\(\int \frac {x^2}{\sqrt {-1+x^6}} \, dx\) [152]

Optimal result

Integrand size = 13, antiderivative size = 18 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

[Out]

1/3*ln(x^3+(x^6-1)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 223, 212} \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \]

[In]

Int[x^2/Sqrt[-1 + x^6],x]

[Out]

ArcTanh[x^3/Sqrt[-1 + x^6]]/3

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]

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \]

[In]

Integrate[x^2/Sqrt[-1 + x^6],x]

[Out]

Log[x^3 + Sqrt[-1 + x^6]]/3

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}\)	\(15\)
trager	\(-\frac {\ln \left (x^{3}-\sqrt {x^{6}-1}\right )}{3}\)	\(17\)
meijerg	\(\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(25\)

[In]

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

[Out]

1/3*ln(x^3+(x^6-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=-\frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \]

[In]

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

[Out]

-1/3*log(-x^3 + sqrt(x^6 - 1))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\begin {cases} \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} & \text {for}\: \left |{x^{6}}\right | > 1 \\- \frac {i \operatorname {asin}{\left (x^{3} \right )}}{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((acosh(x**3)/3, Abs(x**6) > 1), (-I*asin(x**3)/3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.83 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \]

[In]

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

[Out]

1/6*log(sqrt(x^6 - 1)/x^3 + 1) - 1/6*log(sqrt(x^6 - 1)/x^3 - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67 \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\frac {1}{6} \, \sqrt {x^{6} - 1} x^{3} + \frac {1}{6} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 1} \right |}\right ) \]

[In]

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

[Out]

1/6*sqrt(x^6 - 1)*x^3 + 1/6*log(abs(-x^3 + sqrt(x^6 - 1)))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {-1+x^6}} \, dx=\int \frac {x^2}{\sqrt {x^6-1}} \,d x \]

[In]

int(x^2/(x^6 - 1)^(1/2),x)

[Out]

int(x^2/(x^6 - 1)^(1/2), x)