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

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

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

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Rubi [A]  time = 0.0092697, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0034013, size = 22, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\frac{x^2}{\sqrt{a^4-x^4}}\right ) \]

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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Maple [A]  time = 0.01, size = 19, normalized size = 0.9 \begin{align*}{\frac{1}{2}\arctan \left ({{x}^{2}{\frac{1}{\sqrt{{a}^{4}-{x}^{4}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48826, size = 24, normalized size = 1.09 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{\sqrt{a^{4} - x^{4}}}{x^{2}}\right ) \end{align*}

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 = 1.76094, size = 53, normalized size = 2.41 \begin{align*} -\arctan \left (-\frac{a^{2} - \sqrt{a^{4} - x^{4}}}{x^{2}}\right ) \end{align*}

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 [A]  time = 1.06362, size = 31, normalized size = 1.41 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{x^{2}}{a^{2}} \right )}}{2} & \text{for}\: \frac{\left |{x^{4}}\right |}{\left |{a^{4}}\right |} > 1 \\\frac{\operatorname{asin}{\left (\frac{x^{2}}{a^{2}} \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

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)/Abs(a**4) > 1), (asin(x**2/a**2)/2, True))

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Giac [A]  time = 1.07695, size = 14, normalized size = 0.64 \begin{align*} \frac{1}{2} \, \arcsin \left (\frac{x^{2}}{a^{2}}\right ) \end{align*}

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)