3.1035 \(\int \frac{1}{x \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx\)

Optimal. Leaf size=8 \[ \frac{\log (x)}{\sqrt{a}} \]

[Out]

Log[x]/Sqrt[a]

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Rubi [A]  time = 0.001119, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2, 12, 29} \[ \frac{\log (x)}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

Log[x]/Sqrt[a]

Rule 2

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*a^p, x] /; FreeQ[{a, b, n, p}, x] && EqQ[b, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

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

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{a+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac{1}{\sqrt{a} x} \, dx\\ &=\frac{\int \frac{1}{x} \, dx}{\sqrt{a}}\\ &=\frac{\log (x)}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0004301, size = 8, normalized size = 1. \[ \frac{\log (x)}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a + (2 + 2*c - 2*(1 + c))*x^4]),x]

[Out]

Log[x]/Sqrt[a]

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Maple [A]  time = 0.04, size = 7, normalized size = 0.9 \begin{align*}{\ln \left ( x \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)/a^(1/2)

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Maxima [A]  time = 0.944304, size = 8, normalized size = 1. \begin{align*} \frac{\log \left (x\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/sqrt(a)

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Fricas [A]  time = 1.25804, size = 22, normalized size = 2.75 \begin{align*} \frac{\log \left (x\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/sqrt(a)

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Sympy [A]  time = 0.061173, size = 7, normalized size = 0.88 \begin{align*} \frac{\log{\left (x \right )}}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/sqrt(a)

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Giac [A]  time = 1.21322, size = 9, normalized size = 1.12 \begin{align*} \frac{\log \left ({\left | x \right |}\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x))/sqrt(a)