3.14.32 \(\int \frac {(1+x^6) (-1+2 x^6) (-1+x^4+2 x^6)^{5/4}}{x^{10} (-1-x^4+2 x^6)} \, dx\)

Optimal. Leaf size=106 \[ \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^6+x^4-1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^6+x^4-1}}\right )+\frac {\sqrt [4]{2 x^6+x^4-1} \left (20 x^{12}+38 x^{10}+104 x^8-20 x^6-19 x^4+5\right )}{45 x^9} \]

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Rubi [F]  time = 2.88, 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^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)),x]

[Out]

((3 + I*Sqrt[7])*Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(Sqrt[-1 - I*Sqrt[7]] - 2*x), x])/(2*Sqrt[-1 - I*Sqrt[7]]
) + ((3 - I*Sqrt[7])*Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(Sqrt[-1 + I*Sqrt[7]] - 2*x), x])/(2*Sqrt[-1 + I*Sqrt
[7]]) + Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(-1 + x), x]/4 + Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/x^10, x] - De
fer[Int][(-1 + x^4 + 2*x^6)^(5/4)/x^6, x] + Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/x^4, x] + Defer[Int][(-1 + x^4
+ 2*x^6)^(5/4)/x^2, x] - Defer[Int][(-1 + x^4 + 2*x^6)^(5/4)/(1 + x), x]/4 + ((3 + I*Sqrt[7])*Defer[Int][(-1
+ x^4 + 2*x^6)^(5/4)/(Sqrt[-1 - I*Sqrt[7]] + 2*x), x])/(2*Sqrt[-1 - I*Sqrt[7]]) + ((3 - I*Sqrt[7])*Defer[Int][
(-1 + x^4 + 2*x^6)^(5/4)/(Sqrt[-1 + I*Sqrt[7]] + 2*x), x])/(2*Sqrt[-1 + I*Sqrt[7]])

Rubi steps

\begin {align*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2}+\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (-1+x^2\right )}+\frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x^2} \, dx+\frac {1}{2} \int \frac {\left (-5-6 x^2\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+x^2+2 x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (-1+x)}-\frac {\left (-1+x^4+2 x^6\right )^{5/4}}{2 (1+x)}\right ) \, dx+\frac {1}{2} \int \left (\frac {\left (-6+2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2}+\frac {\left (-6-2 i \sqrt {7}\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+i \sqrt {7}+4 x^2} \, dx+\left (-3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1-i \sqrt {7}+4 x^2} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\left (-3-i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1-i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1+i \sqrt {7}\right ) \left (\sqrt {-1-i \sqrt {7}}+2 x\right )}\right ) \, dx+\left (-3+i \sqrt {7}\right ) \int \left (\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}-2 x\right )}+\frac {\sqrt {-1+i \sqrt {7}} \left (-1+x^4+2 x^6\right )^{5/4}}{2 \left (1-i \sqrt {7}\right ) \left (\sqrt {-1+i \sqrt {7}}+2 x\right )}\right ) \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{-1+x} \, dx-\frac {1}{4} \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{1+x} \, dx+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3-i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1+i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1+i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}-2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\frac {\left (3+i \sqrt {7}\right ) \int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{\sqrt {-1-i \sqrt {7}}+2 x} \, dx}{2 \sqrt {-1-i \sqrt {7}}}+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^{10}} \, dx-\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^6} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^4} \, dx+\int \frac {\left (-1+x^4+2 x^6\right )^{5/4}}{x^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^6\right ) \left (-1+2 x^6\right ) \left (-1+x^4+2 x^6\right )^{5/4}}{x^{10} \left (-1-x^4+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)),x]

[Out]

Integrate[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)), x]

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IntegrateAlgebraic [A]  time = 2.76, size = 106, normalized size = 1.00 \begin {gather*} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^6+x^4-1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^6+x^4-1}}\right )+\frac {\sqrt [4]{2 x^6+x^4-1} \left (20 x^{12}+38 x^{10}+104 x^8-20 x^6-19 x^4+5\right )}{45 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^6)*(-1 + 2*x^6)*(-1 + x^4 + 2*x^6)^(5/4))/(x^10*(-1 - x^4 + 2*x^6)),x]

[Out]

((-1 + x^4 + 2*x^6)^(1/4)*(5 - 19*x^4 - 20*x^6 + 104*x^8 + 38*x^10 + 20*x^12))/(45*x^9) + 2^(1/4)*ArcTan[(2^(1
/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)] - 2^(1/4)*ArcTanh[(2^(1/4)*x)/(-1 + x^4 + 2*x^6)^(1/4)]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {5}{4}} {\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (2 \, x^{6} - x^{4} - 1\right )} x^{10}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x, algorithm="giac")

