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]

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

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Rubi [A]  time = 0.0244563, antiderivative size = 30, normalized size of antiderivative = 1., 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0150563, 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]

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

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Maple [B]  time = 0.008, size = 52, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{b}}}} \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52684, size = 188, normalized size = 6.27 \begin{align*} \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 ] \end{align*}

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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Sympy [A]  time = 1.65845, size = 22, normalized size = 0.73 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{2 \sqrt{b}} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + \frac{b}{x^{4}}} x^{3}}\,{d x} \end{align*}

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]

integrate(1/(sqrt(a + b/x^4)*x^3), x)