\(\int \frac {(-2+x^6) (1-x^4+x^6)}{\sqrt [4]{1+x^6} (1+2 x^6+x^8+x^{12})} \, dx\) [2677]
Optimal result
Integrand size = 40, antiderivative size = 241 \[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )
\]
[Out]
1/4*(4-2*2^(1/2))^(1/2)*arctan((2-2^(1/2))^(1/2)*x*(x^6+1)^(1/4)/(-x^2+(x^6+1)^(1/2)))-1/4*(4+2*2^(1/2))^(1/2)
*arctan((2+2^(1/2))^(1/2)*x*(x^6+1)^(1/4)/(-x^2+(x^6+1)^(1/2)))+1/4*(4-2*2^(1/2))^(1/2)*arctanh((2-2^(1/2))^(1
/2)*x*(x^6+1)^(1/4)/(x^2+(x^6+1)^(1/2)))-1/4*(4+2*2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1/2)*x*(x^6+1)^(1/4)/(x^
2+(x^6+1)^(1/2)))
Rubi [F]
\[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx
\]
[In]
Int[((-2 + x^6)*(1 - x^4 + x^6))/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)),x]
[Out]
x*Hypergeometric2F1[1/6, 1/4, 7/6, -x^6] - 3*Defer[Int][1/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)), x] + 2*D
efer[Int][x^4/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)), x] - 3*Defer[Int][x^6/((1 + x^6)^(1/4)*(1 + 2*x^6 +
x^8 + x^12)), x] - Defer[Int][x^8/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)), x] - Defer[Int][x^10/((1 + x^6)^
(1/4)*(1 + 2*x^6 + x^8 + x^12)), x]
Rubi steps \begin{align*}
\text {integral}& = \int \left (\frac {1}{\sqrt [4]{1+x^6}}-\frac {3-2 x^4+3 x^6+x^8+x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt [4]{1+x^6}} \, dx-\int \frac {3-2 x^4+3 x^6+x^8+x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx \\ & = x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )-\int \left (\frac {3}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}-\frac {2 x^4}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}+\frac {3 x^6}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}+\frac {x^8}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}+\frac {x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )}\right ) \, dx \\ & = x \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{4},\frac {7}{6},-x^6\right )+2 \int \frac {x^4}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-3 \int \frac {1}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-3 \int \frac {x^6}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-\int \frac {x^8}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx-\int \frac {x^{10}}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx \\
\end{align*}
Mathematica [A] (verified)
Time = 8.55 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93
\[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2-\sqrt {1+x^6}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{-x^2+\sqrt {1+x^6}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{1+x^6}}{x^2+\sqrt {1+x^6}}\right )}{2 \sqrt {2}}
\]
[In]
Integrate[((-2 + x^6)*(1 - x^4 + x^6))/((1 + x^6)^(1/4)*(1 + 2*x^6 + x^8 + x^12)),x]
[Out]
(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(1 + x^6)^(1/4))/(x^2 - Sqrt[1 + x^6])] + Sqrt[2 - Sqrt[2]]*Arc
