\(\int x^8 \sqrt {-1+4 x^6} \, dx\) [1380]

Optimal result

Integrand size = 15, antiderivative size = 58 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=-\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{192} \text {arctanh}\left (\frac {2 x^3}{\sqrt {-1+4 x^6}}\right ) \]

[Out]

-1/192*arctanh(2*x^3/(4*x^6-1)^(1/2))-1/96*x^3*(4*x^6-1)^(1/2)+1/12*x^9*(4*x^6-1)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 285, 327, 223, 212} \[ \int x^8 \sqrt {-1+4 x^6} \, dx=-\frac {1}{192} \text {arctanh}\left (\frac {2 x^3}{\sqrt {4 x^6-1}}\right )+\frac {1}{12} \sqrt {4 x^6-1} x^9-\frac {1}{96} \sqrt {4 x^6-1} x^3 \]

[In]

Int[x^8*Sqrt[-1 + 4*x^6],x]

[Out]

-1/96*(x^3*Sqrt[-1 + 4*x^6]) + (x^9*Sqrt[-1 + 4*x^6])/12 - ArcTanh[(2*x^3)/Sqrt[-1 + 4*x^6]]/192

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^2 \sqrt {-1+4 x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+4 x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{96} \text {Subst}\left (\int \frac {1}{\sqrt {-1+4 x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{96} \text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+4 x^6}}\right ) \\ & = -\frac {1}{96} x^3 \sqrt {-1+4 x^6}+\frac {1}{12} x^9 \sqrt {-1+4 x^6}-\frac {1}{192} \tanh ^{-1}\left (\frac {2 x^3}{\sqrt {-1+4 x^6}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\frac {1}{96} x^3 \sqrt {-1+4 x^6} \left (-1+8 x^6\right )-\frac {1}{192} \log \left (2 x^3+\sqrt {-1+4 x^6}\right ) \]

[In]

Integrate[x^8*Sqrt[-1 + 4*x^6],x]

[Out]

(x^3*Sqrt[-1 + 4*x^6]*(-1 + 8*x^6))/96 - Log[2*x^3 + Sqrt[-1 + 4*x^6]]/192

Maple [A] (verified)

Time = 4.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71

method	result	size
trager	\(\frac {x^{3} \left (8 x^{6}-1\right ) \sqrt {4 x^{6}-1}}{96}-\frac {\ln \left (2 x^{3}+\sqrt {4 x^{6}-1}\right )}{192}\)	\(41\)
pseudoelliptic	\(-\frac {\ln \left (2 x^{3}+\sqrt {4 x^{6}-1}\right )}{192}+\frac {\left (8 x^{9}-x^{3}\right ) \sqrt {4 x^{6}-1}}{96}\)	\(42\)
risch	\(\frac {x^{3} \left (8 x^{6}-1\right ) \sqrt {4 x^{6}-1}}{96}-\frac {\sqrt {-\operatorname {signum}\left (4 x^{6}-1\right )}\, \arcsin \left (2 x^{3}\right )}{192 \sqrt {\operatorname {signum}\left (4 x^{6}-1\right )}}\)	\(53\)
meijerg	\(-\frac {i \sqrt {\operatorname {signum}\left (4 x^{6}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{3} \left (-24 x^{6}+3\right ) \sqrt {-4 x^{6}+1}}{3}+\frac {i \sqrt {\pi }\, \arcsin \left (2 x^{3}\right )}{2}\right )}{96 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (4 x^{6}-1\right )}}\)	\(67\)

[In]

int(x^8*(4*x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/96*x^3*(8*x^6-1)*(4*x^6-1)^(1/2)-1/192*ln(2*x^3+(4*x^6-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\frac {1}{96} \, {\left (8 \, x^{9} - x^{3}\right )} \sqrt {4 \, x^{6} - 1} + \frac {1}{192} \, \log \left (-2 \, x^{3} + \sqrt {4 \, x^{6} - 1}\right ) \]

