\(\int x^{14} \sqrt {-1+x^6} \, dx\) [614]

Optimal result

Integrand size = 13, antiderivative size = 48 \[ \int x^{14} \sqrt {-1+x^6} \, dx=\frac {1}{144} \sqrt {-1+x^6} \left (-3 x^3-2 x^9+8 x^{15}\right )-\frac {1}{48} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

[Out]

1/144*(x^6-1)^(1/2)*(8*x^15-2*x^9-3*x^3)-1/48*ln(x^3+(x^6-1)^(1/2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 285, 327, 223, 212} \[ \int x^{14} \sqrt {-1+x^6} \, dx=-\frac {1}{48} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right )+\frac {1}{18} \sqrt {x^6-1} x^{15}-\frac {1}{72} \sqrt {x^6-1} x^9-\frac {1}{48} \sqrt {x^6-1} x^3 \]

[In]

Int[x^14*Sqrt[-1 + x^6],x]

[Out]

-1/48*(x^3*Sqrt[-1 + x^6]) - (x^9*Sqrt[-1 + x^6])/72 + (x^15*Sqrt[-1 + x^6])/18 - ArcTanh[x^3/Sqrt[-1 + x^6]]/
48

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^4 \sqrt {-1+x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{18} \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{24} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{48} x^3 \sqrt {-1+x^6}-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right ) \\ & = -\frac {1}{48} x^3 \sqrt {-1+x^6}-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{48} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = -\frac {1}{48} x^3 \sqrt {-1+x^6}-\frac {1}{72} x^9 \sqrt {-1+x^6}+\frac {1}{18} x^{15} \sqrt {-1+x^6}-\frac {1}{48} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int x^{14} \sqrt {-1+x^6} \, dx=\frac {1}{144} x^3 \sqrt {-1+x^6} \left (-3-2 x^6+8 x^{12}\right )-\frac {1}{48} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

[In]

Integrate[x^14*Sqrt[-1 + x^6],x]

[Out]

(x^3*Sqrt[-1 + x^6]*(-3 - 2*x^6 + 8*x^12))/144 - Log[x^3 + Sqrt[-1 + x^6]]/48

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83

method	result	size
trager	\(\frac {x^{3} \left (8 x^{12}-2 x^{6}-3\right ) \sqrt {x^{6}-1}}{144}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{48}\)	\(40\)
pseudoelliptic	\(\frac {\sqrt {x^{6}-1}\, \left (8 x^{15}-2 x^{9}-3 x^{3}\right )}{144}-\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{48}\)	\(41\)
risch	\(\frac {x^{3} \left (8 x^{12}-2 x^{6}-3\right ) \sqrt {x^{6}-1}}{144}-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{48 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(50\)
meijerg	\(\frac {i \sqrt {\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {i \sqrt {\pi }\, x^{3} \left (-40 x^{12}+10 x^{6}+15\right ) \sqrt {-x^{6}+1}}{60}-\frac {i \sqrt {\pi }\, \arcsin \left (x^{3}\right )}{4}\right )}{12 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}}\)	\(66\)

[In]

int(x^14*(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/144*x^3*(8*x^12-2*x^6-3)*(x^6-1)^(1/2)-1/48*ln(x^3+(x^6-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int x^{14} \sqrt {-1+x^6} \, dx=\frac {1}{144} \, {\left (8 \, x^{15} - 2 \, x^{9} - 3 \, x^{3}\right )} \sqrt {x^{6} - 1} + \frac {1}{48} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]

[In]

integrate(x^14*(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/144*(8*x^15 - 2*x^9 - 3*x^3)*sqrt(x^6 - 1) + 1/48*log(-x^3 + sqrt(x^6 - 1))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.83 \[ \int x^{14} \sqrt {-1+x^6} \, dx=\begin {cases} \frac {x^{21}}{18 \sqrt {x^{6} - 1}} - \frac {5 x^{15}}{72 \sqrt {x^{6} - 1}} - \frac {x^{9}}{144 \sqrt {x^{6} - 1}} + \frac {x^{3}}{48 \sqrt {x^{6} - 1}} - \frac {\operatorname {acosh}{\left (x^{3} \right )}}{48} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i x^{21}}{18 \sqrt {1 - x^{6}}} + \frac {5 i x^{15}}{72 \sqrt {1 - x^{6}}} + \frac {i x^{9}}{144 \sqrt {1 - x^{6}}} - \frac {i x^{3}}{48 \sqrt {1 - x^{6}}} + \frac {i \operatorname {asin}{\left (x^{3} \right )}}{48} & \text {otherwise} \end {cases} \]

[In]

integrate(x**14*(x**6-1)**(1/2),x)

[Out]

Piecewise((x**21/(18*sqrt(x**6 - 1)) - 5*x**15/(72*sqrt(x**6 - 1)) - x**9/(144*sqrt(x**6 - 1)) + x**3/(48*sqrt
(x**6 - 1)) - acosh(x**3)/48, Abs(x**6) > 1), (-I*x**21/(18*sqrt(1 - x**6)) + 5*I*x**15/(72*sqrt(1 - x**6)) +
I*x**9/(144*sqrt(1 - x**6)) - I*x**3/(48*sqrt(1 - x**6)) + I*asin(x**3)/48, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (40) = 80\).

Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.27 \[ \int x^{14} \sqrt {-1+x^6} \, dx=-\frac {\frac {3 \, \sqrt {x^{6} - 1}}{x^{3}} + \frac {8 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{x^{9}} - \frac {3 \, {\left (x^{6} - 1\right )}^{\frac {5}{2}}}{x^{15}}}{144 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} - \frac {1}{96} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) + \frac {1}{96} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]

[In]

integrate(x^14*(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

-1/144*(3*sqrt(x^6 - 1)/x^3 + 8*(x^6 - 1)^(3/2)/x^9 - 3*(x^6 - 1)^(5/2)/x^15)/(3*(x^6 - 1)/x^6 - 3*(x^6 - 1)^2
/x^12 + (x^6 - 1)^3/x^18 - 1) - 1/96*log(sqrt(x^6 - 1)/x^3 + 1) + 1/96*log(sqrt(x^6 - 1)/x^3 - 1)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.29 \[ \int x^{14} \sqrt {-1+x^6} \, dx=\frac {1}{144} \, {\left (2 \, {\left (4 \, x^{6} - 1\right )} x^{6} - 3\right )} \sqrt {x^{6} - 1} x^{3} - \frac {\log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{96 \, \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^14*(x^6-1)^(1/2),x, algorithm="giac")

[Out]

1/144*(2*(4*x^6 - 1)*x^6 - 3)*sqrt(x^6 - 1)*x^3 - 1/96*(log(sqrt(-1/x^6 + 1) + 1) - log(-sqrt(-1/x^6 + 1) + 1)
)/sgn(x)

Mupad [F(-1)]

Timed out. \[ \int x^{14} \sqrt {-1+x^6} \, dx=\int x^{14}\,\sqrt {x^6-1} \,d x \]

[In]

int(x^14*(x^6 - 1)^(1/2),x)

[Out]

int(x^14*(x^6 - 1)^(1/2), x)