Tan[(Sqrt[2 - Sqrt[2]]*x*(1 + x^6)^(1/4))/(-x^2 + Sqrt[1 + x^6])] + Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqrt[2
]]*x*(1 + x^6)^(1/4))/(x^2 + Sqrt[1 + x^6])] - Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(1 + x^6)^(1/4))
/(x^2 + Sqrt[1 + x^6])])/(2*Sqrt[2])
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 16.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16
| | |
| method | result | size |
| | |
| pseudoelliptic |
\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{6}+1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}}\right )}{4}\) |
\(38\) |
| trager |
\(\text {Expression too large to display}\) |
\(679\) |
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[In]
int((x^6-2)*(x^6-x^4+1)/(x^6+1)^(1/4)/(x^12+x^8+2*x^6+1),x,method=_RETURNVERBOSE)
[Out]
1/4*sum(1/_R^5*(_R^4-1)*ln((-_R*x+(x^6+1)^(1/4))/x),_R=RootOf(_Z^8+1))
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 37.83 (sec) , antiderivative size = 1319, normalized size of antiderivative = 5.47
\[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\text {Too large to display}
\]
[In]
integrate((x^6-2)*(x^6-x^4+1)/(x^6+1)^(1/4)/(x^12+x^8+2*x^6+1),x, algorithm="fricas")
[Out]
1/8*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*((-1)^(7/8)*x^6 + (-1)^(3/8)*(x^8 + x^2))*sqrt(x^6 + 1) + 2*(-(I + 1)
*x^7 - (I - 1)*x^5 - (I + 1)*x)*(x^6 + 1)^(3/4) + sqrt(2)*(2*(-1)^(5/8)*(x^10 + x^4) + (-1)^(1/8)*(x^12 - x^8
+ 2*x^6 + 1)) - 2*(x^6 + 1)^(1/4)*((-1)^(3/4)*(x^9 + x^7 + x^3) + (-1)^(1/4)*(x^9 - x^7 + x^3)))/(x^12 + x^8 +
2*x^6 + 1)) - 1/8*sqrt(2)*(-1)^(1/8)*log(8*(2*sqrt(2)*((-1)^(7/8)*x^6 + (-1)^(3/8)*(x^8 + x^2))*sqrt(x^6 + 1)
- 2*(-(I + 1)*x^7 - (I - 1)*x^5 - (I + 1)*x)*(x^6 + 1)^(3/4) + sqrt(2)*(2*(-1)^(5/8)*(x^10 + x^4) + (-1)^(1/8
)*(x^12 - x^8 + 2*x^6 + 1)) + 2*(x^6 + 1)^(1/4)*((-1)^(3/4)*(x^9 + x^7 + x^3) + (-1)^(1/4)*(x^9 - x^7 + x^3)))
/(x^12 + x^8 + 2*x^6 + 1)) - 1/8*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*(I*(-1)^(7/8)*x^6 + (-1)^(3/8)*(I*x^8
+ I*x^2))*sqrt(x^6 + 1) + 2*(-(I + 1)*x^7 - (I - 1)*x^5 - (I + 1)*x)*(x^6 + 1)^(3/4) + sqrt(2)*(2*(-1)^(5/8)*(
-I*x^10 - I*x^4) + (-1)^(1/8)*(-I*x^12 + I*x^8 - 2*I*x^6 - I)) + 2*(x^6 + 1)^(1/4)*((-1)^(3/4)*(x^9 + x^7 + x^
3) + (-1)^(1/4)*(x^9 - x^7 + x^3)))/(x^12 + x^8 + 2*x^6 + 1)) + 1/8*I*sqrt(2)*(-1)^(1/8)*log(-8*(2*sqrt(2)*(-I
*(-1)^(7/8)*x^6 + (-1)^(3/8)*(-I*x^8 - I*x^2))*sqrt(x^6 + 1) + 2*(-(I + 1)*x^7 - (I - 1)*x^5 - (I + 1)*x)*(x^6
+ 1)^(3/4) + sqrt(2)*(2*(-1)^(5/8)*(I*x^10 + I*x^4) + (-1)^(1/8)*(I*x^12 - I*x^8 + 2*I*x^6 + I)) + 2*(x^6 + 1
)^(1/4)*((-1)^(3/4)*(x^9 + x^7 + x^3) + (-1)^(1/4)*(x^9 - x^7 + x^3)))/(x^12 + x^8 + 2*x^6 + 1)) - (1/8*I + 1/
8)*(-1)^(1/8)*log(-16*(2*((I - 1)*x^7 + (I + 1)*x^5 + (I - 1)*x)*(x^6 + 1)^(3/4) + 2*(-1)^(5/8)*((I + 1)*x^10
+ (I + 1)*x^4) + 2*((I - 1)*(-1)^(7/8)*x^6 + (-1)^(3/8)*(-(I - 1)*x^8 - (I - 1)*x^2))*sqrt(x^6 + 1) + 2*(x^6 +
