3.11.80 \(\int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx\)

Optimal. Leaf size=81 \[ \frac {\tan ^{-1}\left (\frac {\frac {x}{\sqrt [4]{2}}-\frac {1}{\sqrt [4]{2}}}{\sqrt [4]{x^4+6 x^2+1}}\right )}{2\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\frac {x}{\sqrt [4]{2}}-\frac {1}{\sqrt [4]{2}}}{\sqrt [4]{x^4+6 x^2+1}}\right )}{2\ 2^{3/4}} \]

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Rubi [F]  time = 0.08, 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 {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((1 + x)*(1 + 6*x^2 + x^4)^(1/4)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx &=\int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1+x) \sqrt [4]{1+6 x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((1 + x)*(1 + 6*x^2 + x^4)^(1/4)),x]

[Out]

Integrate[1/((1 + x)*(1 + 6*x^2 + x^4)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 0.99, size = 81, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-\frac {1}{\sqrt [4]{2}}+\frac {x}{\sqrt [4]{2}}}{\sqrt [4]{1+6 x^2+x^4}}\right )}{2\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt [4]{2}}+\frac {x}{\sqrt [4]{2}}}{\sqrt [4]{1+6 x^2+x^4}}\right )}{2\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 + x)*(1 + 6*x^2 + x^4)^(1/4)),x]

[Out]

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

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fricas [B]  time = 8.12, size = 382, normalized size = 4.72 \begin {gather*} -\frac {1}{16} \cdot 8^{\frac {3}{4}} \arctan \left (-\frac {8^{\frac {3}{4}} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} + 4 \cdot 8^{\frac {1}{4}} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )} - 2^{\frac {1}{4}} {\left (8^{\frac {3}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} - 2 \, x + 1\right )} + 8^{\frac {1}{4}} {\left (3 \, x^{4} - 4 \, x^{3} + 18 \, x^{2} - 4 \, x + 3\right )}\right )}}{2 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )}}\right ) + \frac {1}{64} \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {3}{4}} {\left (3 \, x^{4} - 4 \, x^{3} + 18 \, x^{2} - 4 \, x + 3\right )} + 8 \, \sqrt {2} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} - 2 \, x + 1\right )} + 16 \, {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) - \frac {1}{64} \cdot 8^{\frac {3}{4}} \log \left (-\frac {8^{\frac {3}{4}} {\left (3 \, x^{4} - 4 \, x^{3} + 18 \, x^{2} - 4 \, x + 3\right )} - 8 \, \sqrt {2} {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{4} + 6 \, x^{2} + 1} {\left (x^{2} - 2 \, x + 1\right )} - 16 \, {\left (x^{4} + 6 \, x^{2} + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*8^(3/4)*arctan(-1/2*(8^(3/4)*(x^4 + 6*x^2 + 1)^(1/4)*(x^3 - 3*x^2 + 3*x - 1) + 4*8^(1/4)*(x^4 + 6*x^2 +
1)^(3/4)*(x - 1) - 2^(1/4)*(8^(3/4)*sqrt(x^4 + 6*x^2 + 1)*(x^2 - 2*x + 1) + 8^(1/4)*(3*x^4 - 4*x^3 + 18*x^2 -
4*x + 3)))/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + 1/64*8^(3/4)*log((8^(3/4)*(3*x^4 - 4*x^3 + 18*x^2 - 4*x + 3) + 8
*sqrt(2)*(x^4 + 6*x^2 + 1)^(1/4)*(x^3 - 3*x^2 + 3*x - 1) + 8*8^(1/4)*sqrt(x^4 + 6*x^2 + 1)*(x^2 - 2*x + 1) + 1
6*(x^4 + 6*x^2 + 1)^(3/4)*(x - 1))/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) - 1/64*8^(3/4)*log(-(8^(3/4)*(3*x^4 - 4*x^
3 + 18*x^2 - 4*x + 3) - 8*sqrt(2)*(x^4 + 6*x^2 + 1)^(1/4)*(x^3 - 3*x^2 + 3*x - 1) + 8*8^(1/4)*sqrt(x^4 + 6*x^2
+ 1)*(x^2 - 2*x + 1) - 16*(x^4 + 6*x^2 + 1)^(3/4)*(x - 1))/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^4 + 6*x^2 + 1)^(1/4)*(x + 1)), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (1+x \right ) \left (x^{4}+6 x^{2}+1\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^4 + 6*x^2 + 1)^(1/4)*(x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x + 1)*(x**4 + 6*x**2 + 1)**(1/4)), x)

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