3.14.53 \(\int \frac {-3+5 x^8}{(1+x^8) \sqrt [3]{1-x^3+x^8}} \, dx\) [1353]

Optimal. Leaf size=97 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^8-x^3+1)^(1/3)))-ln(x+(x^8-x^3+1)^(1/3))+1/2*ln(x^2-x*(x^8-x^3+1)^(1/3)+(x^
8-x^3+1)^(2/3))

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Rubi [F]
time = 1.21, 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 {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3 + 5*x^8)/((1 + x^8)*(1 - x^3 + x^8)^(1/3)),x]

[Out]

5*Defer[Int][(1 - x^3 + x^8)^(-1/3), x] - (-1)^(1/8)*Defer[Int][1/(((-1)^(1/8) - x)*(1 - x^3 + x^8)^(1/3)), x]
- (-1)^(3/8)*Defer[Int][1/(((-1)^(3/8) - x)*(1 - x^3 + x^8)^(1/3)), x] + (-1)^(5/8)*Defer[Int][1/((-(-1)^(5/8
) - x)*(1 - x^3 + x^8)^(1/3)), x] + (-1)^(7/8)*Defer[Int][1/((-(-1)^(7/8) - x)*(1 - x^3 + x^8)^(1/3)), x] - (-
1)^(1/8)*Defer[Int][1/(((-1)^(1/8) + x)*(1 - x^3 + x^8)^(1/3)), x] - (-1)^(3/8)*Defer[Int][1/(((-1)^(3/8) + x)
*(1 - x^3 + x^8)^(1/3)), x] + (-1)^(5/8)*Defer[Int][1/((-(-1)^(5/8) + x)*(1 - x^3 + x^8)^(1/3)), x] + (-1)^(7/
8)*Defer[Int][1/((-(-1)^(7/8) + x)*(1 - x^3 + x^8)^(1/3)), x]

Rubi steps

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

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Mathematica [A]
time = 1.48, size = 94, normalized size = 0.97 \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 5*x^8)/((1 + x^8)*(1 - x^3 + x^8)^(1/3)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3 + x^8)^(1/3))] - Log[x + (1 - x^3 + x^8)^(1/3)] + Log[x^2 - x*(1 -
x^3 + x^8)^(1/3) + (1 - x^3 + x^8)^(2/3)]/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.30, size = 307, normalized size = 3.16

method	result	size
trager	\(-\ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}-x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{x^{8}+1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}+x^{8}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{8}+1}\right )\)	\(307\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^8-3)/(x^8+1)/(x^8-x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-ln((RootOf(_Z^2-_Z+1)*x^8-x^8+RootOf(_Z^2-_Z+1)^2*x^3+(x^8-x^3+1)^(2/3)*RootOf(_Z^2-_Z+1)*x-RootOf(_Z^2-_Z+1)
*(x^8-x^3+1)^(1/3)*x^2-3*RootOf(_Z^2-_Z+1)*x^3-2*(x^8-x^3+1)^(2/3)*x-(x^8-x^3+1)^(1/3)*x^2+2*x^3+RootOf(_Z^2-_
Z+1)-1)/(x^8+1))+RootOf(_Z^2-_Z+1)*ln(-(-RootOf(_Z^2-_Z+1)*x^8+x^8+RootOf(_Z^2-_Z+1)^2*x^3+(x^8-x^3+1)^(2/3)*R
ootOf(_Z^2-_Z+1)*x+2*RootOf(_Z^2-_Z+1)*(x^8-x^3+1)^(1/3)*x^2+(x^8-x^3+1)^(2/3)*x-(x^8-x^3+1)^(1/3)*x^2-x^3-Roo
tOf(_Z^2-_Z+1)+1)/(x^8+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((5*x^8-3)/(x^8+1)/(x^8-x^3+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((5*x^8 - 3)/((x^8 - x^3 + 1)^(1/3)*(x^8 + 1)), x)

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Fricas [A]
time = 1.25, size = 121, normalized size = 1.25 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}}{x^{8} - 9 \, x^{3} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{8} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{8} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^8-3)/(x^8+1)/(x^8-x^3+1)^(1/3),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan((4*sqrt(3)*(x^8 - x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^8 - x^3 + 1)^(2/3)*x + sqrt(3)*(x^8 - x^3
+ 1))/(x^8 - 9*x^3 + 1)) - 1/2*log((x^8 + 3*(x^8 - x^3 + 1)^(1/3)*x^2 + 3*(x^8 - x^3 + 1)^(2/3)*x + 1)/(x^8 +
1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5 x^{8} - 3}{\left (x^{8} + 1\right ) \sqrt [3]{x^{8} - x^{3} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**8-3)/(x**8+1)/(x**8-x**3+1)**(1/3),x)

[Out]

Integral((5*x**8 - 3)/((x**8 + 1)*(x**8 - x**3 + 1)**(1/3)), x)

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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((5*x^8-3)/(x^8+1)/(x^8-x^3+1)^(1/3),x, algorithm="giac")

[Out]

integrate((5*x^8 - 3)/((x^8 - x^3 + 1)^(1/3)*(x^8 + 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {5\,x^8-3}{\left (x^8+1\right )\,{\left (x^8-x^3+1\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^8 - 3)/((x^8 + 1)*(x^8 - x^3 + 1)^(1/3)),x)

[Out]

int((5*x^8 - 3)/((x^8 + 1)*(x^8 - x^3 + 1)^(1/3)), x)

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