\(\int \frac {1+x^6}{\sqrt [4]{x^3+x^5} (1-x^6)} \, dx\) [2855]

Optimal result

Integrand size = 26, antiderivative size = 297 \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\frac {2}{3} \arctan \left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{3 \sqrt [4]{2}}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )+\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{3 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \]

[Out]

2/3*arctan(x/(x^5+x^3)^(1/4))+1/6*arctan(2^(1/4)*x/(x^5+x^3)^(1/4))*2^(3/4)-1/6*arctan(2^(3/4)*x*(x^5+x^3)^(1/
4)/(2^(1/2)*x^2-(x^5+x^3)^(1/2)))*2^(1/4)+1/3*2^(1/2)*arctan(2^(1/2)*x*(x^5+x^3)^(1/4)/(-x^2+(x^5+x^3)^(1/2)))
+2/3*arctanh(x/(x^5+x^3)^(1/4))+1/6*arctanh(2^(1/4)*x/(x^5+x^3)^(1/4))*2^(3/4)+1/6*arctanh((1/2*x^2*2^(3/4)+1/
2*(x^5+x^3)^(1/2)*2^(1/4))/x/(x^5+x^3)^(1/4))*2^(1/4)+1/3*2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2*(x^5+x^3)^(1/2)
*2^(1/2))/x/(x^5+x^3)^(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 = 1.33 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.07 \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\frac {x^{3/4} \sqrt [4]{1+x^2} \left (4 \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )-\sqrt [4]{2} \arctan \left (\frac {2^{3/4} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {2} \sqrt {x}-\sqrt {1+x^2}}\right )+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{-\sqrt {x}+\sqrt {1+x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{2 \sqrt {x}+\sqrt {2} \sqrt {1+x^2}}\right )\right )}{6 \sqrt [4]{x^3+x^5}} \]

[In]

Integrate[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]

[Out]

(x^(3/4)*(1 + x^2)^(1/4)*(4*ArcTan[x^(1/4)/(1 + x^2)^(1/4)] + 2^(3/4)*ArcTan[(2^(1/4)*x^(1/4))/(1 + x^2)^(1/4)
] - 2^(1/4)*ArcTan[(2^(3/4)*x^(1/4)*(1 + x^2)^(1/4))/(Sqrt[2]*Sqrt[x] - Sqrt[1 + x^2])] + 2*Sqrt[2]*ArcTan[(Sq
rt[2]*x^(1/4)*(1 + x^2)^(1/4))/(-Sqrt[x] + Sqrt[1 + x^2])] + 4*ArcTanh[x^(1/4)/(1 + x^2)^(1/4)] + 2^(3/4)*ArcT
anh[(2^(1/4)*x^(1/4))/(1 + x^2)^(1/4)] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x^(1/4)*(1 + x^2)^(1/4))/(Sqrt[x] + Sqrt[1
+ x^2])] + 2^(1/4)*ArcTanh[(2*2^(1/4)*x^(1/4)*(1 + x^2)^(1/4))/(2*Sqrt[x] + Sqrt[2]*Sqrt[1 + x^2])]))/(6*(x^3
+ x^5)^(1/4))

Maple [A] (verified)

Time = 20.70 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.34

method	result	size
pseudoelliptic	\(-\frac {2 \arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{3}-\frac {\ln \left (\frac {-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) 2^{\frac {1}{4}}}{12}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {1}{4}}}{6}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {1}{4}}}{6}+\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}+x}{x}\right )}{3}-\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}-x}{x}\right )}{3}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x}\right ) 2^{\frac {3}{4}}}{6}+\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{12}-\frac {\ln \left (\frac {-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) \sqrt {2}}{6}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{3}-\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{3}\)	\(399\)
trager	\(\text {Expression too large to display}\)	\(1424\)

[In]

int((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x,method=_RETURNVERBOSE)

[Out]

