Integrand size = 19, antiderivative size = 367 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=-\frac {\left (3-\left (1+\frac {1}{x}\right )^2\right ) x^2}{\sqrt {1+4 x+4 x^2+4 x^4}}+\frac {\left (13-9 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right ) x^2}{10 \sqrt {1+4 x+4 x^2+4 x^4}}+\frac {9 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right ) \left (1+\frac {1}{x}\right ) x^2}{10 \left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right ) \sqrt {1+4 x+4 x^2+4 x^4}}-\frac {9 \left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right ) \sqrt {\frac {5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4}{\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right )^2}} x^2 E\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2\ 5^{3/4} \sqrt {1+4 x+4 x^2+4 x^4}}+\frac {3 \left (3-\sqrt {5}\right ) \left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right ) \sqrt {\frac {5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4}{\left (\sqrt {5}+\left (1+\frac {1}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right ),\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4\ 5^{3/4} \sqrt {1+4 x+4 x^2+4 x^4}} \] Output:
-(3-(1+1/x)^2)*x^2/(4*x^4+4*x^2+4*x+1)^(1/2)+1/10*(13-9*(1+1/x)^2)*(1+1/x) *x^2/(4*x^4+4*x^2+4*x+1)^(1/2)+9/10*(5-2*(1+1/x)^2+(1+1/x)^4)*(1+1/x)*x^2/ (5^(1/2)+(1+1/x)^2)/(4*x^4+4*x^2+4*x+1)^(1/2)-9/10*(5^(1/2)+(1+1/x)^2)*((5 -2*(1+1/x)^2+(1+1/x)^4)/(5^(1/2)+(1+1/x)^2)^2)^(1/2)*x^2*EllipticE(sin(2*a rctan(1/5*(1+1/x)*5^(3/4))),1/10*(50+10*5^(1/2))^(1/2))*5^(1/4)/(4*x^4+4*x ^2+4*x+1)^(1/2)+3/20*(3-5^(1/2))*(5^(1/2)+(1+1/x)^2)*((5-2*(1+1/x)^2+(1+1/ x)^4)/(5^(1/2)+(1+1/x)^2)^2)^(1/2)*x^2*InverseJacobiAM(2*arctan(1/5*(1+1/x )*5^(3/4)),1/10*(50+10*5^(1/2))^(1/2))*5^(1/4)/(4*x^4+4*x^2+4*x+1)^(1/2)
Result contains complex when optimal does not.
Time = 15.35 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\frac {19+42 x-16 x^2+36 x^3+\frac {9}{2} \left (-i+\sqrt {-1-2 i}-2 x\right ) \left (-i-\sqrt {-1+2 i}+2 x\right ) \left (-i+\sqrt {-1+2 i}+2 x\right )-\frac {9 i \sqrt {-\frac {2}{5}+\frac {4 i}{5}} \left (-2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (\frac {1}{2} \left (i+\sqrt {-1-2 i}\right )+x\right )^2 \sqrt {\frac {\left (2 i+\sqrt {-1-2 i}-\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1-2 i}-2 x\right )}{\left (-2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (i+\sqrt {-1-2 i}+2 x\right )}} \sqrt {\frac {(1+2 i) \left ((-1+i)+\sqrt {-1-2 i}\right ) \left (1+2 x+2 i x^2\right )}{\left (i+\sqrt {-1-2 i}+2 x\right )^2}} E\left (\arcsin \left (\frac {\sqrt {\frac {\left (2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1+2 i}+2 x\right )}{\sqrt {-1+2 i} \left (i+\sqrt {-1-2 i}+2 x\right )}}}{\sqrt {2}}\right )|\frac {1}{2} \left (5-\sqrt {5}\right )\right )}{(-1+i)+\sqrt {-1-2 i}}+\frac {(6-3 i) \sqrt {-\frac {2}{5}+\frac {4 i}{5}} \sqrt {\frac {\left (2 i+\sqrt {-1-2 i}-\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1-2 i}-2 x\right )}{\left (-2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (i+\sqrt {-1-2 i}+2 x\right )}} \left (1+2 x+2 i x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {\left (2 i+\sqrt {-1-2 i}+\sqrt {-1+2 i}\right ) \left (-i+\sqrt {-1+2 i}+2 x\right )}{\sqrt {-1+2 i} \left (i+\sqrt {-1-2 i}+2 x\right )}}}{\sqrt {2}}\right ),\frac {1}{2} \left (5-\sqrt {5}\right )\right )}{\sqrt {\frac {(1+2 i) \left ((-1+i)+\sqrt {-1-2 i}\right ) \left (1+2 x+2 i x^2\right )}{\left (i+\sqrt {-1-2 i}+2 x\right )^2}}}}{10 \sqrt {1+4 x+4 x^2+4 x^4}} \] Input:
Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-3/2),x]
Output:
(19 + 42*x - 16*x^2 + 36*x^3 + (9*(-I + Sqrt[-1 - 2*I] - 2*x)*(-I - Sqrt[- 1 + 2*I] + 2*x)*(-I + Sqrt[-1 + 2*I] + 2*x))/2 - ((9*I)*Sqrt[-2/5 + (4*I)/ 5]*(-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(2*I + Sqrt[-1 - 2*I] + Sqrt[- 1 + 2*I])*((I + Sqrt[-1 - 2*I])/2 + x)^2*Sqrt[((2*I + Sqrt[-1 - 2*I] - Sqr t[-1 + 2*I])*(-I + Sqrt[-1 - 2*I] - 2*x))/((-2*I + Sqrt[-1 - 2*I] + Sqrt[- 1 + 2*I])*(I + Sqrt[-1 - 2*I] + 2*x))]*Sqrt[((1 + 2*I)*((-1 + I) + Sqrt[-1 - 2*I])*(1 + 2*x + (2*I)*x^2))/(I + Sqrt[-1 - 2*I] + 2*x)^2]*EllipticE[Ar cSin[Sqrt[((2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(-I + Sqrt[-1 + 2*I] + 2*x))/(Sqrt[-1 + 2*I]*(I + Sqrt[-1 - 2*I] + 2*x))]/Sqrt[2]], (5 - Sqrt[5]) /2])/((-1 + I) + Sqrt[-1 - 2*I]) + ((6 - 3*I)*Sqrt[-2/5 + (4*I)/5]*Sqrt[(( 2*I + Sqrt[-1 - 2*I] - Sqrt[-1 + 2*I])*(-I + Sqrt[-1 - 2*I] - 2*x))/((-2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I])*(I + Sqrt[-1 - 2*I] + 2*x))]*(1 + 2*x + (2*I)*x^2)*EllipticF[ArcSin[Sqrt[((2*I + Sqrt[-1 - 2*I] + Sqrt[-1 + 2*I] )*(-I + Sqrt[-1 + 2*I] + 2*x))/(Sqrt[-1 + 2*I]*(I + Sqrt[-1 - 2*I] + 2*x)) ]/Sqrt[2]], (5 - Sqrt[5])/2])/Sqrt[((1 + 2*I)*((-1 + I) + Sqrt[-1 - 2*I])* (1 + 2*x + (2*I)*x^2))/(I + Sqrt[-1 - 2*I] + 2*x)^2])/(10*Sqrt[1 + 4*x + 4 *x^2 + 4*x^4])
