3.6.47 \(\int \frac {\sqrt {1+x+x^2+x^4} (-2-x+2 x^4)}{(1+x+x^4)^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac {\sqrt {x^4+x^2+x+1} x}{x^4+x+1}-\tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\right ) \]

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Rubi [F]  time = 0.59, 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+x^2+x^4} \left (-2-x+2 x^4\right )}{\left (1+x+x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.21, size = 17667, normalized size = 410.86 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.14, size = 43, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1+x+x^2+x^4}}{1+x+x^4}-\tanh ^{-1}\left (\frac {x}{\sqrt {1+x+x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 67, normalized size = 1.56 \begin {gather*} \frac {{\left (x^{4} + x + 1\right )} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x + x + 1}{x^{4} + x + 1}\right ) - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x}{2 \, {\left (x^{4} + x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((x^4 + x + 1)*log((x^4 + 2*x^2 - 2*sqrt(x^4 + x^2 + x + 1)*x + x + 1)/(x^4 + x + 1)) - 2*sqrt(x^4 + x^2 +
x + 1)*x)/(x^4 + x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.24, size = 66, normalized size = 1.53

method	result	size
trager	\(-\frac {x \sqrt {x^{4}+x^{2}+x +1}}{x^{4}+x +1}+\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+x^{2}+x +1}\, x -2 x^{2}-x -1}{x^{4}+x +1}\right )}{2}\)	\(66\)
risch	\(\text {Expression too large to display}\)	\(4900\)
elliptic	\(\text {Expression too large to display}\)	\(4900\)
default	\(\text {Expression too large to display}\)	\(9105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(x^4+x^2+x+1)^(1/2)/(x^4+x+1)+1/2*ln(-(-x^4+2*(x^4+x^2+x+1)^(1/2)*x-2*x^2-x-1)/(x^4+x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {\left (-2\,x^4+x+2\right )\,\sqrt {x^4+x^2+x+1}}{{\left (x^4+x+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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