3.4.17 \(\int \frac {1}{x (1+x^4)} \, dx\) [317]

Optimal. Leaf size=13 \[ \log (x)-\frac {1}{4} \log \left (1+x^4\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {272, 36, 29, 31} \begin {gather*} \log (x)-\frac {1}{4} \log \left (x^4+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + x^4)),x]

[Out]

Log[x] - Log[1 + x^4]/4

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 272

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

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \log (x)-\frac {1}{4} \log \left (1+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + x^4)),x]

[Out]

Log[x] - Log[1 + x^4]/4

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Maple [A]
time = 0.05, size = 12, normalized size = 0.92

method	result	size
default	\(\ln \left (x \right )-\frac {\ln \left (x^{4}+1\right )}{4}\)	\(12\)
norman	\(\ln \left (x \right )-\frac {\ln \left (x^{4}+1\right )}{4}\)	\(12\)
meijerg	\(\ln \left (x \right )-\frac {\ln \left (x^{4}+1\right )}{4}\)	\(12\)
risch	\(\ln \left (x \right )-\frac {\ln \left (x^{4}+1\right )}{4}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^4+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.11, size = 15, normalized size = 1.15 \begin {gather*} -\frac {1}{4} \, \log \left (x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4+1),x, algorithm="maxima")

[Out]

-1/4*log(x^4 + 1) + 1/4*log(x^4)

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Fricas [A]
time = 0.87, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{4} \, \log \left (x^{4} + 1\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4+1),x, algorithm="fricas")

[Out]

-1/4*log(x^4 + 1) + log(x)

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Sympy [A]
time = 0.03, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{4} + 1 \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**4+1),x)

[Out]

log(x) - log(x**4 + 1)/4

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Giac [A]
time = 1.11, size = 15, normalized size = 1.15 \begin {gather*} -\frac {1}{4} \, \log \left (x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4+1),x, algorithm="giac")

[Out]

-1/4*log(x^4 + 1) + 1/4*log(x^4)

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Mupad [B]
time = 0.07, size = 11, normalized size = 0.85 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^4+1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^4 + 1)),x)

[Out]

log(x) - log(x^4 + 1)/4

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