\(\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} (1-x^6)} \, dx\) [2987]
Optimal result
Integrand size = 28, antiderivative size = 390 \[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\frac {4 \left (-x^3+x^5\right )^{3/4}}{3 x^2 \left (-1+x^2\right )}+\frac {\sqrt [4]{2} \arctan \left (\frac {3^{7/8} \sqrt {2-\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \arctan \left (\frac {3^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [4]{-x^3+x^5}}{-3 x^2+3^{3/4} \sqrt {-x^3+x^5}}\right )+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2-\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right )}{3 \sqrt [8]{3 \left (17+12 \sqrt {2}\right )}}+\frac {1}{3} \sqrt [4]{2} \sqrt [8]{\frac {1}{3} \left (17+12 \sqrt {2}\right )} \text {arctanh}\left (\frac {\frac {\sqrt [8]{3} x^2}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {-x^3+x^5}}{\sqrt [8]{3} \sqrt {2+\sqrt {2}}}}{x \sqrt [4]{-x^3+x^5}}\right )
\]
[Out]
4/3*(x^5-x^3)^(3/4)/x^2/(x^2-1)+1/3*arctan(3^(7/8)*(2-2^(1/2))^(1/2)*x*(x^5-x^3)^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^
3)^(1/2)))*2^(1/4)/(51+36*2^(1/2))^(1/8)+1/3*(17/3+4*2^(1/2))^(1/8)*arctan(3^(7/8)*(2+2^(1/2))^(1/2)*x*(x^5-x^
3)^(1/4)/(-3*x^2+3^(3/4)*(x^5-x^3)^(1/2)))*2^(1/4)+1/3*arctanh((3^(1/8)*x^2/(2-2^(1/2))^(1/2)+1/3*(x^5-x^3)^(1
/2)*3^(7/8)/(2-2^(1/2))^(1/2))/x/(x^5-x^3)^(1/4))*2^(1/4)/(51+36*2^(1/2))^(1/8)+1/3*(17/3+4*2^(1/2))^(1/8)*arc
tanh((3^(1/8)*x^2/(2+2^(1/2))^(1/2)+1/3*(x^5-x^3)^(1/2)*3^(7/8)/(2+2^(1/2))^(1/2))/x/(x^5-x^3)^(1/4))*2^(1/4)
Rubi [F(-1)]
Timed out. \[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\text {\$Aborted}
\]
[In]
Int[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
$Aborted
Rubi steps Aborted
Mathematica [A] (verified)
Time = 3.28 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.05
\[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\frac {x^{3/4} \left (12 \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x}+\sqrt [4]{2} 3^{7/8} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+3^{7/8} \sqrt [4]{34+24 \sqrt {2}} \sqrt [4]{-1+x^2} \arctan \left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{-3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )+\sqrt [4]{2} 3^{7/8} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{\sqrt [8]{17+12 \sqrt {2}} \left (3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}\right )}\right )+3^{7/8} \sqrt [4]{34+24 \sqrt {2}} \sqrt [4]{-1+x^2} \text {arctanh}\left (\frac {\sqrt [4]{2} 3^{7/8} \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x} \sqrt [4]{-1+x^2}}{3 \sqrt {x}+3^{3/4} \sqrt {-1+x^2}}\right )\right )}{9 \sqrt [8]{17+12 \sqrt {2}} \sqrt [4]{x^3 \left (-1+x^2\right )}}
\]
[In]
Integrate[(1 + x^6)/((-x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
