3.6.39 \(\int \frac {(-1+x^3) \sqrt {-1+x^6}}{x^7 (1+x^3)} \, dx\)

Optimal. Leaf size=42 \[ \frac {\left (1-4 x^3\right ) \sqrt {x^6-1}}{6 x^6}-\tan ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right ) \]

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Rubi [F]  time = 0.68, 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 (-1+x^3\right ) \sqrt {-1+x^6}}{x^7 \left (1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^3)*Sqrt[-1 + x^6])/(x^7*(1 + x^3)),x]

[Out]

(-2*Sqrt[-1 + x^6])/3 + Sqrt[-1 + x^6]/(6*x^6) - (2*Sqrt[-1 + x^6])/(3*x^3) + ArcTan[Sqrt[-1 + x^6]]/2 + (2*Ar
cTanh[x^3/Sqrt[-1 + x^6]])/3 + (2*Defer[Int][Sqrt[-1 + x^6]/(1 + x), x])/3 + (4*Defer[Int][Sqrt[-1 + x^6]/(-1
- I*Sqrt[3] + 2*x), x])/3 + (4*Defer[Int][Sqrt[-1 + x^6]/(-1 + I*Sqrt[3] + 2*x), x])/3

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt {-1+x^6}}{x^7 \left (1+x^3\right )} \, dx &=\int \left (-\frac {\sqrt {-1+x^6}}{x^7}+\frac {2 \sqrt {-1+x^6}}{x^4}-\frac {2 \sqrt {-1+x^6}}{x}+\frac {2 \sqrt {-1+x^6}}{3 (1+x)}+\frac {2 (-1+2 x) \sqrt {-1+x^6}}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \int \frac {(-1+2 x) \sqrt {-1+x^6}}{1-x+x^2} \, dx+2 \int \frac {\sqrt {-1+x^6}}{x^4} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x} \, dx-\int \frac {\sqrt {-1+x^6}}{x^7} \, dx\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \int \left (\frac {2 \sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x}+\frac {2 \sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {2 \sqrt {-1+x^6}}{3 x^3}+\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^6}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1-i \sqrt {3}+2 x} \, dx+\frac {4}{3} \int \frac {\sqrt {-1+x^6}}{-1+i \sqrt {3}+2 x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 37, normalized size = 0.88 \begin {gather*} \frac {1}{6} \left (3 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (1-4 x^3\right )}{x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^3)*Sqrt[-1 + x^6])/(x^7*(1 + x^3)),x]

[Out]

(((1 - 4*x^3)*Sqrt[-1 + x^6])/x^6 + 3*ArcTan[Sqrt[-1 + x^6]])/6

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IntegrateAlgebraic [A]  time = 0.18, size = 44, normalized size = 1.05 \begin {gather*} \frac {\left (1-4 x^3\right ) \sqrt {-1+x^6}}{6 x^6}-\tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{-1+x^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)*Sqrt[-1 + x^6])/(x^7*(1 + x^3)),x]

[Out]

((1 - 4*x^3)*Sqrt[-1 + x^6])/(6*x^6) - ArcTan[Sqrt[-1 + x^6]/(-1 + x^3)]

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fricas [A]  time = 0.48, size = 46, normalized size = 1.10 \begin {gather*} \frac {6 \, x^{6} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 4 \, x^{6} - \sqrt {x^{6} - 1} {\left (4 \, x^{3} - 1\right )}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^7/(x^3+1),x, algorithm="fricas")

[Out]

1/6*(6*x^6*arctan(-x^3 + sqrt(x^6 - 1)) - 4*x^6 - sqrt(x^6 - 1)*(4*x^3 - 1))/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^7/(x^3+1),x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - 1)*(x^3 - 1)/((x^3 + 1)*x^7), x)

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maple [A]  time = 0.35, size = 37, normalized size = 0.88

method	result	size
risch	\(-\frac {4 x^{9}-x^{6}-4 x^{3}+1}{6 x^{6} \sqrt {x^{6}-1}}-\frac {\arcsin \left (\frac {1}{x^{3}}\right )}{2}\)	\(37\)
trager	\(-\frac {\left (4 x^{3}-1\right ) \sqrt {x^{6}-1}}{6 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{2}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)*(x^6-1)^(1/2)/x^7/(x^3+1),x,method=_RETURNVERBOSE)

[Out]

-1/6*(4*x^9-x^6-4*x^3+1)/x^6/(x^6-1)^(1/2)-1/2*arcsin(1/x^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)*(x^6-1)^(1/2)/x^7/(x^3+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - 1)*(x^3 - 1)/((x^3 + 1)*x^7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^3-1\right )\,\sqrt {x^6-1}}{x^7\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/(x^7*(x^3 + 1)),x)

[Out]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/(x^7*(x^3 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{7} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)*(x**6-1)**(1/2)/x**7/(x**3+1),x)

[Out]

Integral(sqrt((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1))*(x - 1)*(x**2 + x + 1)/(x**7*(x + 1)*(x**2 - x +
1)), x)

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