Output:
-1/16*(d*x+c)*(c^3-4*a*d^2-c*(d*x+c)^2)/a/c/d/(4*a*d^2+c^3)/(d^2*x^4+4*c*d *x^3+4*c^2*x^2+4*a*c)-1/64*d*(c^3+12*a*d^2+c^(3/2)*(4*a*d^2+c^3)^(1/2))*ar ctan((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)/a/c^(7/4)/(4*a*d^2+c^3)^(3/ 2)/(-c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)+1/64*d*(c^3+12*a*d^2+c^(3/2)*(4*a* d^2+c^3)^(1/2))*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)/a/c^(7/4) /(4*a*d^2+c^3)^(3/2)/(-c^(3/2)+(4*a*d^2+c^3)^(1/2))^(1/2)+1/64*d*(c^3+12*a *d^2-c^(3/2)*(4*a*d^2+c^3)^(1/2))*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)/a/c^(7/4)/(4*a*d^2+c^3)^(3/2)/(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 = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\frac {\frac {4 (c+d x) (4 a d+c x (2 c+d x))}{4 a c+x^2 (2 c+d x)^2}+\text {RootSum}\left [4 a c+4 c^2 \text {$\#$1}^2+4 c d \text {$\#$1}^3+d^2 \text {$\#$1}^4\&,\frac {2 c^3 \log (x-\text {$\#$1})+12 a d^2 \log (x-\text {$\#$1})+2 c^2 d \log (x-\text {$\#$1}) \text {$\#$1}+c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{2 c^2 \text {$\#$1}+3 c d \text {$\#$1}^2+d^2 \text {$\#$1}^3}\&\right ]}{64 a c \left (c^3+4 a d^2\right )} \] Input:
Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-2),x]
Output:
((4*(c + d*x)*(4*a*d + c*x*(2*c + d*x)))/(4*a*c + x^2*(2*c + d*x)^2) + Roo tSum[4*a*c + 4*c^2*#1^2 + 4*c*d*#1^3 + d^2*#1^4 & , (2*c^3*Log[x - #1] + 1 2*a*d^2*Log[x - #1] + 2*c^2*d*Log[x - #1]*#1 + c*d^2*Log[x - #1]*#1^2)/(2* c^2*#1 + 3*c*d*#1^2 + d^2*#1^3) & ])/(64*a*c*(c^3 + 4*a*d^2))
Time = 1.24 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.41, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2458, 1405, 27, 1483, 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.
| \(\displaystyle \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )^2}d\left (\frac {c}{d}+x\right )\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {\int \frac {2 c \left (c^3+d^2 \left (\frac {c}{d}+x\right )^2 c+12 a d^2\right )}{d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}d\left (\frac {c}{d}+x\right )}{32 a c^2 \left (4 a d^2+c^3\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {c^3+d^2 \left (\frac {c}{d}+x\right )^2 c+12 a d^2}{d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}d\left (\frac {c}{d}+x\right )}{16 a c \left (4 a d^2+c^3\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {d \int \frac {\sqrt {2} \sqrt [4]{c} \left (c^3+12 a d^2\right ) \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-d \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \left (\frac {c}{d}+x\right )}{d \left (\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {d \int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+12 a d^2\right )+d \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \left (\frac {c}{d}+x\right )}{d \left (\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}\right )}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}}{16 a c \left (4 a d^2+c^3\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \left (c^3+12 a d^2\right ) \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-d \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+12 a d^2\right )+d \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{2 \sqrt {2} c^{3/4} \sqrt {4 a d^2+c^3} \sqrt {\sqrt {4 a d^2+c^3}+c^{3/2}}}}{16 a c \left (4 a d^2+c^3\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac {c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}}{16 a c \left (c^3+4 a d^2\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-d^2 \left (\frac {c}{d}+x\right )^2 c-4 a d^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}}{16 a c \left (c^3+4 a d^2\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-d^2 \left (\frac {c}{d}+x\right )^2 c-4 a d^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \int \frac {1}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {\left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \int \frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \int \frac {1}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}+\frac {\left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \int \frac {\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}}{16 a c \left (c^3+4 a d^2\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-d^2 \left (\frac {c}{d}+x\right )^2 c-4 a d^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\frac {\left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \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 {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}}{16 a c \left (c^3+4 a d^2\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-d^2 \left (\frac {c}{d}+x\right )^2 c-4 a d^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \int \frac {\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}-\sqrt {2} d \left (\frac {c}{d}+x\right )}{\left (\frac {c}{d}+x\right )^2-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\frac {d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \text {arctanh}\left (\frac {d \left (2 \left (\frac {c}{d}+x\right )-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{\sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\frac {\left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \int \frac {\sqrt {2} d \left (\frac {c}{d}+x\right )+\sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{\left (\frac {c}{d}+x\right )^2+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )}{d}+\frac {\sqrt {c} \sqrt {c^3+4 a d^2}}{d^2}}d\left (\frac {c}{d}+x\right )}{\sqrt {2}}-\frac {d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \text {arctanh}\left (\frac {d \left (2 \left (\frac {c}{d}+x\right )+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{\sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}}{16 a c \left (c^3+4 a d^2\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-d^2 \left (\frac {c}{d}+x\right )^2 c-4 a d^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \text {arctanh}\left (\frac {d \left (2 \left (\frac {c}{d}+x\right )-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{\sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}-\frac {1}{2} d \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \log \left (d^2 \left (\frac {c}{d}+x\right )^2-\sqrt {2} \sqrt [4]{c} d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )+\sqrt {c} \sqrt {c^3+4 a d^2}\right )}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}+\frac {\frac {1}{2} d \left (c^3-\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \log \left (d^2 \left (\frac {c}{d}+x\right )^2+\sqrt {2} \sqrt [4]{c} d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (\frac {c}{d}+x\right )+\sqrt {c} \sqrt {c^3+4 a d^2}\right )-\frac {d \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}} \left (c^3+\sqrt {c^3+4 a d^2} c^{3/2}+12 a d^2\right ) \text {arctanh}\left (\frac {d \left (2 \left (\frac {c}{d}+x\right )+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}{d}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}\right )}{\sqrt {c^{3/2}-\sqrt {c^3+4 a d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {c^3+4 a d^2} \sqrt {c^{3/2}+\sqrt {c^3+4 a d^2}}}}{16 a c \left (c^3+4 a d^2\right )}-\frac {\left (\frac {c}{d}+x\right ) \left (c^3-d^2 \left (\frac {c}{d}+x\right )^2 c-4 a d^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )\right )}\) |
Input:
Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-2),x]
Output:
-1/16*((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(a*c*(c^3 + 4*a*d^2) *(c*(4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4)) + ((-((d*Sqrt[ c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c^3 + 12*a*d^2 + 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]])])/S qrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^ 3 + 4*a*d^2])*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]]) + (-((d*Sqr t[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c^3 + 12*a*d^2 + 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]])])/Sq rt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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]]))/(16*a*c*(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)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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 = 230, normalized size of antiderivative = 0.41
| method | result | size |
| default | \(\frac {\frac {d^{2} x^{3}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {3 c d \,x^{2}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {\left (2 a \,d^{2}+c^{3}\right ) x}{32 c \left (4 a \,d^{2}+c^{3}\right ) a}+\frac {d}{64 a \,d^{2}+16 c^{3}}}{\frac {1}{4} d^{2} x^{4}+c d \,x^{3}+c^{2} x^{2}+a c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{2} \textit {\_Z}^{4}+4 c d \,\textit {\_Z}^{3}+4 c^{2} \textit {\_Z}^{2}+4 a c \right )}{\sum }\frac {\left (\textit {\_R}^{2} c \,d^{2}+2 \textit {\_R} \,c^{2} d +12 a \,d^{2}+2 c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} d^{2}+3 \textit {\_R}^{2} c d +2 \textit {\_R} \,c^{2}}}{64 a c \left (4 a \,d^{2}+c^{3}\right )}\) | \(230\) |
| risch | \(\frac {\frac {d^{2} x^{3}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {3 c d \,x^{2}}{64 a \left (4 a \,d^{2}+c^{3}\right )}+\frac {\left (2 a \,d^{2}+c^{3}\right ) x}{32 c \left (4 a \,d^{2}+c^{3}\right ) a}+\frac {d}{64 a \,d^{2}+16 c^{3}}}{\frac {1}{4} d^{2} x^{4}+c d \,x^{3}+c^{2} x^{2}+a c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d^{2} \textit {\_Z}^{4}+4 c d \,\textit {\_Z}^{3}+4 c^{2} \textit {\_Z}^{2}+4 a c \right )}{\sum }\frac {\left (\frac {d^{2} \textit {\_R}^{2}}{4 a \,d^{2}+c^{3}}+\frac {2 c d \textit {\_R}}{4 a \,d^{2}+c^{3}}+\frac {12 a \,d^{2}+2 c^{3}}{c \left (4 a \,d^{2}+c^{3}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3} d^{2}+3 \textit {\_R}^{2} c d +2 \textit {\_R} \,c^{2}}}{64 a}\) | \(252\) |
Input:
int(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x,method=_RETURNVERBOSE)
Output:
(1/64*d^2/a/(4*a*d^2+c^3)*x^3+3/64/a*c*d/(4*a*d^2+c^3)*x^2+1/32/c*(2*a*d^2 +c^3)/(4*a*d^2+c^3)/a*x+1/16*d/(4*a*d^2+c^3))/(1/4*d^2*x^4+c*d*x^3+c^2*x^2 +a*c)+1/64/a/c/(4*a*d^2+c^3)*sum((_R^2*c*d^2+2*_R*c^2*d+12*a*d^2+2*c^3)/(_ 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. 3222 vs. \(2 (461) = 922\).
