3.7.37 \(\int \frac {(-1+x^3) \sqrt {-1+x^6}}{x (1+x^3)} \, dx\) [637]

Optimal. Leaf size=50 \[ \frac {1}{3} \sqrt {-1+x^6}+\frac {2}{3} \text {ArcTan}\left (x^3+\sqrt {-1+x^6}\right )-\frac {2}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

[Out]

1/3*(x^6-1)^(1/2)+2/3*arctan(x^3+(x^6-1)^(1/2))-2/3*ln(x^3+(x^6-1)^(1/2))

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Rubi [F]
time = 0.36, 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 \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*(1 + x^3)),x]

[Out]

-1/3*Sqrt[-1 + x^6] + ArcTan[Sqrt[-1 + x^6]]/3 + (2*Defer[Int][Sqrt[-1 + x^6]/(1 + x), x])/3 + (4*Defer[Int][S
qrt[-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 \left (1+x^3\right )} \, dx &=\int \left (-\frac {\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-\int \frac {\sqrt {-1+x^6}}{x} \, dx\\ &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )\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 {1}{3} \sqrt {-1+x^6}+\frac {1}{6} \text {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 {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 {1}{3} \sqrt {-1+x^6}+\frac {1}{3} \text {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 {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 {1}{3} \sqrt {-1+x^6}+\frac {1}{3} \tan ^{-1}\left (\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 = 0.09, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{3} \left (\sqrt {-1+x^6}-2 \text {ArcTan}\left (x^3-\sqrt {-1+x^6}\right )-2 \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^6] - 2*ArcTan[x^3 - Sqrt[-1 + x^6]] - 2*ArcTanh[x^3/Sqrt[-1 + x^6]])/3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.74, size = 54, normalized size = 1.08

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}}{3}-\frac {2 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^6-1)^(1/2)-2/3*ln(x^3+(x^6-1)^(1/2))-1/3*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)+(x^6-1)^(1/2))/x^3)

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

[Out]

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

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Fricas [A]
time = 0.35, size = 42, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, \sqrt {x^{6} - 1} + \frac {2}{3} \, \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + \frac {2}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(x^6 - 1) + 2/3*arctan(-x^3 + sqrt(x^6 - 1)) + 2/3*log(-x^3 + sqrt(x^6 - 1))

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

[Out]

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

[Out]

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

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