3.9.100 \(\int \frac {x^5}{(1-x^4)^{3/2}} \, dx\) [900]

Optimal. Leaf size=27 \[ \frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {281, 294, 222} \begin {gather*} \frac {x^2}{2 \sqrt {1-x^4}}-\frac {\text {ArcSin}\left (x^2\right )}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4]) - ArcSin[x^2]/2

Rule 222

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

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]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 \sqrt {1-x^4}}-\frac {1}{2} \sin ^{-1}\left (x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 36, normalized size = 1.33 \begin {gather*} \frac {1}{2} \left (\frac {x^2}{\sqrt {1-x^4}}+\tan ^{-1}\left (\frac {\sqrt {1-x^4}}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 - x^4)^(3/2),x]

[Out]

(x^2/Sqrt[1 - x^4] + ArcTan[Sqrt[1 - x^4]/x^2])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(21)=42\).
time = 0.23, size = 62, normalized size = 2.30

method	result	size
risch	\(-\frac {\arcsin \left (x^{2}\right )}{2}+\frac {x^{2}}{2 \sqrt {-x^{4}+1}}\)	\(22\)
meijerg	\(\frac {i \left (-\frac {i \sqrt {\pi }\, x^{2}}{\sqrt {-x^{4}+1}}+i \sqrt {\pi }\, \arcsin \left (x^{2}\right )\right )}{2 \sqrt {\pi }}\)	\(36\)
trager	\(-\frac {x^{2} \sqrt {-x^{4}+1}}{2 \left (x^{4}-1\right )}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}+1}+x^{2}\right )}{2}\)	\(53\)
default	\(-\frac {\arcsin \left (x^{2}\right )}{2}-\frac {\sqrt {-\left (x^{2}+1\right )^{2}+2 x^{2}+2}}{4 \left (x^{2}+1\right )}-\frac {\sqrt {-\left (x^{2}-1\right )^{2}-2 x^{2}+2}}{4 \left (x^{2}-1\right )}\)	\(62\)
elliptic	\(-\frac {\arcsin \left (x^{2}\right )}{2}-\frac {\sqrt {-\left (x^{2}+1\right )^{2}+2 x^{2}+2}}{4 \left (x^{2}+1\right )}-\frac {\sqrt {-\left (x^{2}-1\right )^{2}-2 x^{2}+2}}{4 \left (x^{2}-1\right )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 31, normalized size = 1.15 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {-x^{4} + 1}} + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{4} + 1}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
time = 0.37, size = 46, normalized size = 1.70 \begin {gather*} -\frac {\sqrt {-x^{4} + 1} x^{2} - 2 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {-x^{4} + 1} - 1}{x^{2}}\right )}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.66, size = 46, normalized size = 1.70 \begin {gather*} \begin {cases} - \frac {i x^{2}}{2 \sqrt {x^{4} - 1}} + \frac {i \operatorname {acosh}{\left (x^{2} \right )}}{2} & \text {for}\: \left |{x^{4}}\right | > 1 \\\frac {x^{2}}{2 \sqrt {1 - x^{4}}} - \frac {\operatorname {asin}{\left (x^{2} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-x**4+1)**(3/2),x)

[Out]

Piecewise((-I*x**2/(2*sqrt(x**4 - 1)) + I*acosh(x**2)/2, Abs(x**4) > 1), (x**2/(2*sqrt(1 - x**4)) - asin(x**2)
/2, True))

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Giac [A]
time = 0.60, size = 28, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {-x^{4} + 1} x^{2}}{2 \, {\left (x^{4} - 1\right )}} - \frac {1}{2} \, \arcsin \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-x^4+1)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^5}{{\left (1-x^4\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(1 - x^4)^(3/2),x)

[Out]

int(x^5/(1 - x^4)^(3/2), x)

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