3.20.81 \(\int \frac {(1+x^2) (1+x^8) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 (-1+x^2)} \, dx\)

Optimal. Leaf size=140 \[ \frac {\sqrt {x^8+x^6+x^4+x^2+1} \left (8 x^8+26 x^6+65 x^4+26 x^2+8\right )}{48 x^6}-\frac {65}{32} \log \left (-2 x^4-x^2+2 \sqrt {x^8+x^6+x^4+x^2+1}-2\right )+2 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x^2}{x^4-2 x^2-\sqrt {x^8+x^6+x^4+x^2+1}+1}\right )+\frac {65 \log (x)}{16} \]

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Rubi [F]  time = 1.14, 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^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^2)*(1 + x^8)*Sqrt[1 + x^2 + x^4 + x^6 + x^8])/(x^7*(-1 + x^2)),x]

[Out]

-Defer[Int][Sqrt[1 + x^2 + x^4 + x^6 + x^8]/x^7, x] - 2*Defer[Int][Sqrt[1 + x^2 + x^4 + x^6 + x^8]/x^5, x] - 2
*Defer[Int][Sqrt[1 + x^2 + x^4 + x^6 + x^8]/x^3, x] + Defer[Subst][Defer[Int][Sqrt[1 + x + x^2 + x^3 + x^4], x
], x, x^2]/2 + 2*Defer[Subst][Defer[Int][Sqrt[1 + x + x^2 + x^3 + x^4]/(-1 + x), x], x, x^2] - Defer[Subst][De
fer[Int][Sqrt[1 + x + x^2 + x^3 + x^4]/x, x], x, x^2]

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \left (1+x^8\right ) \sqrt {1+x^2+x^4+x^6+x^8}}{x^7 \left (-1+x^2\right )} \, dx &=\int \left (-\frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^7}-\frac {2 \sqrt {1+x^2+x^4+x^6+x^8}}{x^5}-\frac {2 \sqrt {1+x^2+x^4+x^6+x^8}}{x^3}-\frac {2 \sqrt {1+x^2+x^4+x^6+x^8}}{x}+x \sqrt {1+x^2+x^4+x^6+x^8}+\frac {4 x \sqrt {1+x^2+x^4+x^6+x^8}}{-1+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^5} \, dx\right )-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^3} \, dx-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x} \, dx+4 \int \frac {x \sqrt {1+x^2+x^4+x^6+x^8}}{-1+x^2} \, dx-\int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^7} \, dx+\int x \sqrt {1+x^2+x^4+x^6+x^8} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {1+x+x^2+x^3+x^4} \, dx,x,x^2\right )-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^5} \, dx-2 \int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^3} \, dx+2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x+x^2+x^3+x^4}}{-1+x} \, dx,x,x^2\right )-\int \frac {\sqrt {1+x^2+x^4+x^6+x^8}}{x^7} \, dx-\operatorname {Subst}\left (\int \frac {\sqrt {1+x+x^2+x^3+x^4}}{x} \, dx,x,x^2\right )\\ \end {align*}

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Mathematica [C]  time = 3.16, size = 687, normalized size = 4.91 \begin {gather*} \frac {\frac {15 (-1)^{3/5} \sqrt {-\frac {\sqrt [5]{-1} \left (\sqrt [5]{-1}-1\right ) \left ((-1)^{4/5}-x^2\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right )^2 \left ((-1)^{2/5}-x^2\right )^2}} \sqrt {\frac {x^2+(-1)^{4/5}+(-1)^{2/5}-\sqrt [5]{-1}+1}{\left (\sqrt [5]{-1}-1\right ) \left ((-1)^{2/5}-x^2\right )}} \left ((-1)^{2/5}-x^2\right )^2 \left (\left (45-32 \sqrt [5]{-1}+38 (-1)^{2/5}-32 (-1)^{3/5}+45 (-1)^{4/5}\right ) F\left (\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )+13 \left (1-\sqrt [5]{-1}+2 (-1)^{3/5}\right ) \Pi \left (\frac {-1+\sqrt [5]{-1}-(-1)^{2/5}}{-1+\sqrt [5]{-1}};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )+\sqrt [5]{-1} \left (13 \left (2-(-1)^{2/5}+(-1)^{3/5}\right ) \Pi \left (\frac {-1+\sqrt [5]{-1}-(-1)^{4/5}}{-1+\sqrt [5]{-1}};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )-64 \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \Pi \left (1-\sqrt [5]{-1}+(-1)^{4/5};\sin ^{-1}\left (\sqrt {-\frac {\sqrt [5]{-1} \left (-1+\sqrt [5]{-1}\right ) \left (x^2+\sqrt [5]{-1}\right )}{\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left ((-1)^{2/5}-x^2\right )}}\right )|\frac {1-\sqrt [5]{-1}+(-1)^{2/5}}{\left (-1+\sqrt [5]{-1}\right )^2}\right )\right )\right )}{\left ((-1)^{2/5}-1\right )^2}+\frac {\left (x^8+x^6+x^4+x^2+1\right ) \left (8 x^8+26 x^6+65 x^4+26 x^2+8\right )}{x^6}}{48 \sqrt {x^8+x^6+x^4+x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 + x^2)*(1 + x^8)*Sqrt[1 + x^2 + x^4 + x^6 + x^8])/(x^7*(-1 + x^2)),x]

