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

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

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {275, 215} \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[1 + x^6],x]

[Out]

ArcSinh[x^3]/3

Rule 215

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

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} \sinh ^{-1}\left (x^3\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.44 \begin {gather*} \frac {1}{3} \sinh ^{-1}\left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[1 + x^6],x]

[Out]

ArcSinh[x^3]/3

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IntegrateAlgebraic [A]  time = 0.10, 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 = 0.46, 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 = 0.45, 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="giac")

[Out]

-1/3*log(-x^3 + sqrt(x^6 + 1))

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maple [A]  time = 0.14, size = 7, normalized size = 0.39

method	result	size
meijerg	\(\frac {\arcsinh \left (x^{3}\right )}{3}\)	\(7\)
trager	\(-\frac {\ln \left (x^{3}-\sqrt {x^{6}+1}\right )}{3}\)	\(17\)

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*arcsinh(x^3)

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maxima [B]  time = 0.51, 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 = 0.75, size = 5, normalized size = 0.28 \begin {gather*} \frac {\operatorname {asinh}{\left (x^{3} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x**3)/3

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