[Out]

integrate((2*x^6 + x^4 - 1)^(5/4)*(2*x^6 - 1)*(x^6 + 1)/((2*x^6 - x^4 - 1)*x^10), x)

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maple [C]  time = 3.45, size = 1584, normalized size = 14.94

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x)

[Out]

1/45*(40*x^18+96*x^16+246*x^14+44*x^12-96*x^10-123*x^8+30*x^6+24*x^4-5)/x^9/(2*x^6+x^4-1)^(3/4)+(-1/2*RootOf(_
Z^4-2)*ln(-(8*RootOf(_Z^4-2)^2*x^18+20*RootOf(_Z^4-2)^2*x^16+8*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*
x^6+3*x^4-1)^(1/4)*RootOf(_Z^4-2)^3*x^13+14*RootOf(_Z^4-2)^2*x^14+8*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x
^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^4-2)^3*x^11-9*x^12*RootOf(_Z^4-2)^2+2*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10
-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^4-2)^3*x^9-20*x^10*RootOf(_Z^4-2)^2-8*(8*x^18+12*x^16+6*x^14-11*x^12-12*
x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^4-2)^3*x^7+8*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^
4-1)^(1/2)*x^8-7*x^8*RootOf(_Z^4-2)^2-4*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*Root
Of(_Z^4-2)^3*x^5+4*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/2)*x^6+6*RootOf(_Z^4-2)^2*x^
6+4*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(3/4)*RootOf(_Z^4-2)*x^3+5*RootOf(_Z^4-2)^2*x^
4+2*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^4-2)^3*x-4*(8*x^18+12*x^16+6*x
^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/2)*x^2-RootOf(_Z^4-2)^2)/(2*x^6+x^4-1)^2/(-1+x)/(1+x)/(2*x^4+x^2+1
))+1/2*RootOf(_Z^2+RootOf(_Z^4-2)^2)*ln((8*RootOf(_Z^4-2)^2*x^18+20*RootOf(_Z^4-2)^2*x^16-8*(8*x^18+12*x^16+6*
x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^13+14*RootOf(
_Z^4-2)^2*x^14-8*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-2)^
2)*RootOf(_Z^4-2)^2*x^11-9*x^12*RootOf(_Z^4-2)^2-2*(8*x^18+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)
^(1/4)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^9-20*x^10*RootOf(_Z^4-2)^2+8*(8*x^18+12*x^16+6*x^14-11
*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^7-8*(8*x^18+12*x^16+
6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/2)*x^8-7*x^8*RootOf(_Z^4-2)^2+4*(8*x^18+12*x^16+6*x^14-11*x^12-
12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x^5-4*(8*x^18+12*x^16+6*x^14
-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/2)*x^6+6*RootOf(_Z^4-2)^2*x^6+4*RootOf(_Z^2+RootOf(_Z^4-2)^2)*(8*x^18
+12*x^16+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(3/4)*x^3+5*RootOf(_Z^4-2)^2*x^4-2*(8*x^18+12*x^16+6*x^14
-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/4)*RootOf(_Z^2+RootOf(_Z^4-2)^2)*RootOf(_Z^4-2)^2*x+4*(8*x^18+12*x^16
+6*x^14-11*x^12-12*x^10-3*x^8+6*x^6+3*x^4-1)^(1/2)*x^2-RootOf(_Z^4-2)^2)/(2*x^6+x^4-1)^2/(-1+x)/(1+x)/(2*x^4+x
^2+1)))/(2*x^6+x^4-1)^(3/4)*((2*x^6+x^4-1)^3)^(1/4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{4} - 1\right )}^{\frac {5}{4}} {\left (2 \, x^{6} - 1\right )} {\left (x^{6} + 1\right )}}{{\left (2 \, x^{6} - x^{4} - 1\right )} x^{10}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)*(2*x^6-1)*(2*x^6+x^4-1)^(5/4)/x^10/(2*x^6-x^4-1),x, algorithm="maxima")

[Out]

integrate((2*x^6 + x^4 - 1)^(5/4)*(2*x^6 - 1)*(x^6 + 1)/((2*x^6 - x^4 - 1)*x^10), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^6+1\right )\,\left (2\,x^6-1\right )\,{\left (2\,x^6+x^4-1\right )}^{5/4}}{x^{10}\,\left (-2\,x^6+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^6 + 1)*(2*x^6 - 1)*(x^4 + 2*x^6 - 1)^(5/4))/(x^10*(x^4 - 2*x^6 + 1)),x)

[Out]

int(-((x^6 + 1)*(2*x^6 - 1)*(x^4 + 2*x^6 - 1)^(5/4))/(x^10*(x^4 - 2*x^6 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)*(2*x**6-1)*(2*x**6+x**4-1)**(5/4)/x**10/(2*x**6-x**4-1),x)

[Out]

Timed out

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