3.308 \(\int \frac{x^3}{(-1+x^4) \sqrt{1+2 x^8}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+1}{\sqrt{3} \sqrt{2 x^8+1}}\right )}{4 \sqrt{3}} \]

[Out]

-ArcTanh[(1 + 2*x^4)/(Sqrt[3]*Sqrt[1 + 2*x^8])]/(4*Sqrt[3])

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Rubi [A]  time = 0.0317954, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1469, 725, 206} \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+1}{\sqrt{3} \sqrt{2 x^8+1}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((-1 + x^4)*Sqrt[1 + 2*x^8]),x]

[Out]

-ArcTanh[(1 + 2*x^4)/(Sqrt[3]*Sqrt[1 + 2*x^8])]/(4*Sqrt[3])

Rule 1469

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0134825, size = 29, normalized size = 0.85 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+1}{\sqrt{6 x^8+3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/((-1 + x^4)*Sqrt[1 + 2*x^8]),x]

[Out]

-ArcTanh[(1 + 2*x^4)/Sqrt[3 + 6*x^8]]/(4*Sqrt[3])

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{{x}^{4}-1}{\frac{1}{\sqrt{2\,{x}^{8}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4-1)/(2*x^8+1)^(1/2),x)

[Out]

int(x^3/(x^4-1)/(2*x^8+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{2 \, x^{8} + 1}{\left (x^{4} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4-1)/(2*x^8+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^3/(sqrt(2*x^8 + 1)*(x^4 - 1)), x)

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Fricas [A]  time = 2.40605, size = 128, normalized size = 3.76 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{2 \, x^{4} - \sqrt{3}{\left (2 \, x^{4} + 1\right )} - \sqrt{2 \, x^{8} + 1}{\left (\sqrt{3} - 3\right )} + 1}{x^{4} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4-1)/(2*x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log((2*x^4 - sqrt(3)*(2*x^4 + 1) - sqrt(2*x^8 + 1)*(sqrt(3) - 3) + 1)/(x^4 - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt{2 x^{8} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4-1)/(2*x**8+1)**(1/2),x)

[Out]

Integral(x**3/((x - 1)*(x + 1)*(x**2 + 1)*sqrt(2*x**8 + 1)), x)

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Giac [B]  time = 1.13792, size = 95, normalized size = 2.79 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (-\frac{{\left | -2 \, \sqrt{2} x^{4} - 2 \, \sqrt{3} + 2 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{8} + 1} \right |}}{2 \,{\left (\sqrt{2} x^{4} - \sqrt{3} - \sqrt{2} - \sqrt{2 \, x^{8} + 1}\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4-1)/(2*x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/12*sqrt(3)*log(-1/2*abs(-2*sqrt(2)*x^4 - 2*sqrt(3) + 2*sqrt(2) + 2*sqrt(2*x^8 + 1))/(sqrt(2)*x^4 - sqrt(3) -
sqrt(2) - sqrt(2*x^8 + 1)))