3.2085 \(\int \frac {1}{\sqrt {a+\frac {b}{x^4}} x^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right )}{2 \sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {335, 275, 217, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{x^2 \sqrt {a+\frac {b}{x^4}}}\right )}{2 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^4]*x^3),x]

[Out]

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

Rule 206

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

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x^4}} x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x^4}} x^2}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^4}} x^2}\right )}{2 \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 52, normalized size = 1.73 \[ -\frac {\sqrt {a x^4+b} \tanh ^{-1}\left (\frac {\sqrt {a x^4+b}}{\sqrt {b}}\right )}{2 \sqrt {b} x^2 \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^4]*x^3),x]

[Out]

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

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fricas [A]  time = 1.05, size = 77, normalized size = 2.57 \[ \left [\frac {\log \left (\frac {a x^{4} - 2 \, \sqrt {b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right )}{4 \, \sqrt {b}}, \frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{2} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{b}\right )}{2 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b/x^4)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 41, normalized size = 1.37 \[ \frac {\arctan \left (\frac {\sqrt {a x^{4} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} - \frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b/x^4)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.01, size = 52, normalized size = 1.73 \[ -\frac {\sqrt {a \,x^{4}+b}\, \ln \left (\frac {2 b +2 \sqrt {a \,x^{4}+b}\, \sqrt {b}}{x^{2}}\right )}{2 \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, \sqrt {b}\, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b/x^4)^(1/2),x)

[Out]

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

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maxima [B]  time = 1.97, size = 45, normalized size = 1.50 \[ \frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} x^{2} - \sqrt {b}}{\sqrt {a + \frac {b}{x^{4}}} x^{2} + \sqrt {b}}\right )}{4 \, \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b/x^4)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^3\,\sqrt {a+\frac {b}{x^4}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x^4)^(1/2)),x)

[Out]

int(1/(x^3*(a + b/x^4)^(1/2)), x)

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sympy [A]  time = 1.63, size = 22, normalized size = 0.73 \[ - \frac {\operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{2}} \right )}}{2 \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b/x**4)**(1/2),x)

[Out]

-asinh(sqrt(b)/(sqrt(a)*x**2))/(2*sqrt(b))

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