Integrand size = 38, antiderivative size = 275 \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=-\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\frac {\sqrt {2-\sqrt {2}} x^2}{-2+\sqrt {2}}-\frac {\sqrt {2-\sqrt {2}} \sqrt {1+x^3}}{-2+\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {-\frac {x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {1+x^3}}{\sqrt {2+\sqrt {2}}}}{x \sqrt [4]{1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )+\sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]
-1/2*(4+2*2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1/2)*x^2/(-2+2^(1/2))-(2-2^( 1/2))^(1/2)*(x^3+1)^(1/2)/(-2+2^(1/2)))/x/(x^3+1)^(1/4))-1/2*(4-2*2^(1/2)) ^(1/2)*arctan((-x^2/(2+2^(1/2))^(1/2)+(x^3+1)^(1/2)/(2+2^(1/2))^(1/2))/x/( x^3+1)^(1/4))+1/2*(4+2*2^(1/2))^(1/2)*arctanh((2-2^(1/2))^(1/2)*x*(x^3+1)^ (1/4)/(x^2+(x^3+1)^(1/2)))+1/2*(4-2*2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1/ 2)*x*(x^3+1)^(1/4)/(x^2+(x^3+1)^(1/2)))
Time = 2.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.80 \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\frac {-\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (-x^2+\sqrt {1+x^3}\right )}{x \sqrt [4]{1+x^3}}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (-x^2+\sqrt {1+x^3}\right )}{x \sqrt [4]{1+x^3}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}} \]
(-(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(-x^2 + Sqrt[1 + x^3]))/( x*(1 + x^3)^(1/4))]) - Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(-x^2 + Sqrt[1 + x^3]))/(x*(1 + x^3)^(1/4))] + Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[ 2 - Sqrt[2]]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])] + Sqrt[2 - Sqrt[2]] *ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])])/Sqr t[2]
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.
| \(\displaystyle \int \frac {\left (x^3+4\right ) \left (x^4+x^3+1\right )}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^6}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}+\frac {5 x^3}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}+\frac {4}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}+\frac {x^7}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}+\frac {4 x^4}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \frac {1}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}dx+5 \int \frac {x^3}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}dx+\int \frac {x^6}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}dx+\int \frac {x^7}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}dx+4 \int \frac {x^4}{\sqrt [4]{x^3+1} \left (x^8+x^6+2 x^3+1\right )}dx\) |
3.29.8.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.16 (sec) , antiderivative size = 693, normalized size of antiderivative = 2.52
-1/16*RootOf(_Z^8+16)^7*ln(-(RootOf(_Z^8+16)^11*x^4+4*RootOf(_Z^8+16)^7*x^ 4+16*RootOf(_Z^8+16)^6*(x^3+1)^(1/4)*x^3-4*RootOf(_Z^8+16)^7*x^3+16*(x^3+1 )^(1/2)*RootOf(_Z^8+16)^5*x^2-4*RootOf(_Z^8+16)^7-16*RootOf(_Z^8+16)^3*x^3 +64*(x^3+1)^(1/2)*RootOf(_Z^8+16)*x^2+128*(x^3+1)^(3/4)*x-16*RootOf(_Z^8+1 6)^3)/(RootOf(_Z^8+16)^4*x^4+4*x^3+4))-1/8*RootOf(_Z^8+16)^5*ln(-(-RootOf( _Z^8+16)^9*x^4-4*(x^3+1)^(1/2)*RootOf(_Z^8+16)^7*x^2+4*RootOf(_Z^8+16)^5*x ^4-4*RootOf(_Z^8+16)^5*x^3+16*(x^3+1)^(1/2)*RootOf(_Z^8+16)^3*x^2+32*(x^3+ 1)^(1/4)*RootOf(_Z^8+16)^2*x^3-4*RootOf(_Z^8+16)^5+64*(x^3+1)^(3/4)*x+16*x ^3*RootOf(_Z^8+16)+16*RootOf(_Z^8+16))/(RootOf(_Z^8+16)^4*x^4-4*x^3-4))+1/ 4*RootOf(_Z^8+16)^3*ln(-(RootOf(_Z^8+16)^11*x^4-4*RootOf(_Z^8+16)^7*x^4-16 *RootOf(_Z^8+16)^6*(x^3+1)^(1/4)*x^3-4*RootOf(_Z^8+16)^7*x^3-16*(x^3+1)^(1 /2)*RootOf(_Z^8+16)^5*x^2-4*RootOf(_Z^8+16)^7+16*RootOf(_Z^8+16)^3*x^3+64* (x^3+1)^(1/2)*RootOf(_Z^8+16)*x^2+128*(x^3+1)^(3/4)*x+16*RootOf(_Z^8+16)^3 )/(RootOf(_Z^8+16)^4*x^4+4*x^3+4))+1/2*RootOf(_Z^8+16)*ln(-(RootOf(_Z^8+16 )^9*x^4-4*(x^3+1)^(1/2)*RootOf(_Z^8+16)^7*x^2+4*RootOf(_Z^8+16)^5*x^4+4*Ro otOf(_Z^8+16)^5*x^3-16*(x^3+1)^(1/2)*RootOf(_Z^8+16)^3*x^2-32*(x^3+1)^(1/4 )*RootOf(_Z^8+16)^2*x^3+4*RootOf(_Z^8+16)^5+64*(x^3+1)^(3/4)*x+16*x^3*Root Of(_Z^8+16)+16*RootOf(_Z^8+16))/(RootOf(_Z^8+16)^4*x^4-4*x^3-4))
Result contains complex when optimal does not.
