3.833 \(\int \frac{1+2 x^8}{x (1+x^8)^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0143167, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 78, 63, 207} \[ -\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0139714, size = 28, normalized size = 1. \[ -\frac{1}{4 \sqrt{x^8+1}}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.025, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{4}{\frac{1}{\sqrt{{x}^{8}+1}}}}+{\frac{1}{4}\ln \left ({ \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.53085, size = 46, normalized size = 1.64 \begin{align*} -\frac{1}{4 \, \sqrt{x^{8} + 1}} - \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.7639, size = 140, normalized size = 5. \begin{align*} -\frac{{\left (x^{8} + 1\right )} \log \left (\sqrt{x^{8} + 1} + 1\right ) -{\left (x^{8} + 1\right )} \log \left (\sqrt{x^{8} + 1} - 1\right ) + 2 \, \sqrt{x^{8} + 1}}{8 \,{\left (x^{8} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 12.2368, size = 37, normalized size = 1.32 \begin{align*} \frac{\log{\left (\sqrt{x^{8} + 1} - 1 \right )}}{8} - \frac{\log{\left (\sqrt{x^{8} + 1} + 1 \right )}}{8} - \frac{1}{4 \sqrt{x^{8} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(x**8 + 1) - 1)/8 - log(sqrt(x**8 + 1) + 1)/8 - 1/(4*sqrt(x**8 + 1))

________________________________________________________________________________________

Giac [A]  time = 1.12574, size = 46, normalized size = 1.64 \begin{align*} -\frac{1}{4 \, \sqrt{x^{8} + 1}} - \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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