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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = 0.00, size = 8, normalized size = 1.00 \[ \frac {\log (x)}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]/Sqrt[a]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 6, normalized size = 0.75 \[ \frac {\log \relax (x)}{\sqrt {a}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 7, normalized size = 0.88 \[ \frac {\log \left ({\left | x \right |}\right )}{\sqrt {a}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 7, normalized size = 0.88 \[ \frac {\ln \relax (x )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.03, size = 6, normalized size = 0.75 \[ \frac {\log \relax (x)}{\sqrt {a}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 4.24, size = 6, normalized size = 0.75 \[ \frac {\ln \relax (x)}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 7, normalized size = 0.88 \[ \frac {\log {\relax (x )}}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/sqrt(a)

________________________________________________________________________________________