3.15.31 \(\int \frac {\sqrt {1+2 x^6} (-1+4 x^6)}{2+x^4+8 x^6+8 x^{12}} \, dx\)

Optimal. Leaf size=101 \[ -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt {2 x^6+1}}{2 \sqrt {2} x^6-x^2+\sqrt {2}}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {2 x^6+1}}{2 \sqrt {2} x^6+x^2+\sqrt {2}}\right )}{4 \sqrt [4]{2}} \]

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Rubi [F]  time = 0.57, 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 {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]

[Out]

Defer[Int][Sqrt[1 + 2*x^6]/(-2 - x^4 - 8*x^6 - 8*x^12), x] + 4*Defer[Int][(x^6*Sqrt[1 + 2*x^6])/(2 + x^4 + 8*x
^6 + 8*x^12), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx &=\int \left (\frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}}+\frac {4 x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}}\right ) \, dx\\ &=4 \int \frac {x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]

[Out]

Integrate[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12), x]

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IntegrateAlgebraic [A]  time = 5.03, size = 101, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[1 + 2*x^6]*(-1 + 4*x^6))/(2 + x^4 + 8*x^6 + 8*x^12),x]

[Out]

-1/4*ArcTan[(2^(3/4)*x*Sqrt[1 + 2*x^6])/(Sqrt[2] - x^2 + 2*Sqrt[2]*x^6)]/2^(1/4) - ArcTanh[(2^(3/4)*x*Sqrt[1 +
2*x^6])/(Sqrt[2] + x^2 + 2*Sqrt[2]*x^6)]/(4*2^(1/4))

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fricas [B]  time = 1.14, size = 1041, normalized size = 10.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x, algorithm="fricas")

[Out]

1/32*8^(3/4)*sqrt(2)*arctan(-1/2*(128*x^24 + 256*x^18 + 32*x^16 + 192*x^12 + 32*x^10 + 2*x^8 + 64*x^6 + 8*x^4
+ 8*sqrt(2)*(16*x^20 + 24*x^14 + 2*x^12 + 12*x^8 + x^6 + 2*x^2) + sqrt(2*x^6 + 1)*(8^(3/4)*sqrt(2)*(24*x^15 +
24*x^9 - x^7 + 6*x^3) + 4*8^(1/4)*sqrt(2)*(16*x^19 + 24*x^13 - 6*x^11 + 12*x^7 - 3*x^5 + 2*x)) - (8^(3/4)*sqrt
(2)*(16*x^20 + 24*x^14 - 6*x^12 + 12*x^8 - 3*x^6 + 2*x^2) + 8^(1/4)*sqrt(2)*(64*x^24 + 128*x^18 - 64*x^16 + 96
*x^12 - 64*x^10 - x^8 + 32*x^6 - 16*x^4 + 4) + 8*(8*x^15 + 8*x^9 + x^7 + 2*x^3 + 4*sqrt(2)*(2*x^11 + x^5))*sqr
t(2*x^6 + 1))*sqrt((16*x^8 + 8*x^2 + sqrt(2)*(8*x^12 + 8*x^6 + x^4 + 2) + sqrt(2*x^6 + 1)*(2*8^(1/4)*sqrt(2)*x
^3 + 8^(3/4)*sqrt(2)*(2*x^7 + x)))/(8*x^12 + 8*x^6 + x^4 + 2)) + 8)/(64*x^24 + 128*x^18 - 112*x^16 + 96*x^12 -
112*x^10 + x^8 + 32*x^6 - 28*x^4 + 4)) - 1/32*8^(3/4)*sqrt(2)*arctan(-1/2*(128*x^24 + 256*x^18 + 32*x^16 + 19
2*x^12 + 32*x^10 + 2*x^8 + 64*x^6 + 8*x^4 + 8*sqrt(2)*(16*x^20 + 24*x^14 + 2*x^12 + 12*x^8 + x^6 + 2*x^2) - sq
rt(2*x^6 + 1)*(8^(3/4)*sqrt(2)*(24*x^15 + 24*x^9 - x^7 + 6*x^3) + 4*8^(1/4)*sqrt(2)*(16*x^19 + 24*x^13 - 6*x^1
1 + 12*x^7 - 3*x^5 + 2*x)) + (8^(3/4)*sqrt(2)*(16*x^20 + 24*x^14 - 6*x^12 + 12*x^8 - 3*x^6 + 2*x^2) + 8^(1/4)*
sqrt(2)*(64*x^24 + 128*x^18 - 64*x^16 + 96*x^12 - 64*x^10 - x^8 + 32*x^6 - 16*x^4 + 4) - 8*(8*x^15 + 8*x^9 + x
^7 + 2*x^3 + 4*sqrt(2)*(2*x^11 + x^5))*sqrt(2*x^6 + 1))*sqrt((16*x^8 + 8*x^2 + sqrt(2)*(8*x^12 + 8*x^6 + x^4 +
2) - sqrt(2*x^6 + 1)*(2*8^(1/4)*sqrt(2)*x^3 + 8^(3/4)*sqrt(2)*(2*x^7 + x)))/(8*x^12 + 8*x^6 + x^4 + 2)) + 8)/
(64*x^24 + 128*x^18 - 112*x^16 + 96*x^12 - 112*x^10 + x^8 + 32*x^6 - 28*x^4 + 4)) - 1/128*8^(3/4)*sqrt(2)*log(
64*(16*x^8 + 8*x^2 + sqrt(2)*(8*x^12 + 8*x^6 + x^4 + 2) + sqrt(2*x^6 + 1)*(2*8^(1/4)*sqrt(2)*x^3 + 8^(3/4)*sqr
t(2)*(2*x^7 + x)))/(8*x^12 + 8*x^6 + x^4 + 2)) + 1/128*8^(3/4)*sqrt(2)*log(64*(16*x^8 + 8*x^2 + sqrt(2)*(8*x^1
2 + 8*x^6 + x^4 + 2) - sqrt(2*x^6 + 1)*(2*8^(1/4)*sqrt(2)*x^3 + 8^(3/4)*sqrt(2)*(2*x^7 + x)))/(8*x^12 + 8*x^6
+ x^4 + 2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{6} - 1\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x, algorithm="giac")

