3.50 \(\int \frac {x}{\sqrt {a^4-x^4}} \, dx\)

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

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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 = {275, 217, 203} \[ \frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a^4 - x^4],x]

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

Rule 203

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

Rule 217

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 275

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} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^4-x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {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 = 0.00, size = 22, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {a^4-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a^4 - x^4],x]

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

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IntegrateAlgebraic [C]  time = 0.09, size = 28, normalized size = 1.27 \[ -\frac {1}{2} i \log \left (\sqrt {a^4-x^4}+i x^2\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[a^4 - x^4],x]

[Out]

(-1/2*I)*Log[I*x^2 + Sqrt[a^4 - x^4]]

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fricas [A]  time = 0.94, size = 25, normalized size = 1.14 \[ -\arctan \left (-\frac {a^{2} - \sqrt {a^{4} - x^{4}}}{x^{2}}\right ) \]

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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giac [A]  time = 1.06, size = 10, normalized size = 0.45 \[ \frac {1}{2} \, \arcsin \left (\frac {x^{2}}{a^{2}}\right ) \]

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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maple [A]  time = 0.34, 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 = 0.96, size = 18, normalized size = 0.82 \[ -\frac {1}{2} \, \arctan \left (\frac {\sqrt {a^{4} - x^{4}}}{x^{2}}\right ) \]

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

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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sympy [A]  time = 1.07, size = 29, normalized size = 1.32 \[ \begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {x^{2}}{a^{2}} \right )}}{2} & \text {for}\: \left |{\frac {x^{4}}{a^{4}}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {x^{2}}{a^{2}} \right )}}{2} & \text {otherwise} \end {cases} \]

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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