\(\int x^6 \sqrt [4]{-1+x^4} \, dx\) [737]

Optimal result

Integrand size = 13, antiderivative size = 57 \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=\frac {1}{32} \sqrt [4]{-1+x^4} \left (-x^3+4 x^7\right )+\frac {3}{64} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {3}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right ) \]

[Out]

1/32*(x^4-1)^(1/4)*(4*x^7-x^3)+3/64*arctan(x/(x^4-1)^(1/4))-3/64*arctanh(x/(x^4-1)^(1/4))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {285, 327, 338, 304, 209, 212} \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=\frac {3}{64} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {3}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{8} \sqrt [4]{x^4-1} x^7-\frac {1}{32} \sqrt [4]{x^4-1} x^3 \]

[In]

Int[x^6*(-1 + x^4)^(1/4),x]

[Out]

-1/32*(x^3*(-1 + x^4)^(1/4)) + (x^7*(-1 + x^4)^(1/4))/8 + (3*ArcTan[x/(-1 + x^4)^(1/4)])/64 - (3*ArcTanh[x/(-1
+ x^4)^(1/4)])/64

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 285

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

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 327

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

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} x^7 \sqrt [4]{-1+x^4}-\frac {1}{8} \int \frac {x^6}{\left (-1+x^4\right )^{3/4}} \, dx \\ & = -\frac {1}{32} x^3 \sqrt [4]{-1+x^4}+\frac {1}{8} x^7 \sqrt [4]{-1+x^4}-\frac {3}{32} \int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx \\ & = -\frac {1}{32} x^3 \sqrt [4]{-1+x^4}+\frac {1}{8} x^7 \sqrt [4]{-1+x^4}-\frac {3}{32} \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = -\frac {1}{32} x^3 \sqrt [4]{-1+x^4}+\frac {1}{8} x^7 \sqrt [4]{-1+x^4}-\frac {3}{64} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {3}{64} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = -\frac {1}{32} x^3 \sqrt [4]{-1+x^4}+\frac {1}{8} x^7 \sqrt [4]{-1+x^4}+\frac {3}{64} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {3}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=\frac {1}{32} x^3 \sqrt [4]{-1+x^4} \left (-1+4 x^4\right )+\frac {3}{64} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {3}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right ) \]

[In]

Integrate[x^6*(-1 + x^4)^(1/4),x]

[Out]

(x^3*(-1 + x^4)^(1/4)*(-1 + 4*x^4))/32 + (3*ArcTan[x/(-1 + x^4)^(1/4)])/64 - (3*ArcTanh[x/(-1 + x^4)^(1/4)])/6
4

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.76 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.58

method	result	size
meijerg	\(\frac {\operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{7} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{4}\right )}{7 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}}}\)	\(33\)
risch	\(\frac {x^{3} \left (4 x^{4}-1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}}{32}-\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {3}{4}} x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{4}\right )}{32 \operatorname {signum}\left (x^{4}-1\right )^{\frac {3}{4}}}\)	\(53\)
pseudoelliptic	\(\frac {16 \left (x^{4}-1\right )^{\frac {1}{4}} x^{7}-4 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+3 \ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )-6 \arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )-3 \ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )}{128 {\left (\left (x^{4}-1\right )^{\frac {1}{4}}-x \right )}^{2} \left (x^{2}+\sqrt {x^{4}-1}\right )^{2} {\left (\left (x^{4}-1\right )^{\frac {1}{4}}+x \right )}^{2}}\)	\(113\)
trager	\(\frac {x^{3} \left (4 x^{4}-1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}}{32}-\frac {3 \ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+2 x^{4}-1\right )}{128}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}\, x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{128}\)	\(134\)

[In]

int(x^6*(x^4-1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/7*signum(x^4-1)^(1/4)/(-signum(x^4-1))^(1/4)*x^7*hypergeom([-1/4,7/4],[11/4],x^4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=\frac {1}{32} \, {\left (4 \, x^{7} - x^{3}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - \frac {3}{64} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{128} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {3}{128} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

integrate(x^6*(x^4-1)^(1/4),x, algorithm="fricas")

[Out]

1/32*(4*x^7 - x^3)*(x^4 - 1)^(1/4) - 3/64*arctan((x^4 - 1)^(1/4)/x) - 3/128*log((x + (x^4 - 1)^(1/4))/x) + 3/1
28*log(-(x - (x^4 - 1)^(1/4))/x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.56 \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=\frac {x^{7} e^{\frac {i \pi }{4}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]

[In]

integrate(x**6*(x**4-1)**(1/4),x)

[Out]

x**7*exp(I*pi/4)*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), x**4)/(4*gamma(11/4))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (45) = 90\).

Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.74 \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=-\frac {\frac {3 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {{\left (x^{4} - 1\right )}^{\frac {5}{4}}}{x^{5}}}{32 \, {\left (\frac {2 \, {\left (x^{4} - 1\right )}}{x^{4}} - \frac {{\left (x^{4} - 1\right )}^{2}}{x^{8}} - 1\right )}} - \frac {3}{64} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{128} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {3}{128} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \]

[In]

integrate(x^6*(x^4-1)^(1/4),x, algorithm="maxima")

[Out]

-1/32*(3*(x^4 - 1)^(1/4)/x + (x^4 - 1)^(5/4)/x^5)/(2*(x^4 - 1)/x^4 - (x^4 - 1)^2/x^8 - 1) - 3/64*arctan((x^4 -
1)^(1/4)/x) - 3/128*log((x^4 - 1)^(1/4)/x + 1) + 3/128*log((x^4 - 1)^(1/4)/x - 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.44 \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=-\frac {1}{32} \, x^{8} {\left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{x} - \frac {3 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )} - \frac {3}{64} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{128} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {3}{128} \, \log \left (-\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) \]

[In]

integrate(x^6*(x^4-1)^(1/4),x, algorithm="giac")

[Out]

-1/32*x^8*((x^4 - 1)^(1/4)*(1/x^4 - 1)/x - 3*(x^4 - 1)^(1/4)/x) - 3/64*arctan((x^4 - 1)^(1/4)/x) - 3/128*log((
x^4 - 1)^(1/4)/x + 1) + 3/128*log(-(x^4 - 1)^(1/4)/x + 1)

Mupad [F(-1)]

Timed out. \[ \int x^6 \sqrt [4]{-1+x^4} \, dx=\int x^6\,{\left (x^4-1\right )}^{1/4} \,d x \]

[In]

int(x^6*(x^4 - 1)^(1/4),x)

[Out]

int(x^6*(x^4 - 1)^(1/4), x)