Integrand size = 15, antiderivative size = 382 \[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx =\text {Too large to display} \] Output:
27/256/(1-2*cos(2/3*arcsin(61/128)))^2/(5-3*x-8*sin(1/3*arcsin(61/128)))+8 1/4096*sec(1/3*arcsin(61/128))^2/(5-3*x-8*cos(1/6*Pi+1/3*arcsin(61/128)))/ (cos(1/6*Pi+1/3*arcsin(61/128))-sin(1/3*arcsin(61/128)))^2+27/256*ln(5-3*x -8*sin(1/3*arcsin(61/128)))*sin(1/3*arcsin(61/128))/(1-2*cos(2/3*arcsin(61 /128)))^3+27/4096*ln(5-3*x-8*cos(1/6*Pi+1/3*arcsin(61/128)))*sec(1/3*arcsi n(61/128))^3*(3^(1/2)*cos(1/3*arcsin(61/128))-sin(1/3*arcsin(61/128)))/(co s(1/3*arcsin(61/128))-3^(1/2)*sin(1/3*arcsin(61/128)))^3-27/4096*ln(5-3*x+ 8*sin(1/3*Pi+1/3*arcsin(61/128)))*sec(1/3*arcsin(61/128))^3*(3^(1/2)*cos(1 /3*arcsin(61/128))+sin(1/3*arcsin(61/128)))/(cos(1/3*arcsin(61/128))+3^(1/ 2)*sin(1/3*arcsin(61/128)))^3+81/4096*sec(1/3*arcsin(61/128))^2/(sin(1/3*a rcsin(61/128))+sin(1/3*Pi+1/3*arcsin(61/128)))^2/(5-3*x+8*sin(1/3*Pi+1/3*a rcsin(61/128)))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\frac {-9+127 x-32 x^2}{469 \left (2+3 x-5 x^2+x^3\right )}-\frac {2}{469} \text {RootSum}\left [2+3 \text {$\#$1}-5 \text {$\#$1}^2+\text {$\#$1}^3\&,\frac {-47 \log (x-\text {$\#$1})+16 \log (x-\text {$\#$1}) \text {$\#$1}}{3-10 \text {$\#$1}+3 \text {$\#$1}^2}\&\right ] \] Input:
Integrate[(2 + 3*x - 5*x^2 + x^3)^(-2),x]
Output:
(-9 + 127*x - 32*x^2)/(469*(2 + 3*x - 5*x^2 + x^3)) - (2*RootSum[2 + 3*#1 - 5*#1^2 + #1^3 & , (-47*Log[x - #1] + 16*Log[x - #1]*#1)/(3 - 10*#1 + 3*# 1^2) & ])/469
Result contains complex when optimal does not.
Time = 4.68 (sec) , antiderivative size = 1162, normalized size of antiderivative = 3.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2481, 2474, 27, 1165, 27, 1200, 2009}
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.
| \(\displaystyle \int \frac {1}{\left (x^3-5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2481 |
| \(\displaystyle \int \frac {1}{\left (\left (x-\frac {5}{3}\right )^3-\frac {16}{3} \left (x-\frac {5}{3}\right )-\frac {61}{27}\right )^2}d\left (x-\frac {5}{3}\right )\) |
| \(\Big \downarrow \) 2474 |
| \(\displaystyle \int \frac {11664}{\left (-6 \left (x-\frac {5}{3}\right )+\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}\right )^2 \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )^2}d\left (x-\frac {5}{3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 11664 \int \frac {1}{\left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )^2 \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )^2}d\left (x-\frac {5}{3}\right )\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle 11664 \left (-\frac {\left (-3 \sqrt {1407}+61 i\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \int -\frac {3888 \left (\frac {3 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) \left (x-\frac {5}{3}\right )}{\sqrt [3]{61+3 i \sqrt {1407}}}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}{\left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )^2 \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}d\left (x-\frac {5}{3}\right )}{34992 \left (61 \sqrt {1407}+4221 i\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )}-\frac {\left (-3 \sqrt {1407}+61 i\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \left (64+\frac {3 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) \left (x-\frac {5}{3}\right )}{\sqrt [3]{61+3 i \sqrt {1407}}}\right )}{108 \left (61 \sqrt {1407}+4221 i\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right ) \left (-6 \left (x-\frac {5}{3}\right )+\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}\right ) \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 11664 \left (\frac {\left (-3 \sqrt {1407}+61 i\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \int \frac {\frac {3 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) \left (x-\frac {5}{3}\right )}{\sqrt [3]{61+3 i \sqrt {1407}}}-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32}{\left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right )^2 \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}d\left (x-\frac {5}{3}\right )}{9 \left (61 \sqrt {1407}+4221 i\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )}-\frac {\left (-3 \sqrt {1407}+61 i\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \left (64+\frac {3 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) \left (x-\frac {5}{3}\right )}{\sqrt [3]{61+3 i \sqrt {1407}}}\right )}{108 \left (61 \sqrt {1407}+4221 i\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right ) \left (-6 \left (x-\frac {5}{3}\right )+\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}\right ) \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle 11664 \left (\frac {\left (61 i-3 \sqrt {1407}\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \int \left (\frac {2 \left (61+3 i \sqrt {1407}\right )^{2/3} \left (3 \left (61+3 i \sqrt {1407}\right ) \left (61 \sqrt [3]{61+3 i \sqrt {1407}}-8 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right )\right ) \left (x-\frac {5}{3}\right )+16 \sqrt [3]{2} \left (2971+549 i \sqrt {1407}\right )+\left (122+6 i \sqrt {1407}\right )^{2/3} \left (7817+183 i \sqrt {1407}\right )+256 i \left (61 i-3 \sqrt {1407}\right ) \sqrt [3]{61+3 i \sqrt {1407}}\right )}{3 \left (512\ 2^{2/3}+32 \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )^2 \left (18 \left (61+3 i \sqrt {1407}\right )^{2/3} \left (x-\frac {5}{3}\right )^2+3\ 2^{2/3} \left (61+3 i \sqrt {1407}+16\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}}\right ) \left (x-\frac {5}{3}\right )+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}-32 \left (61+3 i \sqrt {1407}\right )^{2/3}+512\ 2^{2/3}\right )}+\frac {2 \left (61 \left (61+3 i \sqrt {1407}\right )^{5/3}-16 \left (\sqrt [3]{122+6 i \sqrt {1407}} \left (976+48 i \sqrt {1407}\right )-2^{2/3} \left (4471-183 i \sqrt {1407}\right )\right )\right )}{3 \left (512\ 2^{2/3}+32 \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )^2 \left (-6 \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {5}{3}\right )+\left (2 \left (61+3 i \sqrt {1407}\right )\right )^{2/3}+32 \sqrt [3]{2}\right )}+\frac {32 \sqrt [3]{61+3 i \sqrt {1407}} \left (-61 i+3 \sqrt {1407}\right )}{\left (512 i 2^{2/3}+32 i \left (61+3 i \sqrt {1407}\right )^{2/3}+\left (61 i-3 \sqrt {1407}\right ) \sqrt [3]{122+6 i \sqrt {1407}}\right ) \left (-6 \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {5}{3}\right )+\left (2 \left (61+3 i \sqrt {1407}\right )\right )^{2/3}+32 \sqrt [3]{2}\right )^2}\right )d\left (x-\frac {5}{3}\right )}{9 \left (4221 i+61 \sqrt {1407}\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )}-\frac {\left (61 i-3 \sqrt {1407}\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \left (\frac {3 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) \left (x-\frac {5}{3}\right )}{\sqrt [3]{61+3 i \sqrt {1407}}}+64\right )}{108 \left (4221 i+61 \sqrt {1407}\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right ) \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right ) \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 11664 \left (\frac {\left (61 i-3 \sqrt {1407}\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \left (\frac {2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}} \left (7625-i \left (7817 \sqrt {1407}+16 \sqrt [3]{2 \left (122+6 i \sqrt {1407}\right )} \left (125 i+61 \sqrt {1407}\right )\right )\right ) \text {arctanh}\left (\frac {12 \left (61+3 i \sqrt {1407}\right )^{2/3} \left (x-\frac {5}{3}\right )+32 \sqrt [3]{122+6 i \sqrt {1407}}+2^{2/3} \left (61+3 i \sqrt {1407}\right )}{2 \sqrt {3 \left (32 \left (61+3 i \sqrt {1407}\right )^{4/3}+\sqrt [3]{2} \left (4471-183 i \sqrt {1407}-256 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )}}\right )}{\left (512\ 2^{2/3}+32 \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )^2 \sqrt {3 \left (32 \left (61+3 i \sqrt {1407}\right )^{4/3}+\sqrt [3]{2} \left (4471-183 i \sqrt {1407}-256 \sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )\right )}}-\frac {\left (61 \left (61+3 i \sqrt {1407}\right )^{5/3}-16 \left (\sqrt [3]{122+6 i \sqrt {1407}} \left (976+48 i \sqrt {1407}\right )-2^{2/3} \left (4471-183 i \sqrt {1407}\right )\right )\right ) \log \left (-6 \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {5}{3}\right )+\left (122+6 i \sqrt {1407}\right )^{2/3}+32 \sqrt [3]{2}\right )}{9 \sqrt [3]{61+3 i \sqrt {1407}} \left (512\ 2^{2/3}+32 \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )^2}+\frac {\left (61+3 i \sqrt {1407}\right ) \left (61 \sqrt [3]{61+3 i \sqrt {1407}}-8 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right )\right ) \log \left (18 \left (61+3 i \sqrt {1407}\right )^{2/3} \left (x-\frac {5}{3}\right )^2+3\ 2^{2/3} \left (61+3 i \sqrt {1407}+16\ 2^{2/3} \sqrt [3]{61+3 i \sqrt {1407}}\right ) \left (x-\frac {5}{3}\right )+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}-32 \left (61+3 i \sqrt {1407}\right )^{2/3}+512\ 2^{2/3}\right )}{18 \left (512\ 2^{2/3}+32 \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right )^2}-\frac {16 \left (61+3 i \sqrt {1407}\right )}{3 \left (512\ 2^{2/3}+32 \left (61+3 i \sqrt {1407}\right )^{2/3}+\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{4/3}\right ) \left (-6 \sqrt [3]{61+3 i \sqrt {1407}} \left (x-\frac {5}{3}\right )+\left (122+6 i \sqrt {1407}\right )^{2/3}+32 \sqrt [3]{2}\right )}\right )}{9 \left (4221 i+61 \sqrt {1407}\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right )}-\frac {\left (61 i-3 \sqrt {1407}\right ) \sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )} \left (\frac {3 \left (32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}\right ) \left (x-\frac {5}{3}\right )}{\sqrt [3]{61+3 i \sqrt {1407}}}+64\right )}{108 \left (4221 i+61 \sqrt {1407}\right ) \left (32-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}\right ) \left (\frac {32 \sqrt [3]{2}+\left (122+6 i \sqrt {1407}\right )^{2/3}}{\sqrt [3]{61+3 i \sqrt {1407}}}-6 \left (x-\frac {5}{3}\right )\right ) \left (-18 \left (x-\frac {5}{3}\right )^2-6 \left (\frac {16}{\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}}+\sqrt [3]{\frac {1}{2} \left (61+3 i \sqrt {1407}\right )}\right ) \left (x-\frac {5}{3}\right )-\sqrt [3]{2} \left (61+3 i \sqrt {1407}\right )^{2/3}-\frac {512}{\left (\frac {1}{2} \left (61+3 i \sqrt {1407}\right )\right )^{2/3}}+32\right )}\right )\) |
Input:
Int[(2 + 3*x - 5*x^2 + x^3)^(-2),x]
Output:
11664*(-1/108*((61*I - 3*Sqrt[1407])*((61 + (3*I)*Sqrt[1407])/2)^(1/3)*(64 + (3*(32*2^(1/3) + (122 + (6*I)*Sqrt[1407])^(2/3))*(-5/3 + x))/(61 + (3*I )*Sqrt[1407])^(1/3)))/((4221*I + 61*Sqrt[1407])*(32 - 2^(1/3)*(61 + (3*I)* Sqrt[1407])^(2/3))*((32*2^(1/3) + (122 + (6*I)*Sqrt[1407])^(2/3))/(61 + (3 *I)*Sqrt[1407])^(1/3) - 6*(-5/3 + x))*(32 - 512/((61 + (3*I)*Sqrt[1407])/2 )^(2/3) - 2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/3) - 6*(16/((61 + (3*I)*Sqrt[ 1407])/2)^(1/3) + ((61 + (3*I)*Sqrt[1407])/2)^(1/3))*(-5/3 + x) - 18*(-5/3 + x)^2)) + ((61*I - 3*Sqrt[1407])*((61 + (3*I)*Sqrt[1407])/2)^(1/3)*((-16 *(61 + (3*I)*Sqrt[1407]))/(3*(512*2^(2/3) + 