Integrand size = 59, antiderivative size = 94 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\frac {(1+2 x) \sqrt {1+2 x^2+4 x^3+x^4}}{2 \left (-1+2 x^2\right )}-\text {arctanh}\left (\frac {x (2+x) \left (7-x+27 x^2+33 x^3\right )}{\left (2+37 x^2+31 x^3\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \]
-arctanh(x*(2+x)*(33*x^3+27*x^2-x+7)/(31*x^3+37*x^2+2)/(x^4+4*x^3+2*x^2+1) ^(1/2))+1/2*(1+2*x)*(x^4+4*x^3+2*x^2+1)^(1/2)/(2*x^2-1)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 16.14 (sec) , antiderivative size = 5141, normalized size of antiderivative = 54.69 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\text {Result too large to show} \]
Integrate[(-8 - 8*x - x^2 - 3*x^3 + 7*x^4 + 4*x^5 + 2*x^6)/((-1 + 2*x^2)^2 *Sqrt[1 + 2*x^2 + 4*x^3 + x^4]),x]
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.
| \(\displaystyle \int \frac {2 x^6+4 x^5+7 x^4-3 x^3-x^2-8 x-8}{\left (2 x^2-1\right )^2 \sqrt {x^4+4 x^3+2 x^2+1}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2}{2 \sqrt {x^4+4 x^3+2 x^2+1}}+\frac {x}{\sqrt {x^4+4 x^3+2 x^2+1}}+\frac {2 x+15}{4 \left (2 x^2-1\right ) \sqrt {x^4+4 x^3+2 x^2+1}}+\frac {-17 x-13}{2 \left (2 x^2-1\right )^2 \sqrt {x^4+4 x^3+2 x^2+1}}+\frac {9}{4 \sqrt {x^4+4 x^3+2 x^2+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {9}{4} \int \frac {1}{\sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {x^4+4 x^3+2 x^2+1}}dx+\int \frac {x}{\sqrt {x^4+4 x^3+2 x^2+1}}dx+\frac {1}{2} \int \frac {x^2}{\sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {13}{4} \int \frac {1}{\left (2 x+\sqrt {2}\right )^2 \sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {1}{8} \left (15+\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {13}{8} \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {1}{8} \left (15-\sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2} x+1\right ) \sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {13}{8} \int \frac {1}{\left (\sqrt {2} x+1\right ) \sqrt {x^4+4 x^3+2 x^2+1}}dx-\frac {17}{2} \int \frac {x}{\left (2 x^2-1\right )^2 \sqrt {x^4+4 x^3+2 x^2+1}}dx\) |
Int[(-8 - 8*x - x^2 - 3*x^3 + 7*x^4 + 4*x^5 + 2*x^6)/((-1 + 2*x^2)^2*Sqrt[ 1 + 2*x^2 + 4*x^3 + x^4]),x]
3.3.78.3.1 Defintions of rubi rules used
Timed out.
hanged
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (88) = 176\).
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.90 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\frac {{\left (2 \, x^{2} - 1\right )} \log \left (\frac {1025 \, x^{10} + 6138 \, x^{9} + 12307 \, x^{8} + 10188 \, x^{7} + 4503 \, x^{6} + 3134 \, x^{5} + 1589 \, x^{4} + 140 \, x^{3} + 176 \, x^{2} - {\left (1023 \, x^{8} + 4104 \, x^{7} + 5084 \, x^{6} + 2182 \, x^{5} + 805 \, x^{4} + 624 \, x^{3} + 10 \, x^{2} + 28 \, x\right )} \sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} + 2}{32 \, x^{10} - 80 \, x^{8} + 80 \, x^{6} - 40 \, x^{4} + 10 \, x^{2} - 1}\right ) + \sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x + 1\right )}}{2 \, {\left (2 \, x^{2} - 1\right )}} \]
integrate((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2 +1)^(1/2),x, algorithm="fricas")
1/2*((2*x^2 - 1)*log((1025*x^10 + 6138*x^9 + 12307*x^8 + 10188*x^7 + 4503* x^6 + 3134*x^5 + 1589*x^4 + 140*x^3 + 176*x^2 - (1023*x^8 + 4104*x^7 + 508 4*x^6 + 2182*x^5 + 805*x^4 + 624*x^3 + 10*x^2 + 28*x)*sqrt(x^4 + 4*x^3 + 2 *x^2 + 1) + 2)/(32*x^10 - 80*x^8 + 80*x^6 - 40*x^4 + 10*x^2 - 1)) + sqrt(x ^4 + 4*x^3 + 2*x^2 + 1)*(2*x + 1))/(2*x^2 - 1)