[In]

integrate(x^8*(4*x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/96*(8*x^9 - x^3)*sqrt(4*x^6 - 1) + 1/192*log(-2*x^3 + sqrt(4*x^6 - 1))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.91 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.05 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\begin {cases} \frac {x^{15}}{3 \sqrt {4 x^{6} - 1}} - \frac {x^{9}}{8 \sqrt {4 x^{6} - 1}} + \frac {x^{3}}{96 \sqrt {4 x^{6} - 1}} - \frac {\operatorname {acosh}{\left (2 x^{3} \right )}}{192} & \text {for}\: \left |{x^{6}}\right | > \frac {1}{4} \\- \frac {i x^{15}}{3 \sqrt {1 - 4 x^{6}}} + \frac {i x^{9}}{8 \sqrt {1 - 4 x^{6}}} - \frac {i x^{3}}{96 \sqrt {1 - 4 x^{6}}} + \frac {i \operatorname {asin}{\left (2 x^{3} \right )}}{192} & \text {otherwise} \end {cases} \]

[In]

integrate(x**8*(4*x**6-1)**(1/2),x)

[Out]

Piecewise((x**15/(3*sqrt(4*x**6 - 1)) - x**9/(8*sqrt(4*x**6 - 1)) + x**3/(96*sqrt(4*x**6 - 1)) - acosh(2*x**3)
/192, Abs(x**6) > 1/4), (-I*x**15/(3*sqrt(1 - 4*x**6)) + I*x**9/(8*sqrt(1 - 4*x**6)) - I*x**3/(96*sqrt(1 - 4*x
**6)) + I*asin(2*x**3)/192, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (46) = 92\).

Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=-\frac {\frac {4 \, \sqrt {4 \, x^{6} - 1}}{x^{3}} + \frac {{\left (4 \, x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}}}{96 \, {\left (\frac {8 \, {\left (4 \, x^{6} - 1\right )}}{x^{6}} - \frac {{\left (4 \, x^{6} - 1\right )}^{2}}{x^{12}} - 16\right )}} - \frac {1}{384} \, \log \left (\frac {\sqrt {4 \, x^{6} - 1}}{x^{3}} + 2\right ) + \frac {1}{384} \, \log \left (\frac {\sqrt {4 \, x^{6} - 1}}{x^{3}} - 2\right ) \]

[In]

integrate(x^8*(4*x^6-1)^(1/2),x, algorithm="maxima")

[Out]

-1/96*(4*sqrt(4*x^6 - 1)/x^3 + (4*x^6 - 1)^(3/2)/x^9)/(8*(4*x^6 - 1)/x^6 - (4*x^6 - 1)^2/x^12 - 16) - 1/384*lo
g(sqrt(4*x^6 - 1)/x^3 + 2) + 1/384*log(sqrt(4*x^6 - 1)/x^3 - 2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\frac {1}{96} \, {\left (8 \, x^{6} - 1\right )} \sqrt {4 \, x^{6} - 1} x^{3} - \frac {\log \left (\sqrt {-\frac {1}{x^{6}} + 4} + 2\right ) - \log \left (-\sqrt {-\frac {1}{x^{6}} + 4} + 2\right )}{384 \, \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^8*(4*x^6-1)^(1/2),x, algorithm="giac")

[Out]

1/96*(8*x^6 - 1)*sqrt(4*x^6 - 1)*x^3 - 1/384*(log(sqrt(-1/x^6 + 4) + 2) - log(-sqrt(-1/x^6 + 4) + 2))/sgn(x)

Mupad [F(-1)]

Timed out. \[ \int x^8 \sqrt {-1+4 x^6} \, dx=\int x^8\,\sqrt {4\,x^6-1} \,d x \]

[In]

int(x^8*(4*x^6 - 1)^(1/2),x)

[Out]

int(x^8*(4*x^6 - 1)^(1/2), x)