-2/3*arctan((x^3*(x^2+1))^(1/4)/x)-1/12*ln((-(x^3*(x^2+1))^(1/4)*2^(3/4)*x+2^(1/2)*x^2+(x^3*(x^2+1))^(1/2))/((
x^3*(x^2+1))^(1/4)*2^(3/4)*x+2^(1/2)*x^2+(x^3*(x^2+1))^(1/2)))*2^(1/4)-1/6*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)
+x)/x)*2^(1/4)-1/6*arctan((2^(1/4)*(x^3*(x^2+1))^(1/4)-x)/x)*2^(1/4)+1/3*ln(((x^3*(x^2+1))^(1/4)+x)/x)-1/3*ln(
((x^3*(x^2+1))^(1/4)-x)/x)-1/6*arctan(1/2*(x^3*(x^2+1))^(1/4)/x*2^(3/4))*2^(3/4)+1/12*ln((-2^(1/4)*x-(x^3*(x^2
+1))^(1/4))/(2^(1/4)*x-(x^3*(x^2+1))^(1/4)))*2^(3/4)-1/6*ln((-(x^3*(x^2+1))^(1/4)*2^(1/2)*x+x^2+(x^3*(x^2+1))^
(1/2))/((x^3*(x^2+1))^(1/4)*2^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2)))*2^(1/2)-1/3*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2
)+x)/x)*2^(1/2)-1/3*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)-x)/x)*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 35.51 (sec) , antiderivative size = 1113, normalized size of antiderivative = 3.75 \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="fricas")

[Out]

1/24*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + 2^(3/4)*(x^4 + 2*x^3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*x
+ 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/24*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(x^4
+ 2*x^3 + x^2) - 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) + 1/24*I*2^(3/4)*log
(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(I*x^4 + 2*I*x^3 + I*x^2) + 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^
5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/24*I*2^(3/4)*log(-(4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(-I*x^4
- 2*I*x^3 - I*x^2) - 4*I*2^(1/4)*sqrt(x^5 + x^3)*x - 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - (1/12*I - 1/1
2)*sqrt(2)*log((4*I*(x^5 + x^3)^(1/4)*x^2 + (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*(-(I - 1)*x^4 + (I -
1)*x^3 - (I - 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) + (1/12*I - 1/12)*sqrt(2)*log((4*I*(x^5 + x^3
)^(1/4)*x^2 - (2*I + 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*((I - 1)*x^4 - (I - 1)*x^3 + (I - 1)*x^2) + 4*(x^5
+ x^3)^(3/4))/(x^4 + x^3 + x^2)) + (1/12*I + 1/12)*sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4)*x^2 - (2*I - 2)*sqrt(2
)*sqrt(x^5 + x^3)*x + sqrt(2)*((I + 1)*x^4 - (I + 1)*x^3 + (I + 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^
2)) - (1/12*I + 1/12)*sqrt(2)*log((-4*I*(x^5 + x^3)^(1/4)*x^2 + (2*I - 2)*sqrt(2)*sqrt(x^5 + x^3)*x + sqrt(2)*
(-(I + 1)*x^4 + (I + 1)*x^3 - (I + 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) - (1/24*I - 1/24)*2^(1/4)
*log(-2*(4*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + (2*I + 2)*2^(3/4)*sqrt(x^5 + x^3)*x - 2^(1/4)*((I - 1)*x^4 - (2*I
- 2)*x^3 + (I - 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) + (1/24*I - 1/24)*2^(1/4)*log(-2*(4*I*sqr
t(2)*(x^5 + x^3)^(1/4)*x^2 - (2*I + 2)*2^(3/4)*sqrt(x^5 + x^3)*x - 2^(1/4)*(-(I - 1)*x^4 + (2*I - 2)*x^3 - (I
- 1)*x^2) + 4*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) + (1/24*I + 1/24)*2^(1/4)*log(-2*(-4*I*sqrt(2)*(x^5 + x^
3)^(1/4)*x^2 - (2*I - 2)*2^(3/4)*sqrt(x^5 + x^3)*x - 2^(1/4)*(-(I + 1)*x^4 + (2*I + 2)*x^3 - (I + 1)*x^2) + 4*
(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) - (1/24*I + 1/24)*2^(1/4)*log(-2*(-4*I*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 +
(2*I - 2)*2^(3/4)*sqrt(x^5 + x^3)*x - 2^(1/4)*((I + 1)*x^4 - (2*I + 2)*x^3 + (I + 1)*x^2) + 4*(x^5 + x^3)^(3/
4))/(x^4 + 2*x^3 + x^2)) + 1/3*arctan(2*((x^5 + x^3)^(1/4)*x^2 + (x^5 + x^3)^(3/4))/(x^4 - x^3 + x^2)) + 1/3*l
og((x^4 + x^3 + 2*(x^5 + x^3)^(1/4)*x^2 + x^2 + 2*sqrt(x^5 + x^3)*x + 2*(x^5 + x^3)^(3/4))/(x^4 - x^3 + 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. \[ \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx=\int -\frac {x^6+1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^6-1\right )} \,d x \]

[In]

int(-(x^6 + 1)/((x^3 + x^5)^(1/4)*(x^6 - 1)),x)

[Out]

int(-(x^6 + 1)/((x^3 + x^5)^(1/4)*(x^6 - 1)), x)