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

Optimal. Leaf size=101 \[ \frac {4 \sqrt [4]{1+x^3}}{x}-2 \sqrt {2} \text {ArcTan}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]

[Out]

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

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Rubi [F]
time = 0.90, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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 \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};-x^3\right )}{x}-2 x \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-x^3\right )+\frac {1}{2} x^2 \, _2F_1\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 \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};-x^3\right )}{x}-2 x \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-x^3\right )+\frac {1}{2} x^2 \, _2F_1\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*}

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Mathematica [A]
time = 3.10, size = 95, normalized size = 0.94 \begin {gather*} \frac {4 \sqrt [4]{1+x^3}}{x}-2 \sqrt {2} \text {ArcTan}\left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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])]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 7.35, size = 221, normalized size = 2.19

method	result	size
trager	\(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{3}+1}\, x^{2}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )\)	\(221\)
risch	\(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}+\frac {\left (-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \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 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{7}+3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{4}+3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{3}+1\right ) \left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2}}\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x^{7}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{7}+4 \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 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \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}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )^{\frac {1}{4}} x -3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}+3 x^{6}+3 x^{3}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{3}+1\right ) \left (1+x \right )^{2} \left (x^{2}-x +1\right )^{2}}\right )\right ) \left (\left (x^{3}+1\right )^{3}\right )^{\frac {1}{4}}}{\left (x^{3}+1\right )^{\frac {3}{4}}}\)	\(593\)

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (82) = 164\).
time = 10.98, size = 709, normalized size = 7.02 \begin {gather*} \frac {4 \, \sqrt {2} x \arctan \left (-\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} - {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} + \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 4 \, \sqrt {2} x \arctan \left (-\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} - {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} - \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - \sqrt {2} x \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + \sqrt {2} x \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + 8 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(4*sqrt(2)*x*arctan(-(x^8 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + 2*sqrt(2)*(3*x^5 - x^4 - x)*(x^3 + 1)^(3/4) + 2*
sqrt(2)*(x^7 - 3*x^6 - 3*x^3)*(x^3 + 1)^(1/4) + 4*(x^6 + x^5 + x^2)*sqrt(x^3 + 1) - (16*(x^3 + 1)^(3/4)*x^5 +
2*sqrt(2)*(3*x^6 - x^5 - x^2)*sqrt(x^3 + 1) + sqrt(2)*(x^8 + 8*x^7 - x^6 + 8*x^4 - 2*x^3 - 1) + 4*(x^7 + x^6 +
x^3)*(x^3 + 1)^(1/4))*sqrt((x^4 - 2*sqrt(2)*(x^3 + 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 - 2*sqrt(2)*(x^3
+ 1)^(3/4)*x + 1)/(x^4 + x^3 + 1)) + 1)/(x^8 - 14*x^7 + x^6 - 14*x^4 + 2*x^3 + 1)) - 4*sqrt(2)*x*arctan(-(x^8
+ 2*x^7 + x^6 + 2*x^4 + 2*x^3 - 2*sqrt(2)*(3*x^5 - x^4 - x)*(x^3 + 1)^(3/4) - 2*sqrt(2)*(x^7 - 3*x^6 - 3*x^3)*
(x^3 + 1)^(1/4) + 4*(x^6 + x^5 + x^2)*sqrt(x^3 + 1) - (16*(x^3 + 1)^(3/4)*x^5 - 2*sqrt(2)*(3*x^6 - x^5 - x^2)*
sqrt(x^3 + 1) - sqrt(2)*(x^8 + 8*x^7 - x^6 + 8*x^4 - 2*x^3 - 1) + 4*(x^7 + x^6 + x^3)*(x^3 + 1)^(1/4))*sqrt((x
^4 + 2*sqrt(2)*(x^3 + 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 + 2*sqrt(2)*(x^3 + 1)^(3/4)*x + 1)/(x^4 + x^3 +
1)) + 1)/(x^8 - 14*x^7 + x^6 - 14*x^4 + 2*x^3 + 1)) - sqrt(2)*x*log(4*(x^4 + 2*sqrt(2)*(x^3 + 1)^(1/4)*x^3 +
x^3 + 4*sqrt(x^3 + 1)*x^2 + 2*sqrt(2)*(x^3 + 1)^(3/4)*x + 1)/(x^4 + x^3 + 1)) + sqrt(2)*x*log(4*(x^4 - 2*sqrt(
2)*(x^3 + 1)^(1/4)*x^3 + x^3 + 4*sqrt(x^3 + 1)*x^2 - 2*sqrt(2)*(x^3 + 1)^(3/4)*x + 1)/(x^4 + x^3 + 1)) + 8*(x^
3 + 1)^(1/4))/x

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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