3.15.89 \(\int \frac {(4+x^6) \sqrt [4]{-2-x^4+x^6}}{x^2 (-2+x^6)} \, dx\)

Optimal. Leaf size=119 \[ \frac {2 \sqrt [4]{x^6-x^4-2}}{x}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{\sqrt {x^6-x^4-2}-x^2}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{x^2+\sqrt {x^6-x^4-2}}\right )}{\sqrt {2}} \]

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Rubi [F]  time = 1.75, 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 (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/2*Defer[Int][(-2 - x^4 + x^6)^(1/4)/(2^(1/6) - x), x]/2^(1/6) - 2*Defer[Int][(-2 - x^4 + x^6)^(1/4)/x^2, x]
- Defer[Int][(-2 - x^4 + x^6)^(1/4)/(2^(1/6) + x), x]/(2*2^(1/6)) - ((-1)^(2/3)*Defer[Int][(-2 - x^4 + x^6)^(
1/4)/(2^(1/6) - (-1)^(1/3)*x), x])/(2*2^(1/6)) - ((-1)^(2/3)*Defer[Int][(-2 - x^4 + x^6)^(1/4)/(2^(1/6) + (-1)
^(1/3)*x), x])/(2*2^(1/6)) + ((-1)^(1/3)*Defer[Int][(-2 - x^4 + x^6)^(1/4)/(2^(1/6) - (-1)^(2/3)*x), x])/(2*2^
(1/6)) + ((-1)^(1/3)*Defer[Int][(-2 - x^4 + x^6)^(1/4)/(2^(1/6) + (-1)^(2/3)*x), x])/(2*2^(1/6))

Rubi steps

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

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^6\right ) \sqrt [4]{-2-x^4+x^6}}{x^2 \left (-2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.03, size = 119, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{x^6-x^4-2}}{x}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{\sqrt {x^6-x^4-2}-x^2}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6-x^4-2}}{x^2+\sqrt {x^6-x^4-2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 - x^4 + x^6)^(1/4))/x + ArcTan[(Sqrt[2]*x*(-2 - x^4 + x^6)^(1/4))/(-x^2 + Sqrt[-2 - x^4 + x^6])]/Sqrt[2
] - ArcTanh[(Sqrt[2]*x*(-2 - x^4 + x^6)^(1/4))/(x^2 + Sqrt[-2 - x^4 + x^6])]/Sqrt[2]

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fricas [B]  time = 101.29, size = 780, normalized size = 6.55 \begin {gather*} -\frac {4 \, \sqrt {2} x \arctan \left (-\frac {x^{12} - 4 \, x^{6} + 2 \, \sqrt {2} {\left (x^{7} - 4 \, x^{5} - 2 \, x\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - 4 \, x^{7} - 6 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} - {\left (16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} - 4 \, x^{6} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} + \sqrt {2} {\left (x^{12} - 10 \, x^{10} + 8 \, x^{8} - 4 \, x^{6} + 20 \, x^{4} + 4\right )} + 4 \, {\left (x^{9} - 2 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2}{x^{6} - 2}} + 4}{x^{12} - 16 \, x^{10} + 16 \, x^{8} - 4 \, x^{6} + 32 \, x^{4} + 4}\right ) - 4 \, \sqrt {2} x \arctan \left (-\frac {x^{12} - 4 \, x^{6} - 2 \, \sqrt {2} {\left (x^{7} - 4 \, x^{5} - 2 \, x\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - 4 \, x^{7} - 6 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} - {\left (16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} - 4 \, x^{6} - 2 \, x^{2}\right )} \sqrt {x^{6} - x^{4} - 2} - \sqrt {2} {\left (x^{12} - 10 \, x^{10} + 8 \, x^{8} - 4 \, x^{6} + 20 \, x^{4} + 4\right )} + 4 \, {\left (x^{9} - 2 \, x^{3}\right )} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2}{x^{6} - 2}} + 4}{x^{12} - 16 \, x^{10} + 16 \, x^{8} - 4 \, x^{6} + 32 \, x^{4} + 4}\right ) + \sqrt {2} x \log \left (\frac {4 \, {\left (x^{6} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2\right )}}{x^{6} - 2}\right ) - \sqrt {2} x \log \left (\frac {4 \, {\left (x^{6} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - x^{4} - 2} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - x^{4} - 2\right )}^{\frac {3}{4}} x - 2\right )}}{x^{6} - 2}\right ) - 16 \, {\left (x^{6} - x^{4} - 2\right )}^{\frac {1}{4}}}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*sqrt(2)*x*arctan(-(x^12 - 4*x^6 + 2*sqrt(2)*(x^7 - 4*x^5 - 2*x)*(x^6 - x^4 - 2)^(3/4) + 2*sqrt(2)*(3*x
^9 - 4*x^7 - 6*x^3)*(x^6 - x^4 - 2)^(1/4) + 4*(x^8 - 2*x^2)*sqrt(x^6 - x^4 - 2) - (16*(x^6 - x^4 - 2)^(3/4)*x^
5 + 2*sqrt(2)*(x^8 - 4*x^6 - 2*x^2)*sqrt(x^6 - x^4 - 2) + sqrt(2)*(x^12 - 10*x^10 + 8*x^8 - 4*x^6 + 20*x^4 + 4
) + 4*(x^9 - 2*x^3)*(x^6 - x^4 - 2)^(1/4))*sqrt((x^6 + 2*sqrt(2)*(x^6 - x^4 - 2)^(1/4)*x^3 + 4*sqrt(x^6 - x^4
- 2)*x^2 + 2*sqrt(2)*(x^6 - x^4 - 2)^(3/4)*x - 2)/(x^6 - 2)) + 4)/(x^12 - 16*x^10 + 16*x^8 - 4*x^6 + 32*x^4 +
4)) - 4*sqrt(2)*x*arctan(-(x^12 - 4*x^6 - 2*sqrt(2)*(x^7 - 4*x^5 - 2*x)*(x^6 - x^4 - 2)^(3/4) - 2*sqrt(2)*(3*x
^9 - 4*x^7 - 6*x^3)*(x^6 - x^4 - 2)^(1/4) + 4*(x^8 - 2*x^2)*sqrt(x^6 - x^4 - 2) - (16*(x^6 - x^4 - 2)^(3/4)*x^
5 - 2*sqrt(2)*(x^8 - 4*x^6 - 2*x^2)*sqrt(x^6 - x^4 - 2) - sqrt(2)*(x^12 - 10*x^10 + 8*x^8 - 4*x^6 + 20*x^4 + 4
) + 4*(x^9 - 2*x^3)*(x^6 - x^4 - 2)^(1/4))*sqrt((x^6 - 2*sqrt(2)*(x^6 - x^4 - 2)^(1/4)*x^3 + 4*sqrt(x^6 - x^4
- 2)*x^2 - 2*sqrt(2)*(x^6 - x^4 - 2)^(3/4)*x - 2)/(x^6 - 2)) + 4)/(x^12 - 16*x^10 + 16*x^8 - 4*x^6 + 32*x^4 +
4)) + sqrt(2)*x*log(4*(x^6 + 2*sqrt(2)*(x^6 - x^4 - 2)^(1/4)*x^3 + 4*sqrt(x^6 - x^4 - 2)*x^2 + 2*sqrt(2)*(x^6
- x^4 - 2)^(3/4)*x - 2)/(x^6 - 2)) - sqrt(2)*x*log(4*(x^6 - 2*sqrt(2)*(x^6 - x^4 - 2)^(1/4)*x^3 + 4*sqrt(x^6 -
x^4 - 2)*x^2 - 2*sqrt(2)*(x^6 - x^4 - 2)^(3/4)*x - 2)/(x^6 - 2)) - 16*(x^6 - x^4 - 2)^(1/4))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.81, size = 1372, normalized size = 11.53

