\(\int \frac {\sqrt [3]{-1+x^4} (3+x^4)}{x^2 (-1-x^3+x^4)} \, dx\) [1182]

Optimal result

Integrand size = 30, antiderivative size = 87 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{-1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx \]

[In]

Int[((-1 + x^4)^(1/3)*(3 + x^4))/(x^2*(-1 - x^3 + x^4)),x]

[Out]

(3*(-1 + x^4)^(1/3)*Hypergeometric2F1[-1/3, -1/4, 3/4, x^4])/(x*(1 - x^4)^(1/3)) - 3*Defer[Int][(x*(-1 + x^4)^
(1/3))/(-1 - x^3 + x^4), x] + 4*Defer[Int][(x^2*(-1 + x^4)^(1/3))/(-1 - x^3 + x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{-1+x^4}}{x^2}+\frac {x (-3+4 x) \sqrt [3]{-1+x^4}}{-1-x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{-1+x^4}}{x^2} \, dx\right )+\int \frac {x (-3+4 x) \sqrt [3]{-1+x^4}}{-1-x^3+x^4} \, dx \\ & = -\frac {\left (3 \sqrt [3]{-1+x^4}\right ) \int \frac {\sqrt [3]{1-x^4}}{x^2} \, dx}{\sqrt [3]{1-x^4}}+\int \left (-\frac {3 x \sqrt [3]{-1+x^4}}{-1-x^3+x^4}+\frac {4 x^2 \sqrt [3]{-1+x^4}}{-1-x^3+x^4}\right ) \, dx \\ & = \frac {3 \sqrt [3]{-1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{x \sqrt [3]{1-x^4}}-3 \int \frac {x \sqrt [3]{-1+x^4}}{-1-x^3+x^4} \, dx+4 \int \frac {x^2 \sqrt [3]{-1+x^4}}{-1-x^3+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{-1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]

[In]

Integrate[((-1 + x^4)^(1/3)*(3 + x^4))/(x^2*(-1 - x^3 + x^4)),x]

[Out]

(3*(-1 + x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^4)^(1/3))] + Log[-x + (-1 + x^4)^(1/3)] - L
og[x^2 + x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/2

Maple [A] (verified)

Time = 9.94 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{4}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x +2 \ln \left (\frac {-x +\left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right ) x -\ln \left (\frac {x^{2}+x \left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{4}-1\right )^{\frac {1}{3}}}{2 x}\)	\(89\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}-6 \ln \left (-\frac {-1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}+2154600 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -2586522 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+340561 x^{4}+255269 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+118456 x^{3}+1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-340561}{x^{4}-x^{3}-1}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+\ln \left (\frac {4264416 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}-7995780 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -1531614 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-1684266 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+59850 x^{4}+431087 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+91770 x^{3}-4264416 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-59850}{x^{4}-x^{3}-1}\right )-\ln \left (-\frac {-1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{4}+2154600 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{4}+4118136 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -2586522 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-1524150 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+340561 x^{4}+255269 \left (x^{4}-1\right )^{\frac {2}{3}} x -686356 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+118456 x^{3}+1149120 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-160116 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-340561}{x^{4}-x^{3}-1}\right )\)	\(595\)
risch	\(\text {Expression too large to display}\)	\(754\)

[In]

int((x^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 2.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (-\frac {14106128635054532 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 89654043956484782 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (35416555940707109 \, x^{4} + 2357401720008016 \, x^{3} - 35416555940707109\right )}}{3 \, {\left (51678794422160641 \, x^{4} + 201291873609016 \, x^{3} - 51678794422160641\right )}}\right ) + x \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - x^{3} - 1}\right ) + 6 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{2 \, x} \]

[In]

integrate((x^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x*arctan(-1/3*(14106128635054532*sqrt(3)*(x^4 - 1)^(1/3)*x^2 - 89654043956484782*sqrt(3)*(x^4 -
1)^(2/3)*x - sqrt(3)*(35416555940707109*x^4 + 2357401720008016*x^3 - 35416555940707109))/(51678794422160641*x
^4 + 201291873609016*x^3 - 51678794422160641)) + x*log((x^4 - x^3 + 3*(x^4 - 1)^(1/3)*x^2 - 3*(x^4 - 1)^(2/3)*
x - 1)/(x^4 - x^3 - 1)) + 6*(x^4 - 1)^(1/3))/x

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**4-1)**(1/3)*(x**4+3)/x**2/(x**4-x**3-1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{3} - 1\right )} x^{2}} \,d x } \]

[In]

integrate((x^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x, algorithm="maxima")

[Out]

integrate((x^4 + 3)*(x^4 - 1)^(1/3)/((x^4 - x^3 - 1)*x^2), x)

Giac [F]

\[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} - x^{3} - 1\right )} x^{2}} \,d x } \]

[In]

integrate((x^4-1)^(1/3)*(x^4+3)/x^2/(x^4-x^3-1),x, algorithm="giac")

[Out]

integrate((x^4 + 3)*(x^4 - 1)^(1/3)/((x^4 - x^3 - 1)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{-1+x^4} \left (3+x^4\right )}{x^2 \left (-1-x^3+x^4\right )} \, dx=\int -\frac {{\left (x^4-1\right )}^{1/3}\,\left (x^4+3\right )}{x^2\,\left (-x^4+x^3+1\right )} \,d x \]

[In]

int(-((x^4 - 1)^(1/3)*(x^4 + 3))/(x^2*(x^3 - x^4 + 1)),x)

[Out]

int(-((x^4 - 1)^(1/3)*(x^4 + 3))/(x^2*(x^3 - x^4 + 1)), x)