[Out]

integrate((4*x^6 - 1)*sqrt(2*x^6 + 1)/(8*x^12 + 8*x^6 + x^4 + 2), x)

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maple [C]  time = 2.42, size = 187, normalized size = 1.85

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-4 \sqrt {2 x^{6}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{-4 x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-2}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{2}+4 \sqrt {2 x^{6}+1}\, x -2 \RootOf \left (\textit {\_Z}^{4}+2\right )}{4 x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+2}\right )}{8}\)	\(187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x,method=_RETURNVERBOSE)

[Out]

1/8*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln(-(4*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^6+RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf
(_Z^4+2)^2)*x^2-4*(2*x^6+1)^(1/2)*x+2*RootOf(_Z^2+RootOf(_Z^4+2)^2))/(-4*x^6+RootOf(_Z^4+2)^2*x^2-2))+1/8*Root
Of(_Z^4+2)*ln((-4*RootOf(_Z^4+2)*x^6+RootOf(_Z^4+2)^3*x^2+4*(2*x^6+1)^(1/2)*x-2*RootOf(_Z^4+2))/(4*x^6+RootOf(
_Z^4+2)^2*x^2+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{6} - 1\right )} \sqrt {2 \, x^{6} + 1}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)^(1/2)*(4*x^6-1)/(8*x^12+8*x^6+x^4+2),x, algorithm="maxima")

[Out]

integrate((4*x^6 - 1)*sqrt(2*x^6 + 1)/(8*x^12 + 8*x^6 + x^4 + 2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {2\,x^6+1}\,\left (4\,x^6-1\right )}{8\,x^{12}+8\,x^6+x^4+2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^6 + 1)^(1/2)*(4*x^6 - 1))/(x^4 + 8*x^6 + 8*x^12 + 2),x)

[Out]

int(((2*x^6 + 1)^(1/2)*(4*x^6 - 1))/(x^4 + 8*x^6 + 8*x^12 + 2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right ) \sqrt {2 x^{6} + 1}}{8 x^{12} + 8 x^{6} + x^{4} + 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**6+1)**(1/2)*(4*x**6-1)/(8*x**12+8*x**6+x**4+2),x)

[Out]

Integral((2*x**3 - 1)*(2*x**3 + 1)*sqrt(2*x**6 + 1)/(8*x**12 + 8*x**6 + x**4 + 2), x)

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