3.7.57 \(\int \frac {(1+x^3) \sqrt {-1+x^6}}{x^{13} (-1+x^3)} \, dx\)

Optimal. Leaf size=52 \[ \frac {7}{12} \tan ^{-1}\left (\sqrt {x^6-1}+x^3\right )+\frac {\sqrt {x^6-1} \left (32 x^9+21 x^6+16 x^3+6\right )}{72 x^{12}} \]

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Rubi [F]  time = 0.62, 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^{13} \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^13*(-1 + x^3)),x]

[Out]

(-2*Sqrt[-1 + x^6])/3 + Sqrt[-1 + x^6]/(12*x^12) + (7*Sqrt[-1 + x^6])/(24*x^6) + (2*Sqrt[-1 + x^6])/(3*x^3) -
(2*(-1 + x^6)^(3/2))/(9*x^9) + (7*ArcTan[Sqrt[-1 + x^6]])/24 - (2*ArcTanh[x^3/Sqrt[-1 + x^6]])/3 + (2*Defer[In
t][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^{13} \left (-1+x^3\right )} \, dx &=\int \left (\frac {2 \sqrt {-1+x^6}}{3 (-1+x)}-\frac {\sqrt {-1+x^6}}{x^{13}}-\frac {2 \sqrt {-1+x^6}}{x^{10}}-\frac {2 \sqrt {-1+x^6}}{x^7}-\frac {2 \sqrt {-1+x^6}}{x^4}-\frac {2 \sqrt {-1+x^6}}{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^{10}} \, dx-2 \int \frac {\sqrt {-1+x^6}}{x^7} \, 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^{13}} \, dx\\ &=-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^3} \, dx,x,x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\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}}{12 x^{12}}+\frac {\sqrt {-1+x^6}}{3 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right )-\frac {1}{6} \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}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}-\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{3} \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}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {1}{3} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {2}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\frac {1}{24} \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 {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}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {2 \sqrt {-1+x^6}}{3 x^3}-\frac {2 \left (-1+x^6\right )^{3/2}}{9 x^9}+\frac {7}{24} \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 = 0.14, size = 47, normalized size = 0.90 \begin {gather*} \frac {1}{72} \left (21 \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (32 x^9+21 x^6+16 x^3+6\right )}{x^{12}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)*Sqrt[-1 + x^6])/(x^13*(-1 + x^3)),x]

[Out]

((Sqrt[-1 + x^6]*(6 + 16*x^3 + 21*x^6 + 32*x^9))/x^12 + 21*ArcTan[Sqrt[-1 + x^6]])/72

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IntegrateAlgebraic [A]  time = 0.21, size = 54, normalized size = 1.04 \begin {gather*} \frac {\sqrt {-1+x^6} \left (6+16 x^3+21 x^6+32 x^9\right )}{72 x^{12}}-\frac {7}{12} \tan ^{-1}\left (x^3-\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)*Sqrt[-1 + x^6])/(x^13*(-1 + x^3)),x]

[Out]

(Sqrt[-1 + x^6]*(6 + 16*x^3 + 21*x^6 + 32*x^9))/(72*x^12) - (7*ArcTan[x^3 - Sqrt[-1 + x^6]])/12

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fricas [A]  time = 0.46, size = 55, normalized size = 1.06 \begin {gather*} \frac {42 \, x^{12} \arctan \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 32 \, x^{12} + {\left (32 \, x^{9} + 21 \, x^{6} + 16 \, x^{3} + 6\right )} \sqrt {x^{6} - 1}}{72 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^6-1)^(1/2)/x^13/(x^3-1),x, algorithm="fricas")

[Out]

1/72*(42*x^12*arctan(-x^3 + sqrt(x^6 - 1)) + 32*x^12 + (32*x^9 + 21*x^6 + 16*x^3 + 6)*sqrt(x^6 - 1))/x^12

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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^{13}}\,{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^13/(x^3-1),x, algorithm="giac")

[Out]

integrate(sqrt(x^6 - 1)*(x^3 + 1)/((x^3 - 1)*x^13), x)

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maple [A]  time = 0.39, size = 47, normalized size = 0.90

method	result	size
risch	\(\frac {32 x^{15}+21 x^{12}-16 x^{9}-15 x^{6}-16 x^{3}-6}{72 x^{12} \sqrt {x^{6}-1}}-\frac {7 \arcsin \left (\frac {1}{x^{3}}\right )}{24}\)	\(47\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (32 x^{9}+21 x^{6}+16 x^{3}+6\right )}{72 x^{12}}+\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)*(x^6-1)^(1/2)/x^13/(x^3-1),x,method=_RETURNVERBOSE)

[Out]

1/72*(32*x^15+21*x^12-16*x^9-15*x^6-16*x^3-6)/x^12/(x^6-1)^(1/2)-7/24*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^{13}}\,{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^13/(x^3-1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 - 1)*(x^3 + 1)/((x^3 - 1)*x^13), 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^{13}\,\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^13*(x^3 - 1)),x)

[Out]

int(((x^3 + 1)*(x^6 - 1)^(1/2))/(x^13*(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^{13} \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**13/(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**13*(x - 1)*(x**2 + x +
1)), x)

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