Time = 0.16 (sec) , antiderivative size = 3222, normalized size of antiderivative = 5.70 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x, algorithm="fricas")
Output:
Too large to include
Time = 80.84 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\frac {4 a c d + 3 c^{2} d x^{2} + c d^{2} x^{3} + x \left (4 a d^{2} + 2 c^{3}\right )}{256 a^{3} c^{2} d^{2} + 64 a^{2} c^{5} + x^{4} \cdot \left (64 a^{2} c d^{4} + 16 a c^{4} d^{2}\right ) + x^{3} \cdot \left (256 a^{2} c^{2} d^{3} + 64 a c^{5} d\right ) + x^{2} \cdot \left (256 a^{2} c^{3} d^{2} + 64 a c^{6}\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (1073741824 a^{9} c^{7} d^{6} + 805306368 a^{8} c^{10} d^{4} + 201326592 a^{7} c^{13} d^{2} + 16777216 a^{6} c^{16}\right ) + t^{2} \cdot \left (491520 a^{5} c^{5} d^{4} + 122880 a^{4} c^{8} d^{2} + 8192 a^{3} c^{11}\right ) + 81 a^{2} d^{4} + 18 a c^{3} d^{2} + c^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 67108864 t^{3} a^{7} c^{7} d^{8} - 58720256 t^{3} a^{6} c^{10} d^{6} - 18874368 t^{3} a^{5} c^{13} d^{4} - 2621440 t^{3} a^{4} c^{16} d^{2} - 131072 t^{3} a^{3} c^{19} + 27648 t a^{4} c^{2} d^{8} - 9216 t a^{3} c^{5} d^{6} - 5440 t a^{2} c^{8} d^{4} - 736 t a c^{11} d^{2} - 32 t c^{14} + 324 a^{2} c d^{7} + 81 a c^{4} d^{5} + 5 c^{7} d^{3}}{324 a^{2} d^{8} + 81 a c^{3} d^{6} + 5 c^{6} d^{4}} \right )} \right )\right )} \] Input:
integrate(1/(d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**2,x)
Output:
(4*a*c*d + 3*c**2*d*x**2 + c*d**2*x**3 + x*(4*a*d**2 + 2*c**3))/(256*a**3* c**2*d**2 + 64*a**2*c**5 + x**4*(64*a**2*c*d**4 + 16*a*c**4*d**2) + x**3*( 256*a**2*c**2*d**3 + 64*a*c**5*d) + x**2*(256*a**2*c**3*d**2 + 64*a*c**6)) + RootSum(_t**4*(1073741824*a**9*c**7*d**6 + 805306368*a**8*c**10*d**4 + 201326592*a**7*c**13*d**2 + 16777216*a**6*c**16) + _t**2*(491520*a**5*c**5 *d**4 + 122880*a**4*c**8*d**2 + 8192*a**3*c**11) + 81*a**2*d**4 + 18*a*c** 3*d**2 + c**6, Lambda(_t, _t*log(x + (-67108864*_t**3*a**7*c**7*d**8 - 587 20256*_t**3*a**6*c**10*d**6 - 18874368*_t**3*a**5*c**13*d**4 - 2621440*_t* *3*a**4*c**16*d**2 - 131072*_t**3*a**3*c**19 + 27648*_t*a**4*c**2*d**8 - 9 216*_t*a**3*c**5*d**6 - 5440*_t*a**2*c**8*d**4 - 736*_t*a*c**11*d**2 - 32* _t*c**14 + 324*a**2*c*d**7 + 81*a*c**4*d**5 + 5*c**7*d**3)/(324*a**2*d**8 + 81*a*c**3*d**6 + 5*c**6*d**4))))
\[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\int { \frac {1}{{\left (d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c\right )}^{2}} \,d x } \] Input:
integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x, algorithm="maxima")
Output:
1/16*(c*d^2*x^3 + 3*c^2*d*x^2 + 4*a*c*d + 2*(c^3 + 2*a*d^2)*x)/(4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2* d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2) + 1/16*integrate((c*d^2*x^2 + 2* c^2*d*x + 2*c^3 + 12*a*d^2)/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)/ (a*c^4 + 4*a^2*c*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 1057 vs. \(2 (461) = 922\).
