3.27.98 \(\int \frac {x^6 (-4+x^3)}{(-1+x^3)^{3/4} (1-2 x^3+x^6+x^8)} \, dx\)

Optimal. Leaf size=245 \[ \frac {1}{2} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.42, 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 {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^6*(-4 + x^3))/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)),x]

[Out]

(x^2*(1 - x^3)^(3/4)*Hypergeometric2F1[2/3, 3/4, 5/3, x^3])/(2*(-1 + x^3)^(3/4)) - Defer[Int][x/((-1 + x^3)^(3
/4)*(1 - 2*x^3 + x^6 + x^8)), x] + 2*Defer[Int][x^4/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)), x] - 4*Defer[I
nt][x^6/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)), x] - Defer[Int][x^7/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x
^8)), x]

Rubi steps

\begin {align*} \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx &=\int \left (\frac {x}{\left (-1+x^3\right )^{3/4}}+\frac {x \left (-1+2 x^3-4 x^5-x^6\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}\right ) \, dx\\ &=\int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x \left (-1+2 x^3-4 x^5-x^6\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\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}}+\int \left (-\frac {x}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}+\frac {2 x^4}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}-\frac {4 x^6}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}-\frac {x^7}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )}\right ) \, dx\\ &=\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 {x^4}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-4 \int \frac {x^6}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-\int \frac {x}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx-\int \frac {x^7}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-2 x^3+x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^6*(-4 + x^3))/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)),x]

[Out]

Integrate[(x^6*(-4 + x^3))/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.00, size = 225, normalized size = 0.92 \begin {gather*} \frac {1}{2} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^6*(-4 + x^3))/((-1 + x^3)^(3/4)*(1 - 2*x^3 + x^6 + x^8)),x]

[Out]

(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])])/2 + (Sqrt[2 + Sqrt[
2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^3)^(1/4))/(-x^2 + Sqrt[-1 + x^3])])/2 - (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sq
rt[2 - Sqrt[2]]*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])])/2 - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]
*x*(-1 + x^3)^(1/4))/(x^2 + Sqrt[-1 + x^3])])/2

________________________________________________________________________________________

fricas [B]  time = 0.69, size = 1425, normalized size = 5.82

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3-4)/(x^3-1)^(3/4)/(x^8+x^6-2*x^3+1),x, algorithm="fricas")

[Out]

1/4*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*arctan(-(x*sqrt(sqrt(2) + 2) - x*sqrt(-sqrt(2) +
2) - 2*x*sqrt((2*x^2 + (x^3 - 1)^(1/4)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x
^3 - 1))/x^2) + 2*sqrt(2)*(x^3 - 1)^(1/4))/(x*sqrt(sqrt(2) + 2) + x*sqrt(-sqrt(2) + 2))) + 1/4*(sqrt(2)*sqrt(s
qrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*arctan((x*sqrt(sqrt(2) + 2) - x*sqrt(-sqrt(2) + 2) + 2*x*sqrt((2*x^2
- (x^3 - 1)^(1/4)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^3 - 1))/x^2) - 2*sq
rt(2)*(x^3 - 1)^(1/4))/(x*sqrt(sqrt(2) + 2) + x*sqrt(-sqrt(2) + 2))) - 1/4*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2
)*sqrt(-sqrt(2) + 2))*arctan((x*sqrt(sqrt(2) + 2) + x*sqrt(-sqrt(2) + 2) - 2*x*sqrt((2*x^2 + (x^3 - 1)^(1/4)*(
sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^3 - 1))/x^2) + 2*sqrt(2)*(x^3 - 1)^(1/4
))/(x*sqrt(sqrt(2) + 2) - x*sqrt(-sqrt(2) + 2))) - 1/4*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2)
)*arctan(-(x*sqrt(sqrt(2) + 2) + x*sqrt(-sqrt(2) + 2) + 2*x*sqrt((2*x^2 - (x^3 - 1)^(1/4)*(sqrt(2)*x*sqrt(sqrt
(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^3 - 1))/x^2) - 2*sqrt(2)*(x^3 - 1)^(1/4))/(x*sqrt(sqrt(2)
+ 2) - x*sqrt(-sqrt(2) + 2))) - 1/16*(sqrt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*log(2*(2*x^2 + (
x^3 - 1)^(1/4)*(sqrt(2)*x*sqrt(sqrt(2) + 2) + sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^3 - 1))/x^2) + 1/16*(sq
rt(2)*sqrt(sqrt(2) + 2) + sqrt(2)*sqrt(-sqrt(2) + 2))*log(2*(2*x^2 - (x^3 - 1)^(1/4)*(sqrt(2)*x*sqrt(sqrt(2) +
2) + sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^3 - 1))/x^2) - 1/16*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-
sqrt(2) + 2))*log(2*(2*x^2 + (x^3 - 1)^(1/4)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*
sqrt(x^3 - 1))/x^2) + 1/16*(sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2)*sqrt(-sqrt(2) + 2))*log(2*(2*x^2 - (x^3 - 1)^(
1/4)*(sqrt(2)*x*sqrt(sqrt(2) + 2) - sqrt(2)*x*sqrt(-sqrt(2) + 2)) + 2*sqrt(x^3 - 1))/x^2) + 1/2*sqrt(sqrt(2) +
2)*arctan(-(x*sqrt(-sqrt(2) + 2) - 2*x*sqrt((x^2 + (x^3 - 1)^(1/4)*x*sqrt(-sqrt(2) + 2) + sqrt(x^3 - 1))/x^2)
+ 2*(x^3 - 1)^(1/4))/(x*sqrt(sqrt(2) + 2))) + 1/2*sqrt(sqrt(2) + 2)*arctan((x*sqrt(-sqrt(2) + 2) + 2*x*sqrt((
x^2 - (x^3 - 1)^(1/4)*x*sqrt(-sqrt(2) + 2) + sqrt(x^3 - 1))/x^2) - 2*(x^3 - 1)^(1/4))/(x*sqrt(sqrt(2) + 2))) +
1/2*sqrt(-sqrt(2) + 2)*arctan(-(x*sqrt(sqrt(2) + 2) - 2*x*sqrt((x^2 + (x^3 - 1)^(1/4)*x*sqrt(sqrt(2) + 2) + s
qrt(x^3 - 1))/x^2) + 2*(x^3 - 1)^(1/4))/(x*sqrt(-sqrt(2) + 2))) + 1/2*sqrt(-sqrt(2) + 2)*arctan((x*sqrt(sqrt(2
) + 2) + 2*x*sqrt((x^2 - (x^3 - 1)^(1/4)*x*sqrt(sqrt(2) + 2) + sqrt(x^3 - 1))/x^2) - 2*(x^3 - 1)^(1/4))/(x*sqr
t(-sqrt(2) + 2))) - 1/8*sqrt(sqrt(2) + 2)*log((x^2 + (x^3 - 1)^(1/4)*x*sqrt(sqrt(2) + 2) + sqrt(x^3 - 1))/x^2)
+ 1/8*sqrt(sqrt(2) + 2)*log((x^2 - (x^3 - 1)^(1/4)*x*sqrt(sqrt(2) + 2) + sqrt(x^3 - 1))/x^2) - 1/8*sqrt(-sqrt
(2) + 2)*log((x^2 + (x^3 - 1)^(1/4)*x*sqrt(-sqrt(2) + 2) + sqrt(x^3 - 1))/x^2) + 1/8*sqrt(-sqrt(2) + 2)*log((x
^2 - (x^3 - 1)^(1/4)*x*sqrt(-sqrt(2) + 2) + sqrt(x^3 - 1))/x^2)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 4\right )} x^{6}}{{\left (x^{8} + x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3-4)/(x^3-1)^(3/4)/(x^8+x^6-2*x^3+1),x, algorithm="giac")

