3.4.63 \(\int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx\) [363]

Optimal. Leaf size=18 \[ \frac {1}{5} \tanh ^{-1}\left (\frac {x^5}{\sqrt {-2+x^{10}}}\right ) \]

[Out]

1/5*arctanh(x^5/(x^10-2)^(1/2))

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Rubi [A]
time = 0.00, 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 = {281, 223, 212} \begin {gather*} \frac {1}{5} \tanh ^{-1}\left (\frac {x^5}{\sqrt {x^{10}-2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[-2 + x^10],x]

[Out]

ArcTanh[x^5/Sqrt[-2 + x^10]]/5

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*} \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{\sqrt {-2+x^2}} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^5}{\sqrt {-2+x^{10}}}\right )\\ &=\frac {1}{5} \tanh ^{-1}\left (\frac {x^5}{\sqrt {-2+x^{10}}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{5} \tanh ^{-1}\left (\frac {\sqrt {-2+x^{10}}}{x^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[-2 + x^10],x]

[Out]

ArcTanh[Sqrt[-2 + x^10]/x^5]/5

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.12, size = 31, normalized size = 1.72 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\text {ArcCosh}\left [\frac {\sqrt {2} x^5}{2}\right ]}{5},\text {Abs}\left [x^{10}\right ]>2\right \}\right \},-\frac {I \text {ArcSin}\left [\frac {\sqrt {2} x^5}{2}\right ]}{5}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^4/Sqrt[-2 + x^10],x]')

[Out]

Piecewise[{{ArcCosh[Sqrt[2] x ^ 5 / 2] / 5, Abs[x ^ 10] > 2}}, -I ArcSin[Sqrt[2] x ^ 5 / 2] / 5]

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Maple [A]
time = 0.11, size = 15, normalized size = 0.83

method	result	size
trager	\(\frac {\ln \left (x^{5}+\sqrt {x^{10}-2}\right )}{5}\)	\(15\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {x^{10}}{2}\right )}\, \arcsin \left (\frac {x^{5} \sqrt {2}}{2}\right )}{5 \sqrt {\mathrm {signum}\left (-1+\frac {x^{10}}{2}\right )}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^10-2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*ln(x^5+(x^10-2)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
time = 0.26, size = 33, normalized size = 1.83 \begin {gather*} \frac {1}{10} \, \log \left (\frac {\sqrt {x^{10} - 2}}{x^{5}} + 1\right ) - \frac {1}{10} \, \log \left (\frac {\sqrt {x^{10} - 2}}{x^{5}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^10-2)^(1/2),x, algorithm="maxima")

[Out]

1/10*log(sqrt(x^10 - 2)/x^5 + 1) - 1/10*log(sqrt(x^10 - 2)/x^5 - 1)

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Fricas [A]
time = 0.33, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{5} \, \log \left (-x^{5} + \sqrt {x^{10} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^10-2)^(1/2),x, algorithm="fricas")

[Out]

-1/5*log(-x^5 + sqrt(x^10 - 2))

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Sympy [A]
time = 0.46, size = 32, normalized size = 1.78 \begin {gather*} \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {2} x^{5}}{2} \right )}}{5} & \text {for}\: \left |{x^{10}}\right | > 2 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {2} x^{5}}{2} \right )}}{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(x**10-2)**(1/2),x)

[Out]

Piecewise((acosh(sqrt(2)*x**5/2)/5, Abs(x**10) > 2), (-I*asin(sqrt(2)*x**5/2)/5, True))

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Giac [A]
time = 0.00, size = 20, normalized size = 1.11 \begin {gather*} -\frac {\ln \left |\sqrt {x^{10}-2}-x^{5}\right |}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^10-2)^(1/2),x)

[Out]

-1/5*log(abs(-x^5 + sqrt(x^10 - 2)))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x^4}{\sqrt {x^{10}-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^10 - 2)^(1/2),x)

[Out]

int(x^4/(x^10 - 2)^(1/2), x)

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