(x^(3/4)*(12*(17 + 12*Sqrt[2])^(1/8)*x^(1/4) + 2^(1/4)*3^(7/8)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*x^(1/4
)*(-1 + x^2)^(1/4))/((17 + 12*Sqrt[2])^(1/8)*(-3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + 3^(7/8)*(34 + 24*Sqrt[2
])^(1/4)*(-1 + x^2)^(1/4)*ArcTan[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x^(1/4)*(-1 + x^2)^(1/4))/(-3*Sqrt[x
] + 3^(3/4)*Sqrt[-1 + x^2])] + 2^(1/4)*3^(7/8)*(-1 + x^2)^(1/4)*ArcTanh[(2^(1/4)*3^(7/8)*x^(1/4)*(-1 + x^2)^(1
/4))/((17 + 12*Sqrt[2])^(1/8)*(3*Sqrt[x] + 3^(3/4)*Sqrt[-1 + x^2]))] + 3^(7/8)*(34 + 24*Sqrt[2])^(1/4)*(-1 + x
^2)^(1/4)*ArcTanh[(2^(1/4)*3^(7/8)*(17 + 12*Sqrt[2])^(1/8)*x^(1/4)*(-1 + x^2)^(1/4))/(3*Sqrt[x] + 3^(3/4)*Sqrt
[-1 + x^2])]))/(9*(17 + 12*Sqrt[2])^(1/8)*(x^3*(-1 + x^2))^(1/4))
Maple [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 68.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17
| | |
| method | result | size |
| | |
| pseudoelliptic |
\(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) \left (x^{5}-x^{3}\right )^{\frac {1}{4}}+4 x}{3 \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\) |
\(65\) |
| risch |
\(\text {Expression too large to display}\) |
\(2154\) |
| trager |
\(\text {Expression too large to display}\) |
\(2163\) |
| | |
|
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|
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[In]
int((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x,method=_RETURNVERBOSE)
[Out]
1/3*(-sum(ln((-_R*x+(x^5-x^3)^(1/4))/x)/_R,_R=RootOf(_Z^8+3))*(x^5-x^3)^(1/4)+4*x)/(x^5-x^3)^(1/4)
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 143.79 (sec) , antiderivative size = 2118, normalized size of antiderivative = 5.43
\[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\text {Too large to display}
\]
[In]
integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="fricas")
[Out]
1/36*(3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(I + 1)*x^4 + (I + 1)*x^2)*log((3^(7/8)*sqrt(2)*(-1)^(1/8)*((109*I + 109)*x
^6 - (264*I + 264)*x^5 - (545*I + 545)*x^4 + (264*I + 264)*x^3 + (109*I + 109)*x^2) - 6*3^(3/8)*sqrt(2)*(-1)^(
5/8)*(-(22*I + 22)*x^6 - (109*I + 109)*x^5 + (110*I + 110)*x^4 + (109*I + 109)*x^3 - (22*I + 22)*x^2) - 12*(x^
5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(44*I*x^2 + 109*I*x - 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*sqrt
(2)*(-1)^(7/8)*(-(109*I - 109)*x^3 + (132*I - 132)*x^2 + (109*I - 109)*x) + 3^(5/8)*sqrt(2)*(-1)^(3/8)*((44*I
- 44)*x^3 + (109*I - 109)*x^2 - (44*I - 44)*x)) - 12*(x^5 - x^3)^(1/4)*(3^(1/4)*(-1)^(3/4)*(109*I*x^4 - 132*I*
x^3 - 109*I*x^2) + 3^(3/4)*(-1)^(1/4)*(-44*I*x^4 - 109*I*x^3 + 44*I*x^2)))/(x^6 + x^4 + x^2)) + 3^(7/8)*sqrt(2
)*(-1)^(1/8)*((I - 1)*x^4 - (I - 1)*x^2)*log((3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(109*I - 109)*x^6 + (264*I - 264)*x
