\(\int \frac {(4+x^3) (-1-x^3+x^4)}{x^2 (1+x^3)^{3/4} (1+x^3+x^4)} \, dx\) [1419]

Optimal result

Integrand size = 38, antiderivative size = 101 \[ \int \frac {\left (4+x^3\right ) \left (-1-x^3+x^4\right )}{x^2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{1+x^3}}{x}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*Hypergeometric2F1[-1/3, 3/4, 2/3, -x^3])/x - 2*x*Hypergeometric2F1[1/3, 3/4, 4/3, -x^3] + (x^2*Hypergeometr
ic2F1[2/3, 3/4, 5/3, -x^3])/2 + 2*Defer[Int][1/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] - 2*Defer[Int][x/((1 + x^
3)^(3/4)*(1 + x^3 + x^4)), x] + 8*Defer[Int][x^2/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] + 2*Defer[Int][x^3/((1
+ x^3)^(3/4)*(1 + x^3 + x^4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2}{\left (1+x^3\right )^{3/4}}-\frac {4}{x^2 \left (1+x^3\right )^{3/4}}+\frac {x}{\left (1+x^3\right )^{3/4}}+\frac {2 \left (1-x+4 x^2+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^3\right )^{3/4}} \, dx\right )+2 \int \frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-4 \int \frac {1}{x^2 \left (1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (1+x^3\right )^{3/4}} \, dx \\ & = \frac {4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {2}{3},-x^3\right )}{x}-2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-x^3\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},-x^3\right )+2 \int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}-\frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx \\ & = \frac {4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {2}{3},-x^3\right )}{x}-2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-x^3\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},-x^3\right )+2 \int \frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-2 \int \frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+2 \int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+8 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.47 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94 \[ \int \frac {\left (4+x^3\right ) \left (-1-x^3+x^4\right )}{x^2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{1+x^3}}{x}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]

[In]

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

[Out]

(4*(1 + x^3)^(1/4))/x - 2*Sqrt[2]*ArcTan[(-x^2 + Sqrt[1 + x^3])/(Sqrt[2]*x*(1 + x^3)^(1/4))] - 2*Sqrt[2]*ArcTa
nh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x^2 + Sqrt[1 + x^3])]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.18

method	result	size
trager	\(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )\)	\(220\)
risch	\(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{7}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {3}{4}} x^{3}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2} \left (x^{4}+x^{3}+1\right )}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {3}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{7}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2} \left (x^{4}+x^{3}+1\right )}\right )\right ) {\left (\left (x^{3}+1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{3}+1\right )^{\frac {3}{4}}}\)	\(593\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.84 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.00 \[ \int \frac {\left (4+x^3\right ) \left (-1-x^3+x^4\right )}{x^2 \left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\frac {-\left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} + 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} - i + 1\right )}}{x^{4} + x^{3} + 1}\right ) + \left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} + 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} + i - 1\right )}}{x^{4} + x^{3} + 1}\right ) + \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} + 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} + i + 1\right )}}{x^{4} + x^{3} + 1}\right ) - \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-4 i \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{3} + 1} x^{2} + 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} - i - 1\right )}}{x^{4} + x^{3} + 1}\right ) + 8 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{2 \, x} \]

[In]

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

[Out]

1/2*(-(I + 1)*sqrt(2)*x*log((4*I*(x^3 + 1)^(1/4)*x^3 + (2*I + 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 + 4*(x^3 + 1)^(3/4)
*x + sqrt(2)*((I - 1)*x^4 - (I - 1)*x^3 - I + 1))/(x^4 + x^3 + 1)) + (I + 1)*sqrt(2)*x*log((4*I*(x^3 + 1)^(1/4
)*x^3 - (2*I + 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 + 4*(x^3 + 1)^(3/4)*x + sqrt(2)*(-(I - 1)*x^4 + (I - 1)*x^3 + I -
1))/(x^4 + x^3 + 1)) + (I - 1)*sqrt(2)*x*log((-4*I*(x^3 + 1)^(1/4)*x^3 - (2*I - 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 +
4*(x^3 + 1)^(3/4)*x + sqrt(2)*(-(I + 1)*x^4 + (I + 1)*x^3 + I + 1))/(x^4 + x^3 + 1)) - (I - 1)*sqrt(2)*x*log(
(-4*I*(x^3 + 1)^(1/4)*x^3 + (2*I - 2)*sqrt(2)*sqrt(x^3 + 1)*x^2 + 4*(x^3 + 1)^(3/4)*x + sqrt(2)*((I + 1)*x^4 -
(I + 1)*x^3 - I - 1))/(x^4 + x^3 + 1)) + 8*(x^3 + 1)^(1/4))/x

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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