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

Optimal. Leaf size=18 \[ \frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 217, 206} \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[-1 + x^6],x]

[Out]

ArcTanh[x^3/Sqrt[-1 + x^6]]/3

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[-1 + x^6],x]

[Out]

ArcTanh[x^3/Sqrt[-1 + x^6]]/3

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IntegrateAlgebraic [A]  time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/Sqrt[-1 + x^6],x]

[Out]

Log[x^3 + Sqrt[-1 + x^6]]/3

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fricas [A]  time = 0.43, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(-x^3 + sqrt(x^6 - 1))

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giac [A]  time = 0.40, size = 17, normalized size = 0.94 \begin {gather*} -\frac {1}{3} \, \log \left ({\left | -x^{3} + \sqrt {x^{6} - 1} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(abs(-x^3 + sqrt(x^6 - 1)))

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maple [A]  time = 0.15, size = 17, normalized size = 0.94

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.49, size = 33, normalized size = 1.83 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x^2}{\sqrt {x^6-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^6 - 1)^(1/2),x)

[Out]

int(x^2/(x^6 - 1)^(1/2), x)

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sympy [A]  time = 0.79, size = 19, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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