3.6.74 \(\int \frac {-1+x^2}{(1+x^2) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx\)

Optimal. Leaf size=44 \[ -\frac {\left (\left (x^4+1\right )^5\right )^{9/10} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2} \left (x^4+1\right )^{9/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6688, 6720, 1699, 203} \begin {gather*} -\frac {\sqrt {x^4+1} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2} \sqrt [10]{\left (x^4+1\right )^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/((1 + x^2)*(1 + 5*x^4 + 10*x^8 + 10*x^12 + 5*x^16 + x^20)^(1/10)),x]

[Out]

-((Sqrt[1 + x^4]*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]])/(Sqrt[2]*((1 + x^4)^5)^(1/10)))

Rule 203

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

Rule 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx &=\int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{\left (1+x^4\right )^5}} \, dx\\ &=\frac {\sqrt {1+x^4} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt [10]{\left (1+x^4\right )^5}}\\ &=-\frac {\sqrt {1+x^4} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )}{\sqrt [10]{\left (1+x^4\right )^5}}\\ &=-\frac {\sqrt {1+x^4} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2} \sqrt [10]{\left (1+x^4\right )^5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.11, size = 61, normalized size = 1.39 \begin {gather*} -\frac {\sqrt [4]{-1} \sqrt {x^4+1} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2 \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )}{\sqrt [10]{\left (x^4+1\right )^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^2)/((1 + x^2)*(1 + 5*x^4 + 10*x^8 + 10*x^12 + 5*x^16 + x^20)^(1/10)),x]

[Out]

-(((-1)^(1/4)*Sqrt[1 + x^4]*(EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - 2*EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*x]
, -1]))/((1 + x^4)^5)^(1/10))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 20.56, size = 44, normalized size = 1.00 \begin {gather*} -\frac {\left (\left (1+x^4\right )^5\right )^{9/10} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2} \left (1+x^4\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(1 + 5*x^4 + 10*x^8 + 10*x^12 + 5*x^16 + x^20)^(1/10)),x]

[Out]

-((((1 + x^4)^5)^(9/10)*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]])/(Sqrt[2]*(1 + x^4)^(9/2)))

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 45, normalized size = 1.02 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} x}{x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^20+5*x^16+10*x^12+10*x^8+5*x^4+1)^(1/10),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*arctan(sqrt(2)*(x^20 + 5*x^16 + 10*x^12 + 10*x^8 + 5*x^4 + 1)^(1/10)*x/(x^4 + 1))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^20+5*x^16+10*x^12+10*x^8+5*x^4+1)^(1/10),x, algorithm="giac")

[Out]

integrate((x^2 - 1)/((x^20 + 5*x^16 + 10*x^12 + 10*x^8 + 5*x^4 + 1)^(1/10)*(x^2 + 1)), x)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \left (x^{20}+5 x^{16}+10 x^{12}+10 x^{8}+5 x^{4}+1\right )^{\frac {1}{10}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)/(x^2+1)/(x^20+5*x^16+10*x^12+10*x^8+5*x^4+1)^(1/10),x)

[Out]

int((x^2-1)/(x^2+1)/(x^20+5*x^16+10*x^12+10*x^8+5*x^4+1)^(1/10),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x^20+5*x^16+10*x^12+10*x^8+5*x^4+1)^(1/10),x, algorithm="maxima")

[Out]

integrate((x^2 - 1)/((x^20 + 5*x^16 + 10*x^12 + 10*x^8 + 5*x^4 + 1)^(1/10)*(x^2 + 1)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,{\left (x^{20}+5\,x^{16}+10\,x^{12}+10\,x^8+5\,x^4+1\right )}^{1/10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 1)/((x^2 + 1)*(5*x^4 + 10*x^8 + 10*x^12 + 5*x^16 + x^20 + 1)^(1/10)),x)

[Out]

int((x^2 - 1)/((x^2 + 1)*(5*x^4 + 10*x^8 + 10*x^12 + 5*x^16 + x^20 + 1)^(1/10)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt [10]{\left (x^{4} + 1\right )^{5}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)/(x**2+1)/(x**20+5*x**16+10*x**12+10*x**8+5*x**4+1)**(1/10),x)

[Out]

Integral((x - 1)*(x + 1)/((x**2 + 1)*((x**4 + 1)**5)**(1/10)), x)

________________________________________________________________________________________