[Out]

(((1 + x^2 + x^4 + x^6 + x^8)*(8 + 26*x^2 + 65*x^4 + 26*x^6 + 8*x^8))/x^6 + (15*(-1)^(3/5)*((-1)^(2/5) - x^2)^
2*Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(4/5) - x^2)*((-1)^(1/5) + x^2))/((1 - (-1)^(1/5) + (-1)^(2/5))^2
*((-1)^(2/5) - x^2)^2))]*Sqrt[(1 - (-1)^(1/5) + (-1)^(2/5) + (-1)^(4/5) + x^2)/((-1 + (-1)^(1/5))*((-1)^(2/5)
- x^2))]*((45 - 32*(-1)^(1/5) + 38*(-1)^(2/5) - 32*(-1)^(3/5) + 45*(-1)^(4/5))*EllipticF[ArcSin[Sqrt[-(((-1)^(
1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) + x^2))/((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) - x^2)))]], (1 - (-1)^(1
/5) + (-1)^(2/5))/(-1 + (-1)^(1/5))^2] + 13*(1 - (-1)^(1/5) + 2*(-1)^(3/5))*EllipticPi[(-1 + (-1)^(1/5) - (-1)
^(2/5))/(-1 + (-1)^(1/5)), ArcSin[Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) + x^2))/((1 - (-1)^(1/5) +
(-1)^(2/5))*((-1)^(2/5) - x^2)))]], (1 - (-1)^(1/5) + (-1)^(2/5))/(-1 + (-1)^(1/5))^2] + (-1)^(1/5)*(13*(2 - (
-1)^(2/5) + (-1)^(3/5))*EllipticPi[(-1 + (-1)^(1/5) - (-1)^(4/5))/(-1 + (-1)^(1/5)), ArcSin[Sqrt[-(((-1)^(1/5)
*(-1 + (-1)^(1/5))*((-1)^(1/5) + x^2))/((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) - x^2)))]], (1 - (-1)^(1/5)
+ (-1)^(2/5))/(-1 + (-1)^(1/5))^2] - 64*(1 - (-1)^(1/5) + (-1)^(2/5))*EllipticPi[1 - (-1)^(1/5) + (-1)^(4/5),
ArcSin[Sqrt[-(((-1)^(1/5)*(-1 + (-1)^(1/5))*((-1)^(1/5) + x^2))/((1 - (-1)^(1/5) + (-1)^(2/5))*((-1)^(2/5) - x
^2)))]], (1 - (-1)^(1/5) + (-1)^(2/5))/(-1 + (-1)^(1/5))^2])))/(-1 + (-1)^(2/5))^2)/(48*Sqrt[1 + x^2 + x^4 + x
^6 + x^8])

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IntegrateAlgebraic [A]  time = 0.49, size = 140, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^2+x^4+x^6+x^8} \left (8+26 x^2+65 x^4+26 x^6+8 x^8\right )}{48 x^6}+2 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x^2}{1-2 x^2+x^4-\sqrt {1+x^2+x^4+x^6+x^8}}\right )+\frac {65 \log (x)}{16}-\frac {65}{32} \log \left (-2-x^2-2 x^4+2 \sqrt {1+x^2+x^4+x^6+x^8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^2)*(1 + x^8)*Sqrt[1 + x^2 + x^4 + x^6 + x^8])/(x^7*(-1 + x^2)),x]

[Out]

(Sqrt[1 + x^2 + x^4 + x^6 + x^8]*(8 + 26*x^2 + 65*x^4 + 26*x^6 + 8*x^8))/(48*x^6) + 2*Sqrt[5]*ArcTanh[(Sqrt[5]
*x^2)/(1 - 2*x^2 + x^4 - Sqrt[1 + x^2 + x^4 + x^6 + x^8])] + (65*Log[x])/16 - (65*Log[-2 - x^2 - 2*x^4 + 2*Sqr
t[1 + x^2 + x^4 + x^6 + x^8]])/32

