3.1491 \(\int \frac{x^7}{1+x^8} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{8} \log \left (x^8+1\right ) \]

[Out]

Log[1 + x^8]/8

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Rubi [A]  time = 0.002044, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {260} \[ \frac{1}{8} \log \left (x^8+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 + x^8),x]

[Out]

Log[1 + x^8]/8

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0023796, size = 10, normalized size = 1. \[ \frac{1}{8} \log \left (x^8+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 + x^8),x]

[Out]

Log[1 + x^8]/8

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Maple [A]  time = 0., size = 9, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({x}^{8}+1 \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^8+1),x)

[Out]

1/8*ln(x^8+1)

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Maxima [A]  time = 0.972572, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{8} \, \log \left (x^{8} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(x^8 + 1)

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Fricas [A]  time = 1.24823, size = 24, normalized size = 2.4 \begin{align*} \frac{1}{8} \, \log \left (x^{8} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(x^8 + 1)

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Sympy [A]  time = 0.091294, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (x^{8} + 1 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**8 + 1)/8

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Giac [A]  time = 1.17, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{8} \, \log \left (x^{8} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*log(x^8 + 1)