Time = 0.11 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x, algorithm="giac")
Output:
-1/64*((c*d^2*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d)^2 - 2*c^2*d*( sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) + c/d) + 2*c^3 + 12*a*d^2)*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)) - (c*d^2* (sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d)^2 + 2*c^2*d*(sqrt((c^2*d^2 + 2*sqrt(-a*c)*d^3)/d^4) - c/d) + 2*c^3 + 12*a*d^2)*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)) + (c*d^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) + c/d)^2 - 2*c^2*d*(sqrt((c^2*d^2 - 2*sqrt(-a*c) *d^3)/d^4) + c/d) + 2*c^3 + 12*a*d^2)*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)) - (c*d^2*(sqrt((c^2*d^2 - 2*sqrt(-a*c )*d^3)/d^4) - c/d)^2 + 2*c^2*d*(sqrt((c^2*d^2 - 2*sqrt(-a*c)*d^3)/d^4) - c /d) + 2*c^3 + 12*a*d^2)*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)))/(a*c^4 + 4*a^2*c*d^2) + 1/16*(c*d^2*x^3 + 3*c^2...
Time = 23.64 (sec) , antiderivative size = 5844, normalized size of antiderivative = 10.34 \[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/(4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^2,x)
Output:
(d/(4*(4*a*d^2 + c^3)) + (d^2*x^3)/(16*a*(4*a*d^2 + c^3)) + (x*(2*a*d^2 + c^3))/(8*a*c*(4*a*d^2 + c^3)) + (3*c*d*x^2)/(16*a*(4*a*d^2 + c^3)))/(4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3) - atan(((-(a^3*c^11 + 10*c^3*d^3*(-a^9 *c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2)) /(4096*(a^6*c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^( 1/2)*((((262144*a^4*c^12*d^5 + 2097152*a^5*c^9*d^7 + 4194304*a^6*c^6*d^9)/ (1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)) + (x*(4096*a^3*c^11*d^6 + 32768*a^4*c^8*d^8 + 65536*a^5*c^5*d^10))/(16*(a^2*c^8 + 8*a^3*c^5*d^2 + 16*a^4*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d ^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7* c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) - (4096*a^3*c^8*d^6 + 65536*a^4*c^5*d^8 + 196608*a^5*c^2*d^10)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^2*d^4)))*(-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8* d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6*c^16 + 12*a^7 *c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2) + (64*a*c^7*d^5 + 23 04*a^3*c*d^9 + 704*a^2*c^4*d^7)/(1024*(a^3*c^8 + 8*a^4*c^5*d^2 + 16*a^5*c^ 2*d^4)) + (x*(36*a^2*d^10 + c^6*d^6 + 11*a*c^3*d^8))/(16*(a^2*c^8 + 8*a^3* c^5*d^2 + 16*a^4*c^2*d^4)))*1i + (-(a^3*c^11 + 10*c^3*d^3*(-a^9*c^7)^(1/2) + 15*a^4*c^8*d^2 + 60*a^5*c^5*d^4 + 72*a*d^5*(-a^9*c^7)^(1/2))/(4096*(a^6 *c^16 + 12*a^7*c^13*d^2 + 48*a^8*c^10*d^4 + 64*a^9*c^7*d^6)))^(1/2)*(((...
\[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx=\int \frac {1}{d^{4} x^{8}+8 c \,d^{3} x^{7}+24 c^{2} d^{2} x^{6}+32 c^{3} d \,x^{5}+8 a c \,d^{2} x^{4}+16 c^{4} x^{4}+32 a \,c^{2} d \,x^{3}+32 a \,c^{3} x^{2}+16 a^{2} c^{2}}d x \] Input:
int(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^2,x)
Output:
int(1/(16*a**2*c**2 + 32*a*c**3*x**2 + 32*a*c**2*d*x**3 + 8*a*c*d**2*x**4 + 16*c**4*x**4 + 32*c**3*d*x**5 + 24*c**2*d**2*x**6 + 8*c*d**3*x**7 + d**4 *x**8),x)