^5 + (545*I - 545)*x^4 - (264*I - 264)*x^3 - (109*I - 109)*x^2) - 6*3^(3/8)*sqrt(2)*(-1)^(5/8)*((22*I - 22)*x^
6 + (109*I - 109)*x^5 - (110*I - 110)*x^4 - (109*I - 109)*x^3 + (22*I - 22)*x^2) - 12*(x^5 - x^3)^(3/4)*(109*x
^2 + sqrt(3)*(44*I*x^2 + 109*I*x - 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*sqrt(2)*(-1)^(7/8)*((109*
I + 109)*x^3 - (132*I + 132)*x^2 - (109*I + 109)*x) + 3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(44*I + 44)*x^3 - (109*I +
109)*x^2 + (44*I + 44)*x)) - 12*(x^5 - x^3)^(1/4)*(3^(3/4)*(-1)^(1/4)*(44*I*x^4 + 109*I*x^3 - 44*I*x^2) + 3^(1
/4)*(-1)^(3/4)*(-109*I*x^4 + 132*I*x^3 + 109*I*x^2)))/(x^6 + x^4 + x^2)) + 3^(7/8)*sqrt(2)*(-1)^(1/8)*((I + 1)
*x^4 - (I + 1)*x^2)*log(-(6*3^(3/8)*sqrt(2)*(-1)^(5/8)*((22*I + 22)*x^6 + (109*I + 109)*x^5 - (110*I + 110)*x^
4 - (109*I + 109)*x^3 + (22*I + 22)*x^2) - 3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(109*I + 109)*x^6 + (264*I + 264)*x^5
+ (545*I + 545)*x^4 - (264*I + 264)*x^3 - (109*I + 109)*x^2) + 12*(x^5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(44*I*x
^2 + 109*I*x - 44*I) - 132*x - 109) + 6*sqrt(x^5 - x^3)*(3^(5/8)*sqrt(2)*(-1)^(3/8)*(-(44*I - 44)*x^3 - (109*I
- 109)*x^2 + (44*I - 44)*x) + 3^(1/8)*sqrt(2)*(-1)^(7/8)*((109*I - 109)*x^3 - (132*I - 132)*x^2 - (109*I - 10
9)*x)) + 12*(x^5 - x^3)^(1/4)*(3^(1/4)*(-1)^(3/4)*(109*I*x^4 - 132*I*x^3 - 109*I*x^2) + 3^(3/4)*(-1)^(1/4)*(-4
4*I*x^4 - 109*I*x^3 + 44*I*x^2)))/(x^6 + x^4 + x^2)) + 3^(7/8)*sqrt(2)*(-1)^(1/8)*(-(I - 1)*x^4 + (I - 1)*x^2)
*log(-(6*3^(3/8)*sqrt(2)*(-1)^(5/8)*(-(22*I - 22)*x^6 - (109*I - 109)*x^5 + (110*I - 110)*x^4 + (109*I - 109)*
x^3 - (22*I - 22)*x^2) - 3^(7/8)*sqrt(2)*(-1)^(1/8)*((109*I - 109)*x^6 - (264*I - 264)*x^5 - (545*I - 545)*x^4
+ (264*I - 264)*x^3 + (109*I - 109)*x^2) + 12*(x^5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(44*I*x^2 + 109*I*x - 44*I
) - 132*x - 109) + 6*sqrt(x^5 - x^3)*(3^(5/8)*sqrt(2)*(-1)^(3/8)*((44*I + 44)*x^3 + (109*I + 109)*x^2 - (44*I
+ 44)*x) + 3^(1/8)*sqrt(2)*(-1)^(7/8)*(-(109*I + 109)*x^3 + (132*I + 132)*x^2 + (109*I + 109)*x)) + 12*(x^5 -
x^3)^(1/4)*(3^(3/4)*(-1)^(1/4)*(44*I*x^4 + 109*I*x^3 - 44*I*x^2) + 3^(1/4)*(-1)^(3/4)*(-109*I*x^4 + 132*I*x^3
+ 109*I*x^2)))/(x^6 + x^4 + x^2)) + 2*3^(7/8)*(-1)^(1/8)*(x^4 - x^2)*log(-(3^(7/8)*(-1)^(1/8)*(109*x^6 - 264*x
^5 - 545*x^4 + 264*x^3 + 109*x^2) - 6*3^(3/8)*(-1)^(5/8)*(22*x^6 + 109*x^5 - 110*x^4 - 109*x^3 + 22*x^2) + 6*(
x^5 - x^3)^(3/4)*(109*x^2 + sqrt(3)*(-44*I*x^2 - 109*I*x + 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*(
-1)^(7/8)*(109*x^3 - 132*x^2 - 109*x) + 3^(5/8)*(-1)^(3/8)*(44*x^3 + 109*x^2 - 44*x)) - 6*(x^5 - x^3)^(1/4)*(3
^(1/4)*(-1)^(3/4)*(109*x^4 - 132*x^3 - 109*x^2) + 3^(3/4)*(-1)^(1/4)*(44*x^4 + 109*x^3 - 44*x^2)))/(x^6 + x^4
