3.1025 \(\int \frac{1}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{x}{\sqrt{c x^4}} \]

[Out]

-(x/Sqrt[c*x^4])

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Rubi [A]  time = 0.0014219, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1, 15, 30} \[ -\frac{x}{\sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/Sqrt[c*x^4])

Rule 1

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.001179, size = 12, normalized size = 1. \[ -\frac{x}{\sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/Sqrt[c*x^4])

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Maple [A]  time = 0.041, size = 11, normalized size = 0.9 \begin{align*} -{x{\frac{1}{\sqrt{c{x}^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^4)^(1/2),x)

[Out]

-x/(c*x^4)^(1/2)

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Maxima [A]  time = 0.948303, size = 14, normalized size = 1.17 \begin{align*} -\frac{x}{\sqrt{c x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4)^(1/2),x, algorithm="maxima")

[Out]

-x/sqrt(c*x^4)

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Fricas [A]  time = 1.32357, size = 30, normalized size = 2.5 \begin{align*} -\frac{\sqrt{c x^{4}}}{c x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(c*x^4)/(c*x^3)

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Sympy [A]  time = 0.383548, size = 14, normalized size = 1.17 \begin{align*} - \frac{x}{\sqrt{c} \sqrt{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**4)**(1/2),x)

[Out]

-x/(sqrt(c)*sqrt(x**4))

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Giac [A]  time = 1.17937, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{\sqrt{c} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^4)^(1/2),x, algorithm="giac")

[Out]

-1/(sqrt(c)*x)