\(\int (1-x^4)^{3/2} \, dx\) [804]

Optimal result

Integrand size = 11, antiderivative size = 41 \[ \int \left (1-x^4\right )^{3/2} \, dx=\frac {2}{7} x \sqrt {1-x^4}+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {4}{7} \operatorname {EllipticF}(\arcsin (x),-1) \]

[Out]

1/7*x*(-x^4+1)^(3/2)+4/7*EllipticF(x,I)+2/7*x*(-x^4+1)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 227} \[ \int \left (1-x^4\right )^{3/2} \, dx=\frac {4}{7} \operatorname {EllipticF}(\arcsin (x),-1)+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {2}{7} x \sqrt {1-x^4} \]

[In]

Int[(1 - x^4)^(3/2),x]

[Out]

(2*x*Sqrt[1 - x^4])/7 + (x*(1 - x^4)^(3/2))/7 + (4*EllipticF[ArcSin[x], -1])/7

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {6}{7} \int \sqrt {1-x^4} \, dx \\ & = \frac {2}{7} x \sqrt {1-x^4}+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {4}{7} \int \frac {1}{\sqrt {1-x^4}} \, dx \\ & = \frac {2}{7} x \sqrt {1-x^4}+\frac {1}{7} x \left (1-x^4\right )^{3/2}+\frac {4}{7} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 3.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.37 \[ \int \left (1-x^4\right )^{3/2} \, dx=x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},x^4\right ) \]

[In]

Integrate[(1 - x^4)^(3/2),x]

[Out]

x*Hypergeometric2F1[-3/2, 1/4, 5/4, x^4]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 4.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.29

method	result	size
meijerg	\(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};x^{4}\right )\)	\(12\)
risch	\(\frac {x \left (x^{4}-3\right ) \left (x^{4}-1\right )}{7 \sqrt {-x^{4}+1}}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{7 \sqrt {-x^{4}+1}}\)	\(55\)
default	\(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}+\frac {3 x \sqrt {-x^{4}+1}}{7}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{7 \sqrt {-x^{4}+1}}\)	\(59\)
elliptic	\(-\frac {x^{5} \sqrt {-x^{4}+1}}{7}+\frac {3 x \sqrt {-x^{4}+1}}{7}+\frac {4 \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, F\left (x , i\right )}{7 \sqrt {-x^{4}+1}}\)	\(59\)

[In]

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

[Out]

x*hypergeom([-3/2,1/4],[5/4],x^4)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \left (1-x^4\right )^{3/2} \, dx=-\frac {1}{7} \, {\left (x^{5} - 3 \, x\right )} \sqrt {-x^{4} + 1} + \frac {4}{7} i \, F(\arcsin \left (\frac {1}{x}\right )\,|\,-1) \]

[In]

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

[Out]

-1/7*(x^5 - 3*x)*sqrt(-x^4 + 1) + 4/7*I*elliptic_f(arcsin(1/x), -1)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76 \[ \int \left (1-x^4\right )^{3/2} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

[In]

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

[Out]

x*gamma(1/4)*hyper((-3/2, 1/4), (5/4,), x**4*exp_polar(2*I*pi))/(4*gamma(5/4))

Maxima [F]

\[ \int \left (1-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 1\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((-x^4 + 1)^(3/2), x)

Giac [F]

\[ \int \left (1-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 1\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((-x^4 + 1)^(3/2), x)

Mupad [B] (verification not implemented)

Time = 5.44 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.24 \[ \int \left (1-x^4\right )^{3/2} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ x^4\right ) \]

[In]

int((1 - x^4)^(3/2),x)

[Out]

x*hypergeom([-3/2, 1/4], 5/4, x^4)