3.7.77 \(\int \frac {(-1+x^3) \sqrt {-1+x^6}}{x^4 (1+x^3)} \, dx\)

Optimal. Leaf size=53 \[ \frac {\sqrt {x^6-1}}{3 x^3}+\frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )-\frac {4}{3} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right ) \]

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Rubi [F]  time = 0.60, 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^4 \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^4*(1 + x^3)),x]

[Out]

(2*Sqrt[-1 + x^6])/3 + Sqrt[-1 + x^6]/(3*x^3) - (2*ArcTan[Sqrt[-1 + x^6]])/3 - ArcTanh[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^4 \left (1+x^3\right )} \, dx &=\int \left (-\frac {\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\\ &=-\left (\frac {2}{3} \int \frac {\sqrt {-1+x^6}}{1+x} \, dx\right )-\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} \, dx-\int \frac {\sqrt {-1+x^6}}{x^4} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{x^2} \, dx,x,x^3\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} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{3 x^3}-\frac {1}{3} \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^2}} \, dx,x,x^3\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 {2}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{3 x^3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\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,\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}}{3 x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {1}{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 = 0.14, size = 44, normalized size = 0.83 \begin {gather*} \frac {1}{3} \left (-2 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{x^3}+\log \left (\sqrt {x^6-1}+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.17, size = 57, normalized size = 1.08 \begin {gather*} \frac {\sqrt {-1+x^6}}{3 x^3}+\frac {4}{3} \tan ^{-1}\left (x^3-\sqrt {-1+x^6}\right )-\frac {1}{3} \log \left (-x^3+\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 57, normalized size = 1.08 \begin {gather*} -\frac {4 \, x^{3} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - x^{3} - \sqrt {x^{6} - 1}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{4}}\,{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^4/(x^3+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.54, size = 58, normalized size = 1.09

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{6}-1}}{x^{3}}\right )}{3}\)	\(58\)
risch	\(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }\right )}{3 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^6-1)^(1/2)/x^3+1/3*ln(x^3+(x^6-1)^(1/2))+2/3*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*} \int \frac {\sqrt {x^{6} - 1} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{4}}\,{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^4/(x^3+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - 1)*(x^3 - 1)/((x^3 + 1)*x^4), 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^4\,\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^4*(x^3 + 1)),x)

[Out]

int(((x^3 - 1)*(x^6 - 1)^(1/2))/(x^4*(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^{4} \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**4/(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**4*(x + 1)*(x**2 - x +
1)), x)

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