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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.01, 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

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fricas [A]  time = 0.79, size = 23, normalized size = 1.05 \[ \frac {{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) + 1}{4 \, {\left (x^{4} + 1\right )}} \]

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)

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giac [A]  time = 0.28, size = 18, normalized size = 0.82 \[ \frac {1}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]

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)

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maple [A]  time = 0.01, size = 19, normalized size = 0.86 \[ \frac {\ln \left (x^{4}+1\right )}{4}+\frac {1}{4 x^{4}+4} \]

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)

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maxima [A]  time = 0.98, size = 18, normalized size = 0.82 \[ \frac {1}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \]

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)

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mupad [B]  time = 1.32, size = 18, normalized size = 0.82 \[ \frac {\ln \left (x^4+1\right )}{4}+\frac {1}{4\,\left (x^4+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(2*x^4 + x^8 + 1),x)

[Out]

log(x^4 + 1)/4 + 1/(4*(x^4 + 1))

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sympy [A]  time = 0.11, size = 15, normalized size = 0.68 \[ \frac {\log {\left (x^{4} + 1 \right )}}{4} + \frac {1}{4 x^{4} + 4} \]

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)

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