32*(61 + (3*I)*Sqrt[1407])^(2/ 3) + 2^(1/3)*(61 + (3*I)*Sqrt[1407])^(4/3))*(32*2^(1/3) + (122 + (6*I)*Sqr t[1407])^(2/3) - 6*(61 + (3*I)*Sqrt[1407])^(1/3)*(-5/3 + x))) + (2^(2/3)*( 61 + (3*I)*Sqrt[1407])^(1/3)*(7625 - I*(7817*Sqrt[1407] + 16*(2*(122 + (6* I)*Sqrt[1407]))^(1/3)*(125*I + 61*Sqrt[1407])))*ArcTanh[(2^(2/3)*(61 + (3* I)*Sqrt[1407]) + 32*(122 + (6*I)*Sqrt[1407])^(1/3) + 12*(61 + (3*I)*Sqrt[1 407])^(2/3)*(-5/3 + x))/(2*Sqrt[3*(32*(61 + (3*I)*Sqrt[1407])^(4/3) + 2^(1 /3)*(4471 - (183*I)*Sqrt[1407] - 256*2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/3) ))])])/((512*2^(2/3) + 32*(61 + (3*I)*Sqrt[1407])^(2/3) + 2^(1/3)*(61 + (3 *I)*Sqrt[1407])^(4/3))^2*Sqrt[3*(32*(61 + (3*I)*Sqrt[1407])^(4/3) + 2^(1/3 )*(4471 - (183*I)*Sqrt[1407] - 256*2^(1/3)*(61 + (3*I)*Sqrt[1407])^(2/3))) ]) - ((61*(61 + (3*I)*Sqrt[1407])^(5/3) - 16*((122 + (6*I)*Sqrt[1407])^...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9* a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Simp[1/d^(2*p) Int[Sim p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 *(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d}, x] && NeQ[4*b^3 + 27*a^2 *d, 0] && IntegerQ[p]
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 *d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.19
| method | result | size |
| default | \(\frac {-\frac {32}{469} x^{2}+\frac {127}{469} x -\frac {9}{469}}{x^{3}-5 x^{2}+3 x +2}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +2\right )}{\sum }\frac {\left (-16 \textit {\_R} +47\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )}{469}\) | \(71\) |
| risch | \(\frac {-\frac {32}{469} x^{2}+\frac {127}{469} x -\frac {9}{469}}{x^{3}-5 x^{2}+3 x +2}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +2\right )}{\sum }\frac {\left (-16 \textit {\_R} +47\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )}{469}\) | \(71\) |
Input:
int(1/(x^3-5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
(-32/469*x^2+127/469*x-9/469)/(x^3-5*x^2+3*x+2)+2/469*sum((-16*_R+47)/(3*_ R^2-10*_R+3)*ln(x-_R),_R=RootOf(_Z^3-5*_Z^2+3*_Z+2))
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(x^3-5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
-1/1237940508*(2*(x^3 - 5*x^2 + 3*x + 2)*(309485127*(19659788/204223974060 141*I*sqrt(1407) + 244/103161709)^(1/3)*(-I*sqrt(3) + 1) + 82048*(I*sqrt(3 ) + 1)/(19659788/204223974060141*I*sqrt(1407) + 244/103161709)^(1/3))*log( 122/928455381*(309485127*(19659788/204223974060141*I*sqrt(1407) + 244/1031 61709)^(1/3)*(-I*sqrt(3) + 1) + 82048*(I*sqrt(3) + 1)/(19659788/2042239740 60141*I*sqrt(1407) + 244/103161709)^(1/3))^2 + 27495125/2*(19659788/204223 974060141*I*sqrt(1407) + 244/103161709)^(1/3)*(-I*sqrt(3) + 1) + 9829894*x + 5128000/1407*(I*sqrt(3) + 1)/(19659788/204223974060141*I*sqrt(1407) + 2 44/103161709)^(1/3) - 43076106) + 84465024*x^2 - ((x^3 - 5*x^2 + 3*x + 2)* (309485127*(19659788/204223974060141*I*sqrt(1407) + 244/103161709)^(1/3)*( -I*sqrt(3) + 1) + 82048*(I*sqrt(3) + 1)/(19659788/204223974060141*I*sqrt(1 407) + 244/103161709)^(1/3)) + 1319766*(x^3 - 5*x^2 + 3*x + 2)*sqrt(-1/580 594098252*(309485127*(19659788/204223974060141*I*sqrt(1407) + 244/10316170 9)^(1/3)*(-I*sqrt(3) + 1) + 82048*(I*sqrt(3) + 1)/(19659788/20422397406014 1*I*sqrt(1407) + 244/103161709)^(1/3))^2 + 328192/469))*log(-122/928455381 *(309485127*(19659788/204223974060141*I*sqrt(1407) + 244/103161709)^(1/3)* (-I*sqrt(3) + 1) + 82048*(I*sqrt(3) + 1)/(19659788/204223974060141*I*sqrt( 1407) + 244/103161709)^(1/3))^2 + 1/1407*sqrt(-1/580594098252*(309485127*( 19659788/204223974060141*I*sqrt(1407) + 244/103161709)^(1/3)*(-I*sqrt(3) + 1) + 82048*(I*sqrt(3) + 1)/(19659788/204223974060141*I*sqrt(1407) + 24...
Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\frac {- 32 x^{2} + 127 x - 9}{469 x^{3} - 2345 x^{2} + 1407 x + 938} + \operatorname {RootSum} {\left (103161709 t^{3} - 82048 t - 488, \left ( t \mapsto t \log {\left (\frac {25171456996 t^{2}}{4914947} - \frac {27495125 t}{9829894} + x - \frac {21538053}{4914947} \right )} \right )\right )} \] Input:
integrate(1/(x**3-5*x**2+3*x+2)**2,x)
Output:
(-32*x**2 + 127*x - 9)/(469*x**3 - 2345*x**2 + 1407*x + 938) + RootSum(103 161709*_t**3 - 82048*_t - 488, Lambda(_t, _t*log(25171456996*_t**2/4914947 - 27495125*_t/9829894 + x - 21538053/4914947)))
\[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\int { \frac {1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \] Input:
integrate(1/(x^3-5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
-1/469*(32*x^2 - 127*x + 9)/(x^3 - 5*x^2 + 3*x + 2) - 2/469*integrate((16* x - 47)/(x^3 - 5*x^2 + 3*x + 2), x)
Exception generated. \[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(x^3-5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: -(32 *sageVARx^2-127*sageVARx+9)*1/469/(sageVARx^3-5*sageVARx^2+3*sageVARx+2)+( (-32/61*rootof([[-3,0,80,0,-443],[1,0,-32,0,256,0,-469]])-94)/(-3*(-1/61*r ootof([[-3,0,80,0,-
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {1024\,x}{219961}-\mathrm {root}\left (z^3-\frac {82048\,z}{103161709}-\frac {488}{103161709},z,k\right )\,\left (-\frac {122\,x}{469}+\mathrm {root}\left (z^3-\frac {82048\,z}{103161709}-\frac {488}{103161709},z,k\right )\,\left (32\,x-33\right )+\frac {374}{469}\right )-\frac {3008}{219961}\right )\,\mathrm {root}\left (z^3-\frac {82048\,z}{103161709}-\frac {488}{103161709},z,k\right )\right )-\frac {\frac {32\,x^2}{469}-\frac {127\,x}{469}+\frac {9}{469}}{x^3-5\,x^2+3\,x+2} \] Input:
int(1/(3*x - 5*x^2 + x^3 + 2)^2,x)
Output:
symsum(log((1024*x)/219961 - root(z^3 - (82048*z)/103161709 - 488/10316170 9, z, k)*(root(z^3 - (82048*z)/103161709 - 488/103161709, z, k)*(32*x - 33 ) - (122*x)/469 + 374/469) - 3008/219961)*root(z^3 - (82048*z)/103161709 - 488/103161709, z, k), k, 1, 3) - ((32*x^2)/469 - (127*x)/469 + 9/469)/(3* x - 5*x^2 + x^3 + 2)
\[ \int \frac {1}{\left (2+3 x-5 x^2+x^3\right )^2} \, dx=\int \frac {1}{x^{6}-10 x^{5}+31 x^{4}-26 x^{3}-11 x^{2}+12 x +4}d x \] Input:
int(1/(x^3-5*x^2+3*x+2)^2,x)
Output:
int(1/(x**6 - 10*x**5 + 31*x**4 - 26*x**3 - 11*x**2 + 12*x + 4),x)