3.363 \(\int \frac {1}{x+x^{\sqrt {2}}} \, dx\)

Optimal. Leaf size=24 \[ \log (x)-\left (1+\sqrt {2}\right ) \log \left (x^{\sqrt {2}-1}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {1593, 266, 36, 29, 31} \[ \log (x)-\left (1+\sqrt {2}\right ) \log \left (x^{\sqrt {2}-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x + x^Sqrt[2])^(-1),x]

[Out]

Log[x] - (1 + Sqrt[2])*Log[1 + x^(-1 + Sqrt[2])]

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 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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 24, normalized size = 1.00 \[ \log (x)-\left (1+\sqrt {2}\right ) \log \left (x^{\sqrt {2}-1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x + x^Sqrt[2])^(-1),x]

[Out]

Log[x] - (1 + Sqrt[2])*Log[1 + x^(-1 + Sqrt[2])]

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fricas [A]  time = 0.42, size = 24, normalized size = 1.00 \[ -{\left (\sqrt {2} + 1\right )} \log \left (x + x^{\left (\sqrt {2}\right )}\right ) + {\left (\sqrt {2} + 2\right )} \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x + x^{\left (\sqrt {2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x + x^sqrt(2)), x)

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maple [A]  time = 0.16, size = 39, normalized size = 1.62 \[ \sqrt {2}\, \ln \relax (x )+2 \ln \relax (x )-\sqrt {2}\, \ln \left (x +{\mathrm e}^{\sqrt {2}\, \ln \relax (x )}\right )-\ln \left (x +{\mathrm e}^{\sqrt {2}\, \ln \relax (x )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x+x^(2^(1/2))),x)

[Out]

2^(1/2)*ln(x)+2*ln(x)-ln(x+exp(2^(1/2)*ln(x)))*2^(1/2)-ln(x+exp(2^(1/2)*ln(x)))

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maxima [A]  time = 2.93, size = 31, normalized size = 1.29 \[ \frac {\sqrt {2} \log \relax (x)}{\sqrt {2} - 1} - \frac {\log \left (x + x^{\left (\sqrt {2}\right )}\right )}{\sqrt {2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.25, size = 26, normalized size = 1.08 \[ \ln \relax (x)\,\left (\sqrt {2}+2\right )-\frac {\ln \left (x+x^{\sqrt {2}}\right )}{\sqrt {2}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + x^(2^(1/2))),x)

[Out]

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

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sympy [A]  time = 0.44, size = 32, normalized size = 1.33 \[ - \frac {2 \log {\relax (x )}}{-2 + \sqrt {2}} + \frac {\sqrt {2} \log {\left (x + x^{\sqrt {2}} \right )}}{-2 + \sqrt {2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x)/(-2 + sqrt(2)) + sqrt(2)*log(x + x**(sqrt(2)))/(-2 + sqrt(2))

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