\(\int \frac {x^7}{(1-x^4)^{3/2}} \, dx\) [895]

Optimal result

Integrand size = 15, antiderivative size = 31 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2 \sqrt {1-x^4}}+\frac {\sqrt {1-x^4}}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, 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 = {272, 45} \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=\frac {\sqrt {1-x^4}}{2}+\frac {1}{2 \sqrt {1-x^4}} \]

[In]

Int[x^7/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4]) + Sqrt[1 - x^4]/2

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {-2+x^4}{2 \sqrt {1-x^4}} \]

[In]

Integrate[x^7/(1 - x^4)^(3/2),x]

[Out]

-1/2*(-2 + x^4)/Sqrt[1 - x^4]

Maple [A] (verified)

Time = 4.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55

method	result	size
default	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
risch	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
elliptic	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
pseudoelliptic	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
trager	\(\frac {\left (x^{4}-2\right ) \sqrt {-x^{4}+1}}{2 x^{4}-2}\)	\(24\)
gosper	\(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}-2\right )}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(28\)
meijerg	\(\frac {-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 x^{4}+8\right )}{4 \sqrt {-x^{4}+1}}}{2 \sqrt {\pi }}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=\frac {{\left (x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{4} - 1\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=\frac {x^{4} \sqrt {1 - x^{4}}}{2 x^{4} - 2} - \frac {2 \sqrt {1 - x^{4}}}{2 x^{4} - 2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=\frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.83 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.52 \[ \int \frac {x^7}{\left (1-x^4\right )^{3/2}} \, dx=-\frac {x^4-2}{2\,\sqrt {1-x^4}} \]

[In]

int(x^7/(1 - x^4)^(3/2),x)

[Out]

-(x^4 - 2)/(2*(1 - x^4)^(1/2))