3.10.2 \(\int \frac {1}{x^3 (1-x^4)^{3/2}} \, dx\) [902]

Optimal. Leaf size=34 \[ -\frac {1}{2 x^2 \sqrt {1-x^4}}+\frac {x^2}{\sqrt {1-x^4}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} \frac {x^2}{\sqrt {1-x^4}}-\frac {1}{2 x^2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 - x^4)^(3/2)),x]

[Out]

-1/2*1/(x^2*Sqrt[1 - x^4]) + x^2/Sqrt[1 - x^4]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 25, normalized size = 0.74 \begin {gather*} \frac {-1+2 x^4}{2 x^2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(1 - x^4)^(3/2)),x]

[Out]

(-1 + 2*x^4)/(2*x^2*Sqrt[1 - x^4])

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Maple [A]
time = 0.16, size = 22, normalized size = 0.65

method	result	size
default	\(\frac {2 x^{4}-1}{2 \sqrt {-x^{4}+1}\, x^{2}}\)	\(22\)
meijerg	\(-\frac {-2 x^{4}+1}{2 x^{2} \sqrt {-x^{4}+1}}\)	\(22\)
risch	\(\frac {2 x^{4}-1}{2 \sqrt {-x^{4}+1}\, x^{2}}\)	\(22\)
elliptic	\(\frac {2 x^{4}-1}{2 \sqrt {-x^{4}+1}\, x^{2}}\)	\(22\)
trager	\(-\frac {\left (2 x^{4}-1\right ) \sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right ) x^{2}}\)	\(29\)
gosper	\(-\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (2 x^{4}-1\right )}{2 x^{2} \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 29, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} - \frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 29, normalized size = 0.85 \begin {gather*} -\frac {{\left (2 \, x^{4} - 1\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{6} - x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*x^4 - 1)*sqrt(-x^4 + 1)/(x^6 - x^2)

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Sympy [C] Result contains complex when optimal does not.
time = 0.39, size = 90, normalized size = 2.65 \begin {gather*} \begin {cases} - \frac {2 i x^{4} \sqrt {x^{4} - 1}}{2 x^{6} - 2 x^{2}} + \frac {i \sqrt {x^{4} - 1}}{2 x^{6} - 2 x^{2}} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {2 x^{4} \sqrt {1 - x^{4}}}{2 x^{6} - 2 x^{2}} + \frac {\sqrt {1 - x^{4}}}{2 x^{6} - 2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*x**4*sqrt(x**4 - 1)/(2*x**6 - 2*x**2) + I*sqrt(x**4 - 1)/(2*x**6 - 2*x**2), Abs(x**4) > 1), (-
2*x**4*sqrt(1 - x**4)/(2*x**6 - 2*x**2) + sqrt(1 - x**4)/(2*x**6 - 2*x**2), True))

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Giac [A]
time = 1.70, size = 56, normalized size = 1.65 \begin {gather*} -\frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} + \frac {x^{2}}{4 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}} - \frac {\sqrt {-x^{4} + 1} - 1}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.14, size = 18, normalized size = 0.53 \begin {gather*} \frac {x^4-\frac {1}{2}}{x^2\,\sqrt {1-x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(1 - x^4)^(3/2)),x)

[Out]

(x^4 - 1/2)/(x^2*(1 - x^4)^(1/2))

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