3.9.8 \(\int x \sqrt {-2+x^4} \, dx\) [808]

Optimal. Leaf size=35 \[ \frac {1}{4} x^2 \sqrt {-2+x^4}-\frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {-2+x^4}}\right ) \]

[Out]

-1/2*arctanh(x^2/(x^4-2)^(1/2))+1/4*x^2*(x^4-2)^(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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {281, 201, 223, 212} \begin {gather*} \frac {1}{4} x^2 \sqrt {x^4-2}-\frac {1}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[-2 + x^4],x]

[Out]

(x^2*Sqrt[-2 + x^4])/4 - ArcTanh[x^2/Sqrt[-2 + x^4]]/2

Rule 201

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

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Mathematica [A]
time = 0.08, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^2 \sqrt {-2+x^4}-\frac {1}{2} \tanh ^{-1}\left (\frac {\sqrt {-2+x^4}}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[-2 + x^4],x]

[Out]

(x^2*Sqrt[-2 + x^4])/4 - ArcTanh[Sqrt[-2 + x^4]/x^2]/2

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Maple [A]
time = 0.17, size = 28, normalized size = 0.80

method	result	size
default	\(\frac {x^{2} \sqrt {x^{4}-2}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-2}\right )}{2}\)	\(28\)
trager	\(\frac {x^{2} \sqrt {x^{4}-2}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-2}\right )}{2}\)	\(28\)
risch	\(\frac {x^{2} \sqrt {x^{4}-2}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-2}\right )}{2}\)	\(28\)
elliptic	\(\frac {x^{2} \sqrt {x^{4}-2}}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-2}\right )}{2}\)	\(28\)
meijerg	\(\frac {i \sqrt {\mathrm {signum}\left (-1+\frac {x^{4}}{2}\right )}\, \left (-i \sqrt {\pi }\, x^{2} \sqrt {2}\, \sqrt {-\frac {x^{4}}{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (\frac {x^{2} \sqrt {2}}{2}\right )\right )}{4 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (-1+\frac {x^{4}}{2}\right )}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2*(x^4-2)^(1/2)-1/2*ln(x^2+(x^4-2)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).
time = 0.30, size = 58, normalized size = 1.66 \begin {gather*} -\frac {\sqrt {x^{4} - 2}}{2 \, x^{2} {\left (\frac {x^{4} - 2}{x^{4}} - 1\right )}} - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} - 2}}{x^{2}} + 1\right ) + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} - 2}}{x^{2}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(x^4 - 2)/(x^2*((x^4 - 2)/x^4 - 1)) - 1/4*log(sqrt(x^4 - 2)/x^2 + 1) + 1/4*log(sqrt(x^4 - 2)/x^2 - 1)

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Fricas [A]
time = 0.36, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} - 2} x^{2} + \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(x^4 - 2)*x^2 + 1/2*log(-x^2 + sqrt(x^4 - 2))

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Sympy [C] Result contains complex when optimal does not.
time = 0.74, size = 88, normalized size = 2.51 \begin {gather*} \begin {cases} \frac {x^{6}}{4 \sqrt {x^{4} - 2}} - \frac {x^{2}}{2 \sqrt {x^{4} - 2}} - \frac {\operatorname {acosh}{\left (\frac {\sqrt {2} x^{2}}{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 2 \\- \frac {i x^{6}}{4 \sqrt {2 - x^{4}}} + \frac {i x^{2}}{2 \sqrt {2 - x^{4}}} + \frac {i \operatorname {asin}{\left (\frac {\sqrt {2} x^{2}}{2} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**6/(4*sqrt(x**4 - 2)) - x**2/(2*sqrt(x**4 - 2)) - acosh(sqrt(2)*x**2/2)/2, Abs(x**4) > 2), (-I*x*
*6/(4*sqrt(2 - x**4)) + I*x**2/(2*sqrt(2 - x**4)) + I*asin(sqrt(2)*x**2/2)/2, True))

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Giac [A]
time = 1.19, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, \sqrt {x^{4} - 2} x^{2} + \frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(x^4 - 2)*x^2 + 1/2*log(x^2 - sqrt(x^4 - 2))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x\,\sqrt {x^4-2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(x^4 - 2)^(1/2),x)

[Out]

int(x*(x^4 - 2)^(1/2), x)

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