\(\int \frac {1}{x^7 (1-x^4)^{3/2}} \, dx\) [903]

Optimal result

Integrand size = 15, antiderivative size = 55 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=-\frac {1}{6 x^6 \sqrt {1-x^4}}-\frac {2}{3 x^2 \sqrt {1-x^4}}+\frac {4 x^2}{3 \sqrt {1-x^4}} \]

[Out]

-1/6/x^6/(-x^4+1)^(1/2)-2/3/x^2/(-x^4+1)^(1/2)+4/3*x^2/(-x^4+1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=-\frac {1}{6 \sqrt {1-x^4} x^6}+\frac {4 x^2}{3 \sqrt {1-x^4}}-\frac {2}{3 \sqrt {1-x^4} x^2} \]

[In]

Int[1/(x^7*(1 - x^4)^(3/2)),x]

[Out]

-1/6*1/(x^6*Sqrt[1 - x^4]) - 2/(3*x^2*Sqrt[1 - x^4]) + (4*x^2)/(3*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*} \text {integral}& = -\frac {1}{6 x^6 \sqrt {1-x^4}}+\frac {4}{3} \int \frac {1}{x^3 \left (1-x^4\right )^{3/2}} \, dx \\ & = -\frac {1}{6 x^6 \sqrt {1-x^4}}-\frac {2}{3 x^2 \sqrt {1-x^4}}+\frac {8}{3} \int \frac {x}{\left (1-x^4\right )^{3/2}} \, dx \\ & = -\frac {1}{6 x^6 \sqrt {1-x^4}}-\frac {2}{3 x^2 \sqrt {1-x^4}}+\frac {4 x^2}{3 \sqrt {1-x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=\frac {-1-4 x^4+8 x^8}{6 x^6 \sqrt {1-x^4}} \]

[In]

Integrate[1/(x^7*(1 - x^4)^(3/2)),x]

[Out]

(-1 - 4*x^4 + 8*x^8)/(6*x^6*Sqrt[1 - x^4])

Maple [A] (verified)

Time = 4.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.49

method	result	size
default	\(\frac {8 x^{8}-4 x^{4}-1}{6 \sqrt {-x^{4}+1}\, x^{6}}\)	\(27\)
meijerg	\(-\frac {-8 x^{8}+4 x^{4}+1}{6 x^{6} \sqrt {-x^{4}+1}}\)	\(27\)
risch	\(\frac {8 x^{8}-4 x^{4}-1}{6 \sqrt {-x^{4}+1}\, x^{6}}\)	\(27\)
elliptic	\(\frac {8 x^{8}-4 x^{4}-1}{6 \sqrt {-x^{4}+1}\, x^{6}}\)	\(27\)
pseudoelliptic	\(\frac {8 x^{8}-4 x^{4}-1}{6 \sqrt {-x^{4}+1}\, x^{6}}\)	\(27\)
trager	\(-\frac {\left (8 x^{8}-4 x^{4}-1\right ) \sqrt {-x^{4}+1}}{6 \left (x^{4}-1\right ) x^{6}}\)	\(34\)
gosper	\(-\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (8 x^{8}-4 x^{4}-1\right )}{6 x^{6} \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(38\)

[In]

int(1/x^7/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(8*x^8-4*x^4-1)/(-x^4+1)^(1/2)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=-\frac {{\left (8 \, x^{8} - 4 \, x^{4} - 1\right )} \sqrt {-x^{4} + 1}}{6 \, {\left (x^{10} - x^{6}\right )}} \]

[In]

integrate(1/x^7/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(8*x^8 - 4*x^4 - 1)*sqrt(-x^4 + 1)/(x^10 - x^6)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=\begin {cases} - \frac {8 x^{8} \sqrt {-1 + \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {4 x^{4} \sqrt {-1 + \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {\sqrt {-1 + \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {8 i x^{8} \sqrt {1 - \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {4 i x^{4} \sqrt {1 - \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} + \frac {i \sqrt {1 - \frac {1}{x^{4}}}}{6 x^{8} - 6 x^{4}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x**7/(-x**4+1)**(3/2),x)

[Out]

Piecewise((-8*x**8*sqrt(-1 + x**(-4))/(6*x**8 - 6*x**4) + 4*x**4*sqrt(-1 + x**(-4))/(6*x**8 - 6*x**4) + sqrt(-
1 + x**(-4))/(6*x**8 - 6*x**4), 1/Abs(x**4) > 1), (-8*I*x**8*sqrt(1 - 1/x**4)/(6*x**8 - 6*x**4) + 4*I*x**4*sqr
t(1 - 1/x**4)/(6*x**8 - 6*x**4) + I*sqrt(1 - 1/x**4)/(6*x**8 - 6*x**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=\frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} - \frac {\sqrt {-x^{4} + 1}}{x^{2}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]

[In]

integrate(1/x^7/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

1/2*x^2/sqrt(-x^4 + 1) - sqrt(-x^4 + 1)/x^2 - 1/6*(-x^4 + 1)^(3/2)/x^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (43) = 86\).

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=\frac {x^{6} {\left (\frac {21 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{2}}{x^{4}} + 1\right )}}{48 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}} - \frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} - \frac {7 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}}{16 \, x^{2}} - \frac {{\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}}{48 \, x^{6}} \]

[In]

integrate(1/x^7/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

1/48*x^6*(21*(sqrt(-x^4 + 1) - 1)^2/x^4 + 1)/(sqrt(-x^4 + 1) - 1)^3 - 1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1) - 7/16*
(sqrt(-x^4 + 1) - 1)/x^2 - 1/48*(sqrt(-x^4 + 1) - 1)^3/x^6

Mupad [B] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^7 \left (1-x^4\right )^{3/2}} \, dx=\frac {8\,{\left (x^4-1\right )}^2+12\,x^4-9}{6\,x^6\,\sqrt {1-x^4}} \]

[In]

int(1/(x^7*(1 - x^4)^(3/2)),x)

[Out]

(8*(x^4 - 1)^2 + 12*x^4 - 9)/(6*x^6*(1 - x^4)^(1/2))