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

Optimal. Leaf size=18 \[ -2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.46, 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 {-2-x+2 x^4}{\left (1+x+x^4\right ) \sqrt {1+x+x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 6.21, size = 17638, normalized size = 979.89 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.16, size = 18, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x+x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcTanh[x/Sqrt[1 + x + x^2 + x^4]]

________________________________________________________________________________________

fricas [B]  time = 0.48, size = 35, normalized size = 1.94 \begin {gather*} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} + x^{2} + x + 1} x + x + 1}{x^{4} + x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 6.08, size = 39, normalized size = 2.17

method	result	size
trager	\(-\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+x^{2}+x +1}\, x +2 x^{2}+x +1}{x^{4}+x +1}\right )\)	\(39\)
default	\(\text {Expression too large to display}\)	\(4880\)
elliptic	\(\text {Expression too large to display}\)	\(4880\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {-2\,x^4+x+2}{\left (x^4+x+1\right )\,\sqrt {x^4+x^2+x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________