3.6.100 \(\int \frac {\sqrt {1+x^2+x^6} (-1+2 x^6)}{(1+x^6) (2-x^2+2 x^6)} \, dx\)

Optimal. Leaf size=47 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^6+x^2+1}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^6+x^2+1}}\right ) \]

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Rubi [F]  time = 1.93, 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+x^2+x^6} \left (-1+2 x^6\right )}{\left (1+x^6\right ) \left (2-x^2+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^2+x^6} \left (-1+2 x^6\right )}{\left (1+x^6\right ) \left (2-x^2+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.73, size = 47, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {1+x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   catdef: division by zero

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.37, size = 117, normalized size = 2.49

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{6}+5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}-12 \sqrt {x^{6}+x^{2}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{2 x^{6}-x^{2}+2}\right )}{4}+\frac {\ln \left (-\frac {x^{6}+2 \sqrt {x^{6}+x^{2}+1}\, x +2 x^{2}+1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{2}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*RootOf(_Z^2-6)*ln((2*RootOf(_Z^2-6)*x^6+5*RootOf(_Z^2-6)*x^2-12*(x^6+x^2+1)^(1/2)*x+2*RootOf(_Z^2-6))/(2*x
^6-x^2+2))+1/2*ln(-(x^6+2*(x^6+x^2+1)^(1/2)*x+2*x^2+1)/(x^2+1)/(x^4-x^2+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (2\,x^6-1\right )\,\sqrt {x^6+x^2+1}}{\left (x^6+1\right )\,\left (2\,x^6-x^2+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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