Time = 1.19 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {2504, 27, 7270, 2202, 1576, 27, 1158, 2206, 27, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {1}{\left (4 x^4+4 x^2+4 x+1\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -16 \int \frac {x^2}{16 \left (\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right ) x^4\right )^{3/2}}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {x^2}{\left (\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right ) x^4\right )^{3/2}}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \int \frac {1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2} x^4}d\left (1+\frac {1}{x}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\int \frac {\left (1+\frac {1}{x}\right )^4+6 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )+\int \frac {\left (-4 \left (1+\frac {1}{x}\right )^2-4\right ) \left (1+\frac {1}{x}\right )}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\int \frac {\left (1+\frac {1}{x}\right )^4+6 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )+\frac {1}{2} \int -\frac {4 \left (2+\frac {1}{x}\right )}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )^2\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\int \frac {\left (1+\frac {1}{x}\right )^4+6 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )-2 \int \frac {2+\frac {1}{x}}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )^2\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 1158 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\int \frac {\left (1+\frac {1}{x}\right )^4+6 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^{3/2}}d\left (1+\frac {1}{x}\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\frac {1}{80} \int \frac {24 \left (5-3 \left (1+\frac {1}{x}\right )^2\right )}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\left (13-9 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\frac {3}{10} \int \frac {5-3 \left (1+\frac {1}{x}\right )^2}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\left (13-9 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\frac {3}{10} \left (\left (5-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )+3 \sqrt {5} \int \frac {\sqrt {5}-\left (1+\frac {1}{x}\right )^2}{\sqrt {5} \sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\left (13-9 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\frac {3}{10} \left (\left (5-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )+3 \int \frac {\sqrt {5}-\left (1+\frac {1}{x}\right )^2}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\left (13-9 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\frac {3}{10} \left (3 \int \frac {\sqrt {5}-\left (1+\frac {1}{x}\right )^2}{\sqrt {\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}}d\left (1+\frac {1}{x}\right )+\frac {\left (5-3 \sqrt {5}\right ) \left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right ),\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\left (13-9 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} x^2 \left (\frac {3}{10} \left (\frac {\left (5-3 \sqrt {5}\right ) \left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right ),\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt [4]{5} \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}+3 \left (\frac {\sqrt [4]{5} \left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right ) \sqrt {\frac {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}{\left (\left (\frac {1}{x}+1\right )^2+\sqrt {5}\right )^2}} E\left (2 \arctan \left (\frac {1+\frac {1}{x}}{\sqrt [4]{5}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5} \left (\frac {1}{x}+1\right )}{\left (\frac {1}{x}+1\right )^2+\sqrt {5}}\right )\right )+\frac {2-\frac {1}{x}}{\sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}-\frac {\left (13-9 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \sqrt {\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5}}\right )}{\sqrt {\left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right ) x^4}}\) |