1)^(1/4)*((-1)^(3/4)*(I*x^9 + I*x^7 + I*x^3) + (-1)^(1/4)*(-I*x^9 + I*x^7 - I*x^3)) + (-1)^(1/8)*(-(I + 1)*x^
12 + (I + 1)*x^8 - (2*I + 2)*x^6 - I - 1))/(x^12 + x^8 + 2*x^6 + 1)) + (1/8*I - 1/8)*(-1)^(1/8)*log(-16*(2*((I
- 1)*x^7 + (I + 1)*x^5 + (I - 1)*x)*(x^6 + 1)^(3/4) + 2*(-1)^(5/8)*(-(I - 1)*x^10 - (I - 1)*x^4) + 2*(-(I + 1
)*(-1)^(7/8)*x^6 + (-1)^(3/8)*((I + 1)*x^8 + (I + 1)*x^2))*sqrt(x^6 + 1) + 2*(x^6 + 1)^(1/4)*((-1)^(3/4)*(-I*x
^9 - I*x^7 - I*x^3) + (-1)^(1/4)*(I*x^9 - I*x^7 + I*x^3)) + (-1)^(1/8)*((I - 1)*x^12 - (I - 1)*x^8 + (2*I - 2)
*x^6 + I - 1))/(x^12 + x^8 + 2*x^6 + 1)) - (1/8*I - 1/8)*(-1)^(1/8)*log(-16*(2*((I - 1)*x^7 + (I + 1)*x^5 + (I
- 1)*x)*(x^6 + 1)^(3/4) + 2*(-1)^(5/8)*((I - 1)*x^10 + (I - 1)*x^4) + 2*((I + 1)*(-1)^(7/8)*x^6 + (-1)^(3/8)*
(-(I + 1)*x^8 - (I + 1)*x^2))*sqrt(x^6 + 1) + 2*(x^6 + 1)^(1/4)*((-1)^(3/4)*(-I*x^9 - I*x^7 - I*x^3) + (-1)^(1
/4)*(I*x^9 - I*x^7 + I*x^3)) + (-1)^(1/8)*(-(I - 1)*x^12 + (I - 1)*x^8 - (2*I - 2)*x^6 - I + 1))/(x^12 + x^8 +
2*x^6 + 1)) + (1/8*I + 1/8)*(-1)^(1/8)*log(-16*(2*((I - 1)*x^7 + (I + 1)*x^5 + (I - 1)*x)*(x^6 + 1)^(3/4) + 2
*(-1)^(5/8)*(-(I + 1)*x^10 - (I + 1)*x^4) + 2*(-(I - 1)*(-1)^(7/8)*x^6 + (-1)^(3/8)*((I - 1)*x^8 + (I - 1)*x^2
))*sqrt(x^6 + 1) + 2*(x^6 + 1)^(1/4)*((-1)^(3/4)*(I*x^9 + I*x^7 + I*x^3) + (-1)^(1/4)*(-I*x^9 + I*x^7 - I*x^3)
) + (-1)^(1/8)*((I + 1)*x^12 - (I + 1)*x^8 + (2*I + 2)*x^6 + I + 1))/(x^12 + x^8 + 2*x^6 + 1))
Sympy [F]
\[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^{6} - 2\right ) \left (x^{6} - x^{4} + 1\right )}{\sqrt [4]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (x^{12} + x^{8} + 2 x^{6} + 1\right )}\, dx
\]
[In]
integrate((x**6-2)*(x**6-x**4+1)/(x**6+1)**(1/4)/(x**12+x**8+2*x**6+1),x)
[Out]
Integral((x**6 - 2)*(x**6 - x**4 + 1)/(((x**2 + 1)*(x**4 - x**2 + 1))**(1/4)*(x**12 + x**8 + 2*x**6 + 1)), x)
Maxima [F]
\[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{12} + x^{8} + 2 \, x^{6} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}} \,d x }
\]
[In]
integrate((x^6-2)*(x^6-x^4+1)/(x^6+1)^(1/4)/(x^12+x^8+2*x^6+1),x, algorithm="maxima")
[Out]
integrate((x^6 - x^4 + 1)*(x^6 - 2)/((x^12 + x^8 + 2*x^6 + 1)*(x^6 + 1)^(1/4)), x)
Giac [F]
\[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{12} + x^{8} + 2 \, x^{6} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}} \,d x }
\]
[In]
integrate((x^6-2)*(x^6-x^4+1)/(x^6+1)^(1/4)/(x^12+x^8+2*x^6+1),x, algorithm="giac")
[Out]
integrate((x^6 - x^4 + 1)*(x^6 - 2)/((x^12 + x^8 + 2*x^6 + 1)*(x^6 + 1)^(1/4)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{\sqrt [4]{1+x^6} \left (1+2 x^6+x^8+x^{12}\right )} \, dx=\int \frac {\left (x^6-2\right )\,\left (x^6-x^4+1\right )}{{\left (x^6+1\right )}^{1/4}\,\left (x^{12}+x^8+2\,x^6+1\right )} \,d x
\]
[In]
int(((x^6 - 2)*(x^6 - x^4 + 1))/((x^6 + 1)^(1/4)*(2*x^6 + x^8 + x^12 + 1)),x)
[Out]
int(((x^6 - 2)*(x^6 - x^4 + 1))/((x^6 + 1)^(1/4)*(2*x^6 + x^8 + x^12 + 1)), x)