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fricas [A]  time = 0.61, size = 165, normalized size = 1.18 \begin {gather*} \frac {48 \, \sqrt {5} x^{6} \log \left (-\frac {9 \, x^{8} + 4 \, x^{6} + 14 \, x^{4} - 4 \, \sqrt {5} \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )} + 4 \, x^{2} + 9}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + 195 \, x^{6} \log \left (\frac {2 \, x^{4} + x^{2} + 2 \, \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} + 2}{x^{2}}\right ) + 2 \, {\left (8 \, x^{8} + 26 \, x^{6} + 65 \, x^{4} + 26 \, x^{2} + 8\right )} \sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1}}{96 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^8+1)*(x^8+x^6+x^4+x^2+1)^(1/2)/x^7/(x^2-1),x, algorithm="fricas")

[Out]

1/96*(48*sqrt(5)*x^6*log(-(9*x^8 + 4*x^6 + 14*x^4 - 4*sqrt(5)*sqrt(x^8 + x^6 + x^4 + x^2 + 1)*(x^4 + 1) + 4*x^
2 + 9)/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)) + 195*x^6*log((2*x^4 + x^2 + 2*sqrt(x^8 + x^6 + x^4 + x^2 + 1) + 2)/
x^2) + 2*(8*x^8 + 26*x^6 + 65*x^4 + 26*x^2 + 8)*sqrt(x^8 + x^6 + x^4 + x^2 + 1))/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{8} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^8+1)*(x^8+x^6+x^4+x^2+1)^(1/2)/x^7/(x^2-1),x, algorithm="giac")

[Out]

integrate(sqrt(x^8 + x^6 + x^4 + x^2 + 1)*(x^8 + 1)*(x^2 + 1)/((x^2 - 1)*x^7), x)

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maple [C]  time = 1.33, size = 137, normalized size = 0.98

method	result	size
trager	\(\frac {\sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}\, \left (8 x^{8}+26 x^{6}+65 x^{4}+26 x^{2}+8\right )}{48 x^{6}}+\RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}-5\right )-2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}\right )-\frac {65 \ln \left (\frac {-2-x^{2}-2 x^{4}+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{x^{2}}\right )}{32}\)	\(137\)
risch	\(\frac {65 x^{12}+91 x^{10}+99 x^{8}+99 x^{6}+99 x^{4}+34 x^{2}+8}{48 x^{6} \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}+\frac {\left (8 x^{2}+26\right ) \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{48}+\RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}-5\right )-2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}\right )-\frac {65 \ln \left (\frac {-2-x^{2}-2 x^{4}+2 \sqrt {x^{8}+x^{6}+x^{4}+x^{2}+1}}{x^{2}}\right )}{32}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)*(x^8+1)*(x^8+x^6+x^4+x^2+1)^(1/2)/x^7/(x^2-1),x,method=_RETURNVERBOSE)

[Out]

1/48*(x^8+x^6+x^4+x^2+1)^(1/2)*(8*x^8+26*x^6+65*x^4+26*x^2+8)/x^6+RootOf(_Z^2-5)*ln(-(RootOf(_Z^2-5)*x^4+RootO
f(_Z^2-5)-2*(x^8+x^6+x^4+x^2+1)^(1/2))/(-1+x)^2/(1+x)^2)-65/32*ln((-2-x^2-2*x^4+2*(x^8+x^6+x^4+x^2+1)^(1/2))/x
^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} + x^{6} + x^{4} + x^{2} + 1} {\left (x^{8} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(x^8+1)*(x^8+x^6+x^4+x^2+1)^(1/2)/x^7/(x^2-1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^8 + x^6 + x^4 + x^2 + 1)*(x^8 + 1)*(x^2 + 1)/((x^2 - 1)*x^7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2+1\right )\,\left (x^8+1\right )\,\sqrt {x^8+x^6+x^4+x^2+1}}{x^7\,\left (x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 + 1)*(x^8 + 1)*(x^2 + x^4 + x^6 + x^8 + 1)^(1/2))/(x^7*(x^2 - 1)),x)

[Out]

int(((x^2 + 1)*(x^8 + 1)*(x^2 + x^4 + x^6 + x^8 + 1)^(1/2))/(x^7*(x^2 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x^{4} - x^{3} + x^{2} - x + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (x^{2} + 1\right ) \left (x^{8} + 1\right )}{x^{7} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)*(x**8+1)*(x**8+x**6+x**4+x**2+1)**(1/2)/x**7/(x**2-1),x)

[Out]

Integral(sqrt((x**4 - x**3 + x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1))*(x**2 + 1)*(x**8 + 1)/(x**7*(x - 1)*(
x + 1)), x)

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