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]

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

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Rubi [A]  time = 0.0163587, antiderivative size = 24, normalized size of antiderivative = 1., 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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.0208126, size = 24, normalized size = 1. \[ \log (x)-\left (1+\sqrt{2}\right ) \log \left (x^{\sqrt{2}-1}+1\right ) \]

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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Maple [A]  time = 0.02, size = 39, normalized size = 1.6 \begin{align*} \sqrt{2}\ln \left ( x \right ) +2\,\ln \left ( x \right ) -\ln \left ( x+{{\rm e}^{\sqrt{2}\ln \left ( x \right ) }} \right ) \sqrt{2}-\ln \left ( x+{{\rm e}^{\sqrt{2}\ln \left ( x \right ) }} \right ) \end{align*}

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 = 1.61263, size = 42, normalized size = 1.75 \begin{align*} \frac{\sqrt{2} \log \left (x\right )}{\sqrt{2} - 1} - \frac{\log \left (x + x^{\left (\sqrt{2}\right )}\right )}{\sqrt{2} - 1} \end{align*}

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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Fricas [A]  time = 0.774054, size = 78, normalized size = 3.25 \begin{align*} -{\left (\sqrt{2} + 1\right )} \log \left (x + x^{\left (\sqrt{2}\right )}\right ) +{\left (\sqrt{2} + 2\right )} \log \left (x\right ) \end{align*}

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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Sympy [B]  time = 0.680891, size = 76, normalized size = 3.17 \begin{align*} \frac{140 \sqrt{2} \log{\left (x \right )}}{-338 + 239 \sqrt{2}} - \frac{198 \log{\left (x \right )}}{-338 + 239 \sqrt{2}} + \frac{99 \sqrt{2} \log{\left (x + x^{\sqrt{2}} \right )}}{-338 + 239 \sqrt{2}} - \frac{140 \log{\left (x + x^{\sqrt{2}} \right )}}{-338 + 239 \sqrt{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

140*sqrt(2)*log(x)/(-338 + 239*sqrt(2)) - 198*log(x)/(-338 + 239*sqrt(2)) + 99*sqrt(2)*log(x + x**(sqrt(2)))/(
-338 + 239*sqrt(2)) - 140*log(x + x**(sqrt(2)))/(-338 + 239*sqrt(2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x + x^{\left (\sqrt{2}\right )}}\,{d x} \end{align*}

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)