\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int \frac {2 x^{6} + 4 x^{5} + 7 x^{4} - 3 x^{3} - x^{2} - 8 x - 8}{\sqrt {\left (x + 1\right ) \left (x^{3} + 3 x^{2} - x + 1\right )} \left (2 x^{2} - 1\right )^{2}}\, dx \]
integrate((2*x**6+4*x**5+7*x**4-3*x**3-x**2-8*x-8)/(2*x**2-1)**2/(x**4+4*x **3+2*x**2+1)**(1/2),x)
Integral((2*x**6 + 4*x**5 + 7*x**4 - 3*x**3 - x**2 - 8*x - 8)/(sqrt((x + 1 )*(x**3 + 3*x**2 - x + 1))*(2*x**2 - 1)**2), x)
\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int { \frac {2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x^{2} - 1\right )}^{2}} \,d x } \]
integrate((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2 +1)^(1/2),x, algorithm="maxima")
integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^ 3 + 2*x^2 + 1)*(2*x^2 - 1)^2), x)
\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int { \frac {2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x^{2} - 1\right )}^{2}} \,d x } \]
integrate((2*x^6+4*x^5+7*x^4-3*x^3-x^2-8*x-8)/(2*x^2-1)^2/(x^4+4*x^3+2*x^2 +1)^(1/2),x, algorithm="giac")
integrate((2*x^6 + 4*x^5 + 7*x^4 - 3*x^3 - x^2 - 8*x - 8)/(sqrt(x^4 + 4*x^ 3 + 2*x^2 + 1)*(2*x^2 - 1)^2), x)
Timed out. \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int -\frac {-2\,x^6-4\,x^5-7\,x^4+3\,x^3+x^2+8\,x+8}{{\left (2\,x^2-1\right )}^2\,\sqrt {x^4+4\,x^3+2\,x^2+1}} \,d x \]
int(-(8*x + x^2 + 3*x^3 - 7*x^4 - 4*x^5 - 2*x^6 + 8)/((2*x^2 - 1)^2*(2*x^2 + 4*x^3 + x^4 + 1)^(1/2)),x)
int(-(8*x + x^2 + 3*x^3 - 7*x^4 - 4*x^5 - 2*x^6 + 8)/((2*x^2 - 1)^2*(2*x^2 + 4*x^3 + x^4 + 1)^(1/2)), x)
Time = 0.02 (sec) , antiderivative size = 786, normalized size of antiderivative = 8.36 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx =\text {Too large to display} \]
int((2*x**6 + 4*x**5 + 7*x**4 - 3*x**3 - x**2 - 8*x - 8)/(sqrt(x**4 + 4*x* *3 + 2*x**2 + 1)*(4*x**4 - 4*x**2 + 1)),x)
(2*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*x + sqrt(x**4 + 4*x**3 + 2*x**2 + 1) - 4*log(x**2 + 4*x + 2)*x**2 + 2*log(x**2 + 4*x + 2) + 10*log( - 16*sqrt(x* *4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) - 13*sqrt(x**4 + 4*x**3 + 2*x**2 + 1) + 28*sqrt(2)*x**2 + 32*sqrt(2)*x - 6*sqrt(2) + 35*x**2 + 26*x - 11)*x**2 - 5 *log( - 16*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) - 13*sqrt(x**4 + 4*x** 3 + 2*x**2 + 1) + 28*sqrt(2)*x**2 + 32*sqrt(2)*x - 6*sqrt(2) + 35*x**2 + 2 6*x - 11) + 4*log( - 2*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) - sqrt(x** 4 + 4*x**3 + 2*x**2 + 1) + 4*sqrt(2)*x**2 + 16*sqrt(2)*x + 10*sqrt(2) - 5* x**2 - 20*x - 9)*x**2 - 2*log( - 2*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2 ) - sqrt(x**4 + 4*x**3 + 2*x**2 + 1) + 4*sqrt(2)*x**2 + 16*sqrt(2)*x + 10* sqrt(2) - 5*x**2 - 20*x - 9) - 2*log(sqrt(2) + 2*x)*x**2 + log(sqrt(2) + 2 *x) + 2*log(sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) + 4*sqrt(x**4 + 4*x** 3 + 2*x**2 + 1) - 2*sqrt(2)*x - 4*sqrt(2) - x - 2)*x**2 - log(sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) + 4*sqrt(x**4 + 4*x**3 + 2*x**2 + 1) - 2*sqrt (2)*x - 4*sqrt(2) - x - 2) + 2*log(25*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqr t(2)*x + 6*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) + 44*sqrt(x**4 + 4*x** 3 + 2*x**2 + 1)*x + 38*sqrt(x**4 + 4*x**3 + 2*x**2 + 1) - 7*sqrt(2)*x**3 - 36*sqrt(2)*x**2 - 21*sqrt(2)*x - 10*sqrt(2) - 28*x**3 - 130*x**2 - 84*x + 2)*x**2 - log(25*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2)*x + 6*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*sqrt(2) + 44*sqrt(x**4 + 4*x**3 + 2*x**2 + 1)*x...