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x^6-x^4-2)^(1/4)/x+(-1/2*RootOf(_Z^4+1)^3*ln(-(RootOf(_Z^4+1)^2*x^18-4*RootOf(_Z^4+1)^2*x^16+5*x^14*RootOf(
_Z^4+1)^2-2*RootOf(_Z^4+1)*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^13-8*RootOf(_Z^4+
1)^2*x^12+4*RootOf(_Z^4+1)*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^11+16*x^10*RootOf
(_Z^4+1)^2-2*RootOf(_Z^4+1)*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^9+2*(x^18-3*x^16
+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/2)*x^8-10*x^8*RootOf(_Z^4+1)^2+2*RootOf(_Z^4+1)^3*(x^18-3*x^1
6+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(3/4)*x^3+8*RootOf(_Z^4+1)*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6
*x^8+12*x^6-12*x^4-8)^(1/4)*x^7-2*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/2)*x^6+12*RootO
f(_Z^4+1)^2*x^6-8*RootOf(_Z^4+1)*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^5-16*RootOf
(_Z^4+1)^2*x^4-4*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/2)*x^2-8*RootOf(_Z^4+1)*(x^18-3*
x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x-8*RootOf(_Z^4+1)^2)/(x^6-2)/(x^6-x^4-2)^2)-1/2*RootO
f(_Z^4+1)*ln(-(-RootOf(_Z^4+1)^2*x^18+4*RootOf(_Z^4+1)^2*x^16-2*RootOf(_Z^4+1)^3*(x^18-3*x^16+3*x^14-7*x^12+12
*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^13-5*x^14*RootOf(_Z^4+1)^2+4*RootOf(_Z^4+1)^3*(x^18-3*x^16+3*x^14-7*x^12+
12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^11+8*RootOf(_Z^4+1)^2*x^12-2*RootOf(_Z^4+1)^3*(x^18-3*x^16+3*x^14-7*x^1
2+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^9-16*x^10*RootOf(_Z^4+1)^2+8*RootOf(_Z^4+1)^3*(x^18-3*x^16+3*x^14-7*x
^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x^7+2*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/2)
*x^8+10*x^8*RootOf(_Z^4+1)^2-8*RootOf(_Z^4+1)^3*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4
)*x^5-2*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/2)*x^6-12*RootOf(_Z^4+1)^2*x^6+2*RootOf(_
Z^4+1)*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(3/4)*x^3+16*RootOf(_Z^4+1)^2*x^4-8*RootOf(_Z
^4+1)^3*(x^18-3*x^16+3*x^14-7*x^12+12*x^10-6*x^8+12*x^6-12*x^4-8)^(1/4)*x-4*(x^18-3*x^16+3*x^14-7*x^12+12*x^10
-6*x^8+12*x^6-12*x^4-8)^(1/2)*x^2+8*RootOf(_Z^4+1)^2)/(x^6-2)/(x^6-x^4-2)^2))/(x^6-x^4-2)^(3/4)*((x^6-x^4-2)^3
)^(1/4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**6 + 4)*(x**6 - x**4 - 2)**(1/4)/(x**2*(x**6 - 2)), x)

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