\(\int \frac {(-3+x^4) \sqrt [3]{1+x^4}}{x^2 (1+x^3+x^4)} \, dx\) [1237]

Optimal result

Integrand size = 28, antiderivative size = 90 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]

[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]

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

[In]

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

[Out]

(3*Hypergeometric2F1[-1/3, -1/4, 3/4, -x^4])/x + 3*Defer[Int][(x*(1 + x^4)^(1/3))/(1 + x^3 + x^4), x] + 4*Defe
r[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 {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}+\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 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{x}+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 = 1.00 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {3 \sqrt [3]{1+x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^4}}\right )-\log \left (x+\sqrt [3]{1+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]

[In]

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

Maple [A] (verified)

Time = 7.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \ln \left (\frac {x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {x^{2}-x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{4}+1\right )^{\frac {1}{3}}}{2 x}\)	\(87\)
trager	\(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}-\ln \left (\frac {15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-30 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+14 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-13 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 x^{4}+9 \left (x^{4}+1\right )^{\frac {2}{3}} x +14 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+3 x^{3}+15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+14 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3}{x^{4}+x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{4}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{4}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}+12 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+2 x^{4}-6 \left (x^{4}+1\right )^{\frac {2}{3}} x -3 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-15 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2}{x^{4}+x^{3}+1}\right )\)	\(402\)
risch	\(\text {Expression too large to display}\)	\(778\)

[In]

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

[Out]

1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^4+1)^(1/3))/x)*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 = 1.74 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^2 \left (1+x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} + x^{3} + 1\right )}}{3 \, {\left (x^{4} - x^{3} + 1\right )}}\right ) - x \log \left (\frac {x^{4} + x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + x^{3} + 1}\right ) + 6 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{2 \, x} \]

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)