3.1382 \(\int x^2 \sqrt {-2+x^6} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{6} x^3 \sqrt {x^6-2}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 195, 217, 206} \[ \frac {1}{6} x^3 \sqrt {x^6-2}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[-2 + x^6],x]

[Out]

(x^3*Sqrt[-2 + x^6])/6 - ArcTanh[x^3/Sqrt[-2 + x^6]]/3

Rule 195

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 206

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

Rule 217

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 275

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*} \int x^2 \sqrt {-2+x^6} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \sqrt {-2+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-2+x^6}}\right )\\ &=\frac {1}{6} x^3 \sqrt {-2+x^6}-\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-2+x^6}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.43 \[ \frac {\left (x^6-2\right ) \left (2 \sin ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )+\sqrt {2-x^6} x^3\right )}{6 \sqrt {-\left (x^6-2\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[-2 + x^6],x]

[Out]

((-2 + x^6)*(x^3*Sqrt[2 - x^6] + 2*ArcSin[x^3/Sqrt[2]]))/(6*Sqrt[-(-2 + x^6)^2])

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fricas [A]  time = 0.91, size = 29, normalized size = 0.83 \[ \frac {1}{6} \, \sqrt {x^{6} - 2} x^{3} + \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 30, normalized size = 0.86 \[ \frac {1}{6} \, \sqrt {x^{6} - 2} x^{3} + \frac {1}{3} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(x^6 - 2)*x^3 + 1/3*log(abs(-x^3 + sqrt(x^6 - 2)))

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maple [C]  time = 0.16, size = 47, normalized size = 1.34 \[ \frac {\sqrt {x^{6}-2}\, x^{3}}{6}-\frac {\sqrt {-\mathrm {signum}\left (\frac {x^{6}}{2}-1\right )}\, \arcsin \left (\frac {\sqrt {2}\, x^{3}}{2}\right )}{3 \sqrt {\mathrm {signum}\left (\frac {x^{6}}{2}-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^6-2)^(1/2),x)

[Out]

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

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maxima [B]  time = 0.99, size = 58, normalized size = 1.66 \[ -\frac {\sqrt {x^{6} - 2}}{3 \, x^{3} {\left (\frac {x^{6} - 2}{x^{6}} - 1\right )}} - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 2}}{x^{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int x^2\,\sqrt {x^6-2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^6 - 2)^(1/2),x)

[Out]

int(x^2*(x^6 - 2)^(1/2), x)

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sympy [A]  time = 3.14, size = 90, normalized size = 2.57 \[ \begin {cases} \frac {x^{9}}{6 \sqrt {x^{6} - 2}} - \frac {x^{3}}{3 \sqrt {x^{6} - 2}} - \frac {\operatorname {acosh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{3} & \text {for}\: \frac {\left |{x^{6}}\right |}{2} > 1 \\- \frac {i x^{9}}{6 \sqrt {2 - x^{6}}} + \frac {i x^{3}}{3 \sqrt {2 - x^{6}}} + \frac {i \operatorname {asin}{\left (\frac {\sqrt {2} x^{3}}{2} \right )}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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