Input:
Int[(1 + 4*x + 4*x^2 + 4*x^4)^(-3/2),x]
Output:
-((Sqrt[5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4]*x^2*((2 - x^(-1))/Sqrt[5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4] - ((13 - 9*(1 + x^(-1))^2)*(1 + x^(-1)) )/(10*Sqrt[5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4]) + (3*(3*(-((Sqrt[5 - 2* (1 + x^(-1))^2 + (1 + x^(-1))^4]*(1 + x^(-1)))/(Sqrt[5] + (1 + x^(-1))^2)) + (5^(1/4)*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x ^(-1))^4)/(Sqrt[5] + (1 + x^(-1))^2)^2]*EllipticE[2*ArcTan[(1 + x^(-1))/5^ (1/4)], (5 + Sqrt[5])/10])/Sqrt[5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4]) + ((5 - 3*Sqrt[5])*(Sqrt[5] + (1 + x^(-1))^2)*Sqrt[(5 - 2*(1 + x^(-1))^2 + ( 1 + x^(-1))^4)/(Sqrt[5] + (1 + x^(-1))^2)^2]*EllipticF[2*ArcTan[(1 + x^(-1 ))/5^(1/4)], (5 + Sqrt[5])/10])/(2*5^(1/4)*Sqrt[5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4])))/10))/Sqrt[(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)*x^4])
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_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Leaf count of result is larger than twice the leaf count of optimal. \(2563\) vs. \(2(328)=656\).
Time = 0.88 (sec) , antiderivative size = 2564, normalized size of antiderivative = 6.99
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2564\) |
| risch | \(\text {Expression too large to display}\) | \(2564\) |
| elliptic | \(\text {Expression too large to display}\) | \(2564\) |
Input:
int(1/(4*x^4+4*x^2+4*x+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-8*(-9/20*x^3+1/5*x^2-21/40*x-19/80)/(4*x^4+4*x^2+4*x+1)^(1/2)+3/5*(RootOf (4*_Z^4+4*_Z^2+4*_Z+1,index=1)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))*((Roo tOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x -RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index= 4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,in dex=2)))^(1/2)*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))^2*((RootOf(4*_Z^4+ 4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-RootOf(4*_ Z^4+4*_Z^2+4*_Z+1,index=3))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=3)-RootOf(4 *_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)))^(1 /2)*((RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,ind ex=1))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4))/(RootOf(4*_Z^4+4*_Z^2+4*_Z +1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+ 4*_Z+1,index=2)))^(1/2)/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^ 4+4*_Z^2+4*_Z+1,index=2))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2)-RootOf(4*_ Z^4+4*_Z^2+4*_Z+1,index=1))/((x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=1))*(x-R ootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index= 3))*(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)))^(1/2)*EllipticF(((RootOf(4*_ Z^4+4*_Z^2+4*_Z+1,index=4)-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=2))*(x-RootOf (4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=4)-Root Of(4*_Z^4+4*_Z^2+4*_Z+1,index=1))/(x-RootOf(4*_Z^4+4*_Z^2+4*_Z+1,index=...
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4*x^4+4*x^2+4*x+1)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(4*x^4 + 4*x^2 + 4*x + 1)/(16*x^8 + 32*x^6 + 32*x^5 + 24*x^4 + 32*x^3 + 24*x^2 + 8*x + 1), x)
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (4 x^{4} + 4 x^{2} + 4 x + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(4*x**4+4*x**2+4*x+1)**(3/2),x)
Output:
Integral((4*x**4 + 4*x**2 + 4*x + 1)**(-3/2), x)
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4*x^4+4*x^2+4*x+1)^(3/2),x, algorithm="maxima")
Output:
integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2), x)
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(4*x^4+4*x^2+4*x+1)^(3/2),x, algorithm="giac")
Output:
integrate((4*x^4 + 4*x^2 + 4*x + 1)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (4\,x^4+4\,x^2+4\,x+1\right )}^{3/2}} \,d x \] Input:
int(1/(4*x + 4*x^2 + 4*x^4 + 1)^(3/2),x)
Output:
int(1/(4*x + 4*x^2 + 4*x^4 + 1)^(3/2), x)
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {4 x^{4}+4 x^{2}+4 x +1}}{16 x^{8}+32 x^{6}+32 x^{5}+24 x^{4}+32 x^{3}+24 x^{2}+8 x +1}d x \] Input:
int(1/(4*x^4+4*x^2+4*x+1)^(3/2),x)
Output:
int(sqrt(4*x**4 + 4*x**2 + 4*x + 1)/(16*x**8 + 32*x**6 + 32*x**5 + 24*x**4 + 32*x**3 + 24*x**2 + 8*x + 1),x)