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

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*x^2)/Sqrt[a - b*x^4]]/(2*Sqrt[b])

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Rubi [A]  time = 0.0142488, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {275, 217, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a - b*x^4],x]

[Out]

ArcTan[(Sqrt[b]*x^2)/Sqrt[a - b*x^4]]/(2*Sqrt[b])

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

Mathematica [A]  time = 0.0042364, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a - b*x^4],x]

[Out]

ArcTan[(Sqrt[b]*x^2)/Sqrt[a - b*x^4]]/(2*Sqrt[b])

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Maple [A]  time = 0.005, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53517, size = 182, normalized size = 5.87 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (2 \, b x^{4} - 2 \, \sqrt{-b x^{4} + a} \sqrt{-b} x^{2} - a\right )}{4 \, b}, -\frac{\arctan \left (\frac{\sqrt{-b x^{4} + a} \sqrt{b} x^{2}}{b x^{4} - a}\right )}{2 \, \sqrt{b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*sqrt(-b)*log(2*b*x^4 - 2*sqrt(-b*x^4 + a)*sqrt(-b)*x^2 - a)/b, -1/2*arctan(sqrt(-b*x^4 + a)*sqrt(b)*x^2/
(b*x^4 - a))/sqrt(b)]

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Sympy [A]  time = 1.46137, size = 54, normalized size = 1.74 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \frac{\left |{b x^{4}}\right |}{\left |{a}\right |} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12424, size = 41, normalized size = 1.32 \begin{align*} -\frac{\log \left ({\left | -\sqrt{-b} x^{2} + \sqrt{-b x^{4} + a} \right |}\right )}{2 \, \sqrt{-b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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