Time = 47.89 (sec) , antiderivative size = 1537, normalized size of antiderivative = 5.59 \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\text {Too large to display} \]
-1/4*sqrt(2)*(-1)^(1/8)*log(8*(2*sqrt(2)*sqrt(x^3 + 1)*((-1)^(7/8)*(x^6 - x^5 - x^2) + (-1)^(3/8)*(x^6 + x^5 + x^2)) + 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4) - sqrt(2)*((-1)^(5/8)*(x^8 + 2*x^7 - x^6 + 2*x^4 - 2*x^3 - 1) - ( -1)^(1/8)*(x^8 - 2*x^7 - x^6 - 2*x^4 - 2*x^3 - 1)) - 4*((-1)^(1/4)*x^7 - ( -1)^(3/4)*(x^6 + x^3))*(x^3 + 1)^(1/4))/(x^8 + x^6 + 2*x^3 + 1)) + 1/4*sqr t(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*sqrt(x^3 + 1)*((-1)^(7/8)*(x^6 - x^5 - x ^2) + (-1)^(3/8)*(x^6 + x^5 + x^2)) - 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4 ) - sqrt(2)*((-1)^(5/8)*(x^8 + 2*x^7 - x^6 + 2*x^4 - 2*x^3 - 1) - (-1)^(1/ 8)*(x^8 - 2*x^7 - x^6 - 2*x^4 - 2*x^3 - 1)) + 4*((-1)^(1/4)*x^7 - (-1)^(3/ 4)*(x^6 + x^3))*(x^3 + 1)^(1/4))/(x^8 + x^6 + 2*x^3 + 1)) - 1/4*I*sqrt(2)* (-1)^(1/8)*log(-8*(2*sqrt(2)*sqrt(x^3 + 1)*((-1)^(7/8)*(I*x^6 - I*x^5 - I* x^2) + (-1)^(3/8)*(I*x^6 + I*x^5 + I*x^2)) - 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4) + sqrt(2)*((-1)^(5/8)*(I*x^8 + 2*I*x^7 - I*x^6 + 2*I*x^4 - 2*I*x^ 3 - I) + (-1)^(1/8)*(-I*x^8 + 2*I*x^7 + I*x^6 + 2*I*x^4 + 2*I*x^3 + I)) - 4*((-1)^(1/4)*x^7 - (-1)^(3/4)*(x^6 + x^3))*(x^3 + 1)^(1/4))/(x^8 + x^6 + 2*x^3 + 1)) + 1/4*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*sqrt(x^3 + 1)*((- 1)^(7/8)*(-I*x^6 + I*x^5 + I*x^2) + (-1)^(3/8)*(-I*x^6 - I*x^5 - I*x^2)) - 4*(x^5 - I*x^4 - I*x)*(x^3 + 1)^(3/4) + sqrt(2)*((-1)^(5/8)*(-I*x^8 - 2*I *x^7 + I*x^6 - 2*I*x^4 + 2*I*x^3 + I) + (-1)^(1/8)*(I*x^8 - 2*I*x^7 - I*x^ 6 - 2*I*x^4 - 2*I*x^3 - I)) - 4*((-1)^(1/4)*x^7 - (-1)^(3/4)*(x^6 + x^3...
\[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\int \frac {\left (x^{3} + 4\right ) \left (x^{4} + x^{3} + 1\right )}{\sqrt [4]{\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{8} + x^{6} + 2 x^{3} + 1\right )}\, dx \]
Integral((x**3 + 4)*(x**4 + x**3 + 1)/(((x + 1)*(x**2 - x + 1))**(1/4)*(x* *8 + x**6 + 2*x**3 + 1)), x)
\[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\int { \frac {{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 4\right )}}{{\left (x^{8} + x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {\left (4+x^3\right ) \left (1+x^3+x^4\right )}{\sqrt [4]{1+x^3} \left (1+2 x^3+x^6+x^8\right )} \, dx=\int \frac {\left (x^3+4\right )\,\left (x^4+x^3+1\right )}{{\left (x^3+1\right )}^{1/4}\,\left (x^8+x^6+2\,x^3+1\right )} \,d x \]