3.19.45 \(\int \frac {2-x^4+2 x^8}{\sqrt [4]{1+x^4} (-2+x^4+x^8)} \, dx\) [1845]

Optimal. Leaf size=125 \[ \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}} \]

[Out]

arctan(x/(x^4+1)^(1/4))-2^(1/4)*arctan(1/2*x*2^(3/4)/(x^4+1)^(1/4))-1/4*arctan(2^(1/4)*x/(x^4+1)^(1/4))*2^(3/4
)+arctanh(x/(x^4+1)^(1/4))-2^(1/4)*arctanh(1/2*x*2^(3/4)/(x^4+1)^(1/4))-1/4*arctanh(2^(1/4)*x/(x^4+1)^(1/4))*2
^(3/4)

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Rubi [A]
time = 0.23, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6860, 246, 218, 212, 209, 385} \begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - x^4 + 2*x^8)/((1 + x^4)^(1/4)*(-2 + x^4 + x^8)),x]

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {2-x^4+2 x^8}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{1+x^4}}+\frac {3 \left (2-x^4\right )}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{1+x^4}} \, dx+3 \int \frac {2-x^4}{\sqrt [4]{1+x^4} \left (-2+x^4+x^8\right )} \, dx\\ &=2 \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+3 \int \left (\frac {2}{3 \sqrt [4]{1+x^4} \left (-2+2 x^4\right )}-\frac {8}{3 \sqrt [4]{1+x^4} \left (4+2 x^4\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{1+x^4} \left (-2+2 x^4\right )} \, dx-8 \int \frac {1}{\sqrt [4]{1+x^4} \left (4+2 x^4\right )} \, dx+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+2 \text {Subst}\left (\int \frac {1}{-2+4 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-8 \text {Subst}\left (\int \frac {1}{4-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 125, normalized size = 1.00 \begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \text {ArcTan}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}}+\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - x^4 + 2*x^8)/((1 + x^4)^(1/4)*(-2 + x^4 + x^8)),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-x^{4}+2}{\left (x^{4}+1\right )^{\frac {1}{4}} \left (x^{8}+x^{4}-2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^8-x^4+2)/(x^4+1)^(1/4)/(x^8+x^4-2),x)

[Out]

int((2*x^8-x^4+2)/(x^4+1)^(1/4)/(x^8+x^4-2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-x^4+2)/(x^4+1)^(1/4)/(x^8+x^4-2),x, algorithm="maxima")

[Out]

integrate((2*x^8 - x^4 + 2)/((x^8 + x^4 - 2)*(x^4 + 1)^(1/4)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (95) = 190\).
time = 0.36, size = 255, normalized size = 2.04 \begin {gather*} -\frac {1}{2} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {8^{\frac {3}{4}} \sqrt {2} x \sqrt {\frac {\sqrt {2} x^{2} + 2 \, \sqrt {x^{4} + 1}}{x^{2}}} - 2 \cdot 8^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{8 \, x}\right ) - \frac {1}{2} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} + 1}}{x^{2}}} - 2^{\frac {3}{4}} {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{2 \, x}\right ) - \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (\frac {8^{\frac {1}{4}} x + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 8^{\frac {3}{4}} \log \left (-\frac {8^{\frac {1}{4}} x - 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-x^4+2)/(x^4+1)^(1/4)/(x^8+x^4-2),x, algorithm="fricas")

[Out]

-1/2*8^(3/4)*arctan(1/8*(8^(3/4)*sqrt(2)*x*sqrt((sqrt(2)*x^2 + 2*sqrt(x^4 + 1))/x^2) - 2*8^(3/4)*(x^4 + 1)^(1/
4))/x) - 1/2*2^(3/4)*arctan(1/2*(2^(3/4)*x*sqrt((sqrt(2)*x^2 + sqrt(x^4 + 1))/x^2) - 2^(3/4)*(x^4 + 1)^(1/4))/
x) - 1/8*8^(3/4)*log((8^(1/4)*x + 2*(x^4 + 1)^(1/4))/x) + 1/8*8^(3/4)*log(-(8^(1/4)*x - 2*(x^4 + 1)^(1/4))/x)
- 1/8*2^(3/4)*log((2^(1/4)*x + (x^4 + 1)^(1/4))/x) + 1/8*2^(3/4)*log(-(2^(1/4)*x - (x^4 + 1)^(1/4))/x) - arcta
n((x^4 + 1)^(1/4)/x) + 1/2*log((x + (x^4 + 1)^(1/4))/x) - 1/2*log(-(x - (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 {2 x^{8} - x^{4} + 2}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt [4]{x^{4} + 1} \left (x^{4} + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**8-x**4+2)/(x**4+1)**(1/4)/(x**8+x**4-2),x)

[Out]

Integral((2*x**8 - x**4 + 2)/((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)**(1/4)*(x**4 + 2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-x^4+2)/(x^4+1)^(1/4)/(x^8+x^4-2),x, algorithm="giac")

[Out]

integrate((2*x^8 - x^4 + 2)/((x^8 + x^4 - 2)*(x^4 + 1)^(1/4)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,x^8-x^4+2}{{\left (x^4+1\right )}^{1/4}\,\left (x^8+x^4-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^8 - x^4 + 2)/((x^4 + 1)^(1/4)*(x^4 + x^8 - 2)),x)

[Out]

int((2*x^8 - x^4 + 2)/((x^4 + 1)^(1/4)*(x^4 + x^8 - 2)), x)

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