3.1.50 \(\int \frac {x}{\sqrt {a^4-x^4}} \, dx\) [50]

Optimal. Leaf size=22 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right ) \]

[Out]

1/2*arctan(x^2/(a^4-x^4)^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 22, 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, 223, 209} \begin {gather*} \frac {1}{2} \text {ArcTan}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a^4 - x^4],x]

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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}{\sqrt {a^4-x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a^4-x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^2}{\sqrt {a^4-x^4}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {a^4-x^4}}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a^4 - x^4],x]

[Out]

-1/2*ArcTan[Sqrt[a^4 - x^4]/x^2]

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Maple [A]
time = 0.05, size = 19, normalized size = 0.86

method	result	size
default	\(\frac {\arctan \left (\frac {x^{2}}{\sqrt {a^{4}-x^{4}}}\right )}{2}\)	\(19\)
elliptic	\(\frac {\arctan \left (\frac {x^{2}}{\sqrt {a^{4}-x^{4}}}\right )}{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(x^2/(a^4-x^4)^(1/2))

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Maxima [A]
time = 4.49, size = 18, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {\sqrt {a^{4} - x^{4}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(sqrt(a^4 - x^4)/x^2)

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Fricas [A]
time = 0.97, size = 25, normalized size = 1.14 \begin {gather*} -\arctan \left (-\frac {a^{2} - \sqrt {a^{4} - x^{4}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(-(a^2 - sqrt(a^4 - x^4))/x^2)

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Sympy [C] Result contains complex when optimal does not.
time = 0.45, size = 29, normalized size = 1.32 \begin {gather*} \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {x^{2}}{a^{2}} \right )}}{2} & \text {for}\: \left |{\frac {x^{4}}{a^{4}}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {x^{2}}{a^{2}} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.05, size = 10, normalized size = 0.45 \begin {gather*} \frac {1}{2} \, \arcsin \left (\frac {x^{2}}{a^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(x^2/a^2)

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Mupad [B]
time = 0.09, size = 18, normalized size = 0.82 \begin {gather*} \frac {\mathrm {atan}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a^4 - x^4)^(1/2),x)

[Out]

atan(x^2/(a^4 - x^4)^(1/2))/2

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