\(\int \frac {x}{\sqrt {a-b x^4}} \, dx\) [837]

Optimal result

Integrand size = 14, antiderivative size = 31 \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {281, 223, 209} \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b}} \]

[In]

Int[x/Sqrt[a - b*x^4],x]

[Out]

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

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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=-\frac {i \log \left (i \sqrt {b} x^2+\sqrt {a-b x^4}\right )}{2 \sqrt {b}} \]

[In]

Integrate[x/Sqrt[a - b*x^4],x]

[Out]

((-1/2*I)*Log[I*Sqrt[b]*x^2 + Sqrt[a - b*x^4]])/Sqrt[b]

Maple [A] (verified)

Time = 4.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {\arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {b}}\)	\(24\)
elliptic	\(\frac {\arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {b}}\)	\(24\)
pseudoelliptic	\(-\frac {\arctan \left (\frac {\sqrt {-b \,x^{4}+a}}{x^{2} \sqrt {b}}\right )}{2 \sqrt {b}}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=\left [-\frac {\sqrt {-b} \log \left (2 \, b x^{4} - 2 \, \sqrt {-b x^{4} + a} \sqrt {-b} x^{2} - a\right )}{4 \, b}, -\frac {\arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {b} x^{2}}{b x^{4} - a}\right )}{2 \, \sqrt {b}}\right ] \]

[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)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=\begin {cases} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} & \text {for}\: \left |{\frac {b x^{4}}{a}}\right | > 1 \\\frac {\operatorname {asin}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} & \text {otherwise} \end {cases} \]

[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/a) > 1), (asin(sqrt(b)*x**2/sqrt(a))/(2*sqrt
(b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {b} x^{2}}\right )}{2 \, \sqrt {b}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=-\frac {\log \left ({\left | -\sqrt {-b} x^{2} + \sqrt {-b x^{4} + a} \right |}\right )}{2 \, \sqrt {-b}} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {a-b x^4}} \, dx=\int \frac {x}{\sqrt {a-b\,x^4}} \,d x \]

[In]

int(x/(a - b*x^4)^(1/2),x)

[Out]

int(x/(a - b*x^4)^(1/2), x)