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

Optimal. Leaf size=95 \[ \frac {4 \sqrt [4]{x^3-1}}{x}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right ) \]

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Rubi [F]  time = 1.41, 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*(1 - x^3)^(3/4)*Hypergeometric2F1[-1/3, 3/4, 2/3, x^3])/(x*(-1 + x^3)^(3/4)) - (2*x*(1 - x^3)^(3/4)*Hyperg
eometric2F1[1/3, 3/4, 4/3, x^3])/(-1 + x^3)^(3/4) + (x^2*(1 - x^3)^(3/4)*Hypergeometric2F1[2/3, 3/4, 5/3, x^3]
)/(2*(-1 + x^3)^(3/4)) + 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\\ &=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 {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}-\frac {\left (2 \left (1-x^3\right )^{3/4}\right ) \int \frac {1}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\frac {\left (4 \left (1-x^3\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}\\ &=-\frac {4 \left (1-x^3\right )^{3/4} \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};x^3\right )}{x \left (-1+x^3\right )^{3/4}}-\frac {2 x \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{3/4}}+\frac {x^2 \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};x^3\right )}{2 \left (-1+x^3\right )^{3/4}}+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 [F]  time = 0.46, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 4.24, size = 95, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-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]

IntegrateAlgebraic[((-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[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])] - 2*Sqrt[2]*Ar
cTanh[(Sqrt[2]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])]

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fricas [B]  time = 24.71, size = 459, normalized size = 4.83 \begin {gather*} -\frac {4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - {\left (x^{4} - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1\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}}}{x^{4} - x^{3} + 1}\right ) + 4 \, \sqrt {2} x \arctan \left (\frac {\sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + {\left (x^{4} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - 1\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}}}{x^{4} - x^{3} + 1}\right ) + \sqrt {2} x \log \left (\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}\right ) - \sqrt {2} x \log \left (\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}\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((sqrt(2)*(x^3 - 1)^(1/4)*x^3 + sqrt(2)*(x^3 - 1)^(3/4)*x - (x^4 - sqrt(2)*(x^3 - 1)^(
1/4)*x^3 + x^3 + 2*sqrt(x^3 - 1)*x^2 - sqrt(2)*(x^3 - 1)^(3/4)*x - 1)*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)))/(x^4 - x^3 + 1)) + 4*sqrt(2
)*x*arctan((sqrt(2)*(x^3 - 1)^(1/4)*x^3 + sqrt(2)*(x^3 - 1)^(3/4)*x + (x^4 + sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^3
+ 2*sqrt(x^3 - 1)*x^2 + sqrt(2)*(x^3 - 1)^(3/4)*x - 1)*sqrt((x^4 - 2*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + x^3 + 4*sq
rt(x^3 - 1)*x^2 - 2*sqrt(2)*(x^3 - 1)^(3/4)*x - 1)/(x^4 + x^3 - 1)))/(x^4 - x^3 + 1)) + sqrt(2)*x*log((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((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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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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maple [C]  time = 8.28, size = 221, normalized size = 2.33

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

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)^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))+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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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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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^4 - x^3 + 1))/(x^2*(x^3 - 1)^(3/4)*(x^3 + x^4 - 1)),x)

[Out]

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

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sympy [F(-1)]  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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