3.1.1 \(\int \frac {x}{(-1+x^2)^{3/4}} \, dx\)

Optimal. Leaf size=11 \[ 2 \sqrt [4]{x^2-1} \]

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \begin {gather*} 2 \sqrt [4]{x^2-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(-1 + x^2)^(3/4),x]

[Out]

2*(-1 + x^2)^(1/4)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx &=2 \sqrt [4]{-1+x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{x^2-1} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(-1 + x^2)^(3/4),x]

[Out]

2*(-1 + x^2)^(1/4)

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IntegrateAlgebraic [A]  time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{-1+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(-1 + x^2)^(3/4),x]

[Out]

2*(-1 + x^2)^(1/4)

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fricas [A]  time = 1.01, size = 9, normalized size = 0.82 \begin {gather*} 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-1)^(3/4),x, algorithm="fricas")

[Out]

2*(x^2 - 1)^(1/4)

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giac [A]  time = 0.49, size = 9, normalized size = 0.82 \begin {gather*} 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-1)^(3/4),x, algorithm="giac")

[Out]

2*(x^2 - 1)^(1/4)

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maple [A]  time = 0.40, size = 10, normalized size = 0.91

method	result	size
derivativedivides	\(2 \left (x^{2}-1\right )^{\frac {1}{4}}\)	\(10\)
default	\(2 \left (x^{2}-1\right )^{\frac {1}{4}}\)	\(10\)
trager	\(2 \left (x^{2}-1\right )^{\frac {1}{4}}\)	\(10\)
risch	\(2 \left (x^{2}-1\right )^{\frac {1}{4}}\)	\(10\)
gosper	\(\frac {2 \left (-1+x \right ) \left (1+x \right )}{\left (x^{2}-1\right )^{\frac {3}{4}}}\)	\(16\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {3}{4}} x^{2} \hypergeom \left (\left [\frac {3}{4}, 1\right ], \relax [2], x^{2}\right )}{2 \mathrm {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.72, size = 9, normalized size = 0.82 \begin {gather*} 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2-1)^(3/4),x, algorithm="maxima")

[Out]

2*(x^2 - 1)^(1/4)

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mupad [B]  time = 0.40, size = 9, normalized size = 0.82 \begin {gather*} 2\,{\left (x^2-1\right )}^{1/4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 - 1)^(3/4),x)

[Out]

2*(x^2 - 1)^(1/4)

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sympy [A]  time = 0.17, size = 8, normalized size = 0.73 \begin {gather*} 2 \sqrt [4]{x^{2} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2-1)**(3/4),x)

[Out]

2*(x**2 - 1)**(1/4)

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