3.273 \(\int \frac{x^7}{1+2 x^4+x^8} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{4 \left (x^4+1\right )}+\frac{1}{4} \log \left (x^4+1\right ) \]

[Out]

1/(4*(1 + x^4)) + Log[1 + x^4]/4

________________________________________________________________________________________

Rubi [A]  time = 0.01037, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {28, 266, 43} \[ \frac{1}{4 \left (x^4+1\right )}+\frac{1}{4} \log \left (x^4+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 + 2*x^4 + x^8),x]

[Out]

1/(4*(1 + x^4)) + Log[1 + x^4]/4

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^7}{1+2 x^4+x^8} \, dx &=\int \frac{x^7}{\left (1+x^4\right )^2} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(1+x)^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{1}{(1+x)^2}+\frac{1}{1+x}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{4 \left (1+x^4\right )}+\frac{1}{4} \log \left (1+x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0066118, size = 18, normalized size = 0.82 \[ \frac{1}{4} \left (\frac{1}{x^4+1}+\log \left (x^4+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 + 2*x^4 + x^8),x]

[Out]

((1 + x^4)^(-1) + Log[1 + x^4])/4

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 19, normalized size = 0.9 \begin{align*}{\frac{1}{4\,{x}^{4}+4}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^8+2*x^4+1),x)

[Out]

1/4/(x^4+1)+1/4*ln(x^4+1)

________________________________________________________________________________________

Maxima [A]  time = 0.971477, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8+2*x^4+1),x, algorithm="maxima")

[Out]

1/4/(x^4 + 1) + 1/4*log(x^4 + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.43578, size = 59, normalized size = 2.68 \begin{align*} \frac{{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) + 1}{4 \,{\left (x^{4} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8+2*x^4+1),x, algorithm="fricas")

[Out]

1/4*((x^4 + 1)*log(x^4 + 1) + 1)/(x^4 + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.110301, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 x^{4} + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**8+2*x**4+1),x)

[Out]

log(x**4 + 1)/4 + 1/(4*x**4 + 4)

________________________________________________________________________________________

Giac [A]  time = 1.12836, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{4} \, \log \left (x^{4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(x^8+2*x^4+1),x, algorithm="giac")

[Out]

1/4/(x^4 + 1) + 1/4*log(x^4 + 1)