\(\int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx\) [1542]

Optimal result

Integrand size = 13, antiderivative size = 18 \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\frac {1}{5} \text {arctanh}\left (\frac {x^5}{\sqrt {-2+x^{10}}}\right ) \]

[Out]

1/5*arctanh(x^5/(x^10-2)^(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} \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\frac {1}{5} \text {arctanh}\left (\frac {x^5}{\sqrt {x^{10}-2}}\right ) \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\frac {1}{5} \log \left (x^5+\sqrt {-2+x^{10}}\right ) \]

[In]

Integrate[x^4/Sqrt[-2 + x^10],x]

[Out]

Log[x^5 + Sqrt[-2 + x^10]]/5

Maple [A] (verified)

Time = 3.72 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {\ln \left (x^{5}+\sqrt {x^{10}-2}\right )}{5}\)	\(15\)
trager	\(-\frac {\ln \left (x^{5}-\sqrt {x^{10}-2}\right )}{5}\)	\(17\)
meijerg	\(\frac {\sqrt {-\operatorname {signum}\left (-1+\frac {x^{10}}{2}\right )}\, \arcsin \left (\frac {x^{5} \sqrt {2}}{2}\right )}{5 \sqrt {\operatorname {signum}\left (-1+\frac {x^{10}}{2}\right )}}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=-\frac {1}{5} \, \log \left (-x^{5} + \sqrt {x^{10} - 2}\right ) \]

[In]

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

[Out]

-1/5*log(-x^5 + sqrt(x^10 - 2))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {2} x^{5}}{2} \right )}}{5} & \text {for}\: \left |{x^{10}}\right | > 2 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {2} x^{5}}{2} \right )}}{5} & \text {otherwise} \end {cases} \]

[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))

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 \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\frac {1}{10} \, \log \left (\frac {\sqrt {x^{10} - 2}}{x^{5}} + 1\right ) - \frac {1}{10} \, \log \left (\frac {\sqrt {x^{10} - 2}}{x^{5}} - 1\right ) \]

[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)

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 \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\frac {1}{10} \, \sqrt {x^{10} - 2} x^{5} + \frac {1}{5} \, \log \left ({\left | -x^{5} + \sqrt {x^{10} - 2} \right |}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\sqrt {-2+x^{10}}} \, dx=\int \frac {x^4}{\sqrt {x^{10}-2}} \,d x \]

[In]

int(x^4/(x^10 - 2)^(1/2),x)

[Out]

int(x^4/(x^10 - 2)^(1/2), x)