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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ -\frac {x}{\sqrt {c x^4}} \]

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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fricas [A]  time = 0.81, size = 15, normalized size = 1.25 \[ -\frac {\sqrt {c x^{4}}}{c x^{3}} \]

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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giac [A]  time = 0.15, size = 8, normalized size = 0.67 \[ -\frac {1}{\sqrt {c} x} \]

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)

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maple [A]  time = 0.00, size = 11, normalized size = 0.92 \[ -\frac {x}{\sqrt {c \,x^{4}}} \]

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 = 0.98, size = 10, normalized size = 0.83 \[ -\frac {x}{\sqrt {c x^{4}}} \]

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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mupad [B]  time = 4.30, size = 13, normalized size = 1.08 \[ -\frac {\sqrt {x^4}}{\sqrt {c}\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^4)^(1/2)/(c^(1/2)*x^3)

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sympy [A]  time = 0.47, size = 14, normalized size = 1.17 \[ - \frac {x}{\sqrt {c} \sqrt {x^{4}}} \]

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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