∫ −2−x+2x4 ______________________________________(1+x+x4 )√ ______

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 \[ -2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\right ) \]

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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*}

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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

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IntegrateAlgebraic [A] time = 0.18, size = 18, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+x+1}}\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]]

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fricas [B] time = 0.44, 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))

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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)

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maple [C] time = 5.32, size = 4880, normalized size = 271.11 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[Out]

4*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=4))*((RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4

+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+_Z+1,

index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))^2*((RootOf(_Z^4+_Z^2+_Z

+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=3))/(RootOf(_Z^4+_Z^2+_Z+1,index=3)

-RootOf(_Z^4+_Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)*((RootOf(_Z^4+_Z^2+_Z+1,index=2)-Ro

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Z^2+_Z+1,index=1))/(x-RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2)/(RootOf(_Z^4+_Z^2+_Z+1,index=4)-RootOf(_Z^4+_Z^2+

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=1))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=3))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=4)))

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2),((RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=3))*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^

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index=4)+RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2))+2*sum(_alpha*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2

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)*(x-RootOf(_Z^4+_Z^2+_Z+1,index=2))^2*((RootOf(_Z^4+_Z^2+_Z+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=1))*(x-Roo

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_Z^4+_Z^2+_Z+1,index=3))*(x-RootOf(_Z^4+_Z^2+_Z+1,index=4)))^(1/2)*(-RootOf(_Z^4+_Z^2+_Z+1,index=2)^3*_alpha^3

+RootOf(_Z^4+_Z^2+_Z+1,index=2)^3*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*_alpha^3-RootOf(_Z^4+_Z^2+_Z+1,ind

ex=2)*_alpha^3-RootOf(_Z^4+_Z^2+_Z+1,index=2)^3+RootOf(_Z^4+_Z^2+_Z+1,index=2)*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,

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+1,index=3))*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=4))/(RootOf(_Z^4+_Z^2+_Z+1,index=1)-R

ootOf(_Z^4+_Z^2+_Z+1,index=3))/(-RootOf(_Z^4+_Z^2+_Z+1,index=4)+RootOf(_Z^4+_Z^2+_Z+1,index=2)))^(1/2))+(RootO

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Z^4+_Z^2+_Z+1,index=1)^3*_alpha^2+RootOf(_Z^4+_Z^2+_Z+1,index=1)^2*_alpha^3-RootOf(_Z^4+_Z^2+_Z+1,index=1)*_al

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Z^2+_Z+1,index=2)+284/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)+90/257*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^

2+_Z+1,index=2)+125/257*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=1)-363/257*RootOf(_Z^4+_Z^2

+_Z+1,index=1)*RootOf(_Z^4+_Z^2+_Z+1,index=2)-312/257*_alpha*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_

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(_Z^4+_Z^2+_Z+1,index=1)*RootOf(_Z^4+_Z^2+_Z+1,index=2)-320/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2-222/257*RootO

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^3+53/257*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=2)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2+202/2

57*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=1)*RootOf(_Z^4+_Z^2+_Z+1,index=2)+186/257*RootOf

(_Z^4+_Z^2+_Z+1,index=1)^3+99/257*_alpha^3-336/257*_alpha^2+150/257*_alpha-185/257*RootOf(_Z^4+_Z^2+_Z+1,index

=4)*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2-102/257*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^3+7

3/257*RootOf(_Z^4+_Z^2+_Z+1,index=4)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2+16/257*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*

RootOf(_Z^4+_Z^2+_Z+1,index=1)^3-82/257*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2+187/

257*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)+20/257*RootOf(_Z^4+_Z^2+_Z+1,index=2)*Root

Of(_Z^4+_Z^2+_Z+1,index=1)^3+98/257*RootOf(_Z^4+_Z^2+_Z+1,index=2)*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2-86/257*Roo

tOf(_Z^4+_Z^2+_Z+1,index=2)^2*_alpha^3+151/257*RootOf(_Z^4+_Z^2+_Z+1,index=2)*_alpha^3+58/257*RootOf(_Z^4+_Z^2

+_Z+1,index=2)*_alpha^2+336/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)^3*_alpha^3-150/257*RootOf(_Z^4+_Z^2+_Z+1,index=

1)^3*_alpha^2-150/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2*_alpha^3+332/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)*_alpha^

3-237/257*RootOf(_Z^4+_Z^2+_Z+1,index=1)*_alpha^2-4/257*_alpha*RootOf(_Z^4+_Z^2+_Z+1,index=1)^3-87/257*_alpha*

RootOf(_Z^4+_Z^2+_Z+1,index=1)^2-186/257*_alpha*RootOf(_Z^4+_Z^2+_Z+1,index=1)+239/257*_alpha*RootOf(_Z^4+_Z^2

+_Z+1,index=4)+154/257*_alpha*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2-359/257*_alpha*RootOf(_Z^4+_Z^2+_Z+1,index=2)-2

87/257*_alpha^3*RootOf(_Z^4+_Z^2+_Z+1,index=4)+96/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=4)-136/257*_alpha^2

*RootOf(_Z^4+_Z^2+_Z+1,index=2)^2-4/257*_alpha^2*RootOf(_Z^4+_Z^2+_Z+1,index=1)^2+44/257,((RootOf(_Z^4+_Z^2+_Z

+1,index=2)-RootOf(_Z^4+_Z^2+_Z+1,index=3))*(RootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=4))/(R

ootOf(_Z^4+_Z^2+_Z+1,index=1)-RootOf(_Z^4+_Z^2+_Z+1,index=3))/(-RootOf(_Z^4+_Z^2+_Z+1,index=4)+RootOf(_Z^4+_Z^

2+_Z+1,index=2)))^(1/2))),_alpha=RootOf(_Z^4+_Z+1))

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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)

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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)

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

[Out]

Timed out

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