[Out]

integrate((x^3 - 4)*x^6/((x^8 + x^6 - 2*x^3 + 1)*(x^3 - 1)^(3/4)), x)

________________________________________________________________________________________

maple [C]  time = 7.40, size = 468, normalized size = 1.91

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{8}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{3}-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (-\frac {-2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{8}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{3}-1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{4}+2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{8}+1\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{3}+1}\right )}{2}\)	\(468\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^3-4)/(x^3-1)^(3/4)/(x^8+x^6-2*x^3+1),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^8+1)*ln(-(RootOf(_Z^8+1)^11*x^4+RootOf(_Z^8+1)^7*x^3-RootOf(_Z^8+1)^7+2*(x^3-1)^(1/4)*RootOf(_Z^
8+1)^2*x^3-2*(x^3-1)^(1/2)*RootOf(_Z^8+1)*x^2+2*(x^3-1)^(3/4)*x)/(RootOf(_Z^8+1)^4*x^4-x^3+1))+1/2*RootOf(_Z^8
+1)^3*ln(-(-RootOf(_Z^8+1)^9*x^4+2*(x^3-1)^(1/4)*RootOf(_Z^8+1)^6*x^3+RootOf(_Z^8+1)^5*x^3-2*(x^3-1)^(1/2)*Roo
tOf(_Z^8+1)^3*x^2-RootOf(_Z^8+1)^5+2*(x^3-1)^(3/4)*x)/(RootOf(_Z^8+1)^4*x^4+x^3-1))+1/2*RootOf(_Z^8+1)^7*ln(-(
-2*(x^3-1)^(1/2)*RootOf(_Z^8+1)^7*x^2-2*(x^3-1)^(1/4)*RootOf(_Z^8+1)^6*x^3-RootOf(_Z^8+1)^5*x^4+2*(x^3-1)^(3/4
)*x+RootOf(_Z^8+1)*x^3-RootOf(_Z^8+1))/(RootOf(_Z^8+1)^4*x^4+x^3-1))-1/2*RootOf(_Z^8+1)^5*ln(-(-RootOf(_Z^8+1)
^7*x^4+2*(x^3-1)^(1/2)*RootOf(_Z^8+1)^5*x^2-2*(x^3-1)^(1/4)*RootOf(_Z^8+1)^2*x^3-RootOf(_Z^8+1)^3*x^3+2*(x^3-1
)^(3/4)*x+RootOf(_Z^8+1)^3)/(RootOf(_Z^8+1)^4*x^4-x^3+1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 4\right )} x^{6}}{{\left (x^{8} + x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^3-4)/(x^3-1)^(3/4)/(x^8+x^6-2*x^3+1),x, algorithm="maxima")

[Out]

integrate((x^3 - 4)*x^6/((x^8 + x^6 - 2*x^3 + 1)*(x^3 - 1)^(3/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{3/4}\,\left (x^8+x^6-2\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(x^3 - 4))/((x^3 - 1)^(3/4)*(x^6 - 2*x^3 + x^8 + 1)),x)

[Out]

int((x^6*(x^3 - 4))/((x^3 - 1)^(3/4)*(x^6 - 2*x^3 + x^8 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} \left (x^{3} - 4\right )}{\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{8} + x^{6} - 2 x^{3} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(x**3-4)/(x**3-1)**(3/4)/(x**8+x**6-2*x**3+1),x)

[Out]

Integral(x**6*(x**3 - 4)/(((x - 1)*(x**2 + x + 1))**(3/4)*(x**8 + x**6 - 2*x**3 + 1)), x)

________________________________________________________________________________________