+ x^2)) - 2*3^(7/8)*(-1)^(1/8)*(x^4 - x^2)*log((3^(7/8)*(-1)^(1/8)*(109*x^6 - 264*x^5 - 545*x^4 + 264*x^3 + 10
9*x^2) - 6*3^(3/8)*(-1)^(5/8)*(22*x^6 + 109*x^5 - 110*x^4 - 109*x^3 + 22*x^2) - 6*(x^5 - x^3)^(3/4)*(109*x^2 +
sqrt(3)*(-44*I*x^2 - 109*I*x + 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(1/8)*(-1)^(7/8)*(109*x^3 - 132*x^
2 - 109*x) + 3^(5/8)*(-1)^(3/8)*(44*x^3 + 109*x^2 - 44*x)) + 6*(x^5 - x^3)^(1/4)*(3^(1/4)*(-1)^(3/4)*(109*x^4
- 132*x^3 - 109*x^2) + 3^(3/4)*(-1)^(1/4)*(44*x^4 + 109*x^3 - 44*x^2)))/(x^6 + x^4 + x^2)) - 2*3^(7/8)*(-1)^(1
/8)*(I*x^4 - I*x^2)*log((3^(7/8)*(-1)^(1/8)*(109*I*x^6 - 264*I*x^5 - 545*I*x^4 + 264*I*x^3 + 109*I*x^2) - 6*3^
(3/8)*(-1)^(5/8)*(22*I*x^6 + 109*I*x^5 - 110*I*x^4 - 109*I*x^3 + 22*I*x^2) - 6*(x^5 - x^3)^(3/4)*(109*x^2 + sq
rt(3)*(-44*I*x^2 - 109*I*x + 44*I) - 132*x - 109) - 6*sqrt(x^5 - x^3)*(3^(5/8)*(-1)^(3/8)*(-44*I*x^3 - 109*I*x
^2 + 44*I*x) + 3^(1/8)*(-1)^(7/8)*(-109*I*x^3 + 132*I*x^2 + 109*I*x)) - 6*(x^5 - x^3)^(1/4)*(3^(1/4)*(-1)^(3/4
)*(109*x^4 - 132*x^3 - 109*x^2) + 3^(3/4)*(-1)^(1/4)*(44*x^4 + 109*x^3 - 44*x^2)))/(x^6 + x^4 + x^2)) - 2*3^(7
/8)*(-1)^(1/8)*(-I*x^4 + I*x^2)*log(-(6*3^(3/8)*(-1)^(5/8)*(-22*I*x^6 - 109*I*x^5 + 110*I*x^4 + 109*I*x^3 - 22
*I*x^2) - 3^(7/8)*(-1)^(1/8)*(-109*I*x^6 + 264*I*x^5 + 545*I*x^4 - 264*I*x^3 - 109*I*x^2) + 6*(x^5 - x^3)^(3/4
)*(109*x^2 + sqrt(3)*(-44*I*x^2 - 109*I*x + 44*I) - 132*x - 109) + 6*sqrt(x^5 - x^3)*(3^(1/8)*(-1)^(7/8)*(109*
I*x^3 - 132*I*x^2 - 109*I*x) + 3^(5/8)*(-1)^(3/8)*(44*I*x^3 + 109*I*x^2 - 44*I*x)) + 6*(x^5 - x^3)^(1/4)*(3^(1
/4)*(-1)^(3/4)*(109*x^4 - 132*x^3 - 109*x^2) + 3^(3/4)*(-1)^(1/4)*(44*x^4 + 109*x^3 - 44*x^2)))/(x^6 + x^4 + x
^2)) + 48*(x^5 - x^3)^(3/4))/(x^4 - x^2)
Sympy [F]
\[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=- \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} - x^{3}} - \sqrt [4]{x^{5} - x^{3}}}\, dx
\]
[In]
integrate((x**6+1)/(x**5-x**3)**(1/4)/(-x**6+1),x)
[Out]
-Integral(x**6/(x**6*(x**5 - x**3)**(1/4) - (x**5 - x**3)**(1/4)), x) - Integral(1/(x**6*(x**5 - x**3)**(1/4)
- (x**5 - x**3)**(1/4)), x)
Maxima [F]
\[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x }
\]
[In]
integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="maxima")
[Out]
-integrate((x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
Giac [F]
\[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int { -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}}} \,d x }
\]
[In]
integrate((x^6+1)/(x^5-x^3)^(1/4)/(-x^6+1),x, algorithm="giac")
[Out]
integrate(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1-x^6\right )} \, dx=\int -\frac {x^6+1}{\left (x^6-1\right )\,{\left (x^5-x^3\right )}^{1/4}} \,d x
\]
[In]
int(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)),x)
[Out]
int(-(x^6 + 1)/((x^6 - 1)*(